On the Dirichlet problem for H-systems with small boundary data blow up phenomena and nonexistence results

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On the Dirichlet problem for H-systems with small boundary data: blow up phenomena and nonexistence results∗ Paolo Caldiroli1 and Roberta Musina2 1

Dipartimento di Matematica Universit` a di Torino via Carlo Alberto, 10 – 10123 Torino, Italy e-mail: [email protected] 2

Dipartimento di Matematica ed Informatica Universit` a di Udine via delle Scienze, 206 – 33100 Udine, Italy e-mail: [email protected]

Abstract. Given H: R3 → R of class C 1 and bounded, we consider a sequence (un ) of solutions of the H-system ∆u = 2H(u)ux ∧ uy in the unit open disc D2 satisfying the boundary condition un = γ n on ∂D2 . In the first part of this paper, assuming that (un ) is bounded in H 1 (D2 , R3 ) we study the behavior of (un ) when the boundary data γ n shrink to zero. We show that either un → 0 strongly in H 1 (D2 , R3 ) or un blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on R2 . Under additional assumptions on H we can obtain more precise information on the blow up. In the second part of the article we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum and we detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a “large” solution at a mountain pass level when the boundary datum is small. Summary Introduction 1. Preliminaries 2. Blowing H-bubbles 3. Blowing H-bubbles with low energy 4. Blowing minimal H-bubbles 5. Nonexistence results References ∗ Work supported by Ministero dell’Istruzione, dell’Universit` a e della Ricerca, progetto di ricerca “Metodi variazionali ed equazioni differenziali nonlineari” (cofin. 2003/2004). The second author is supported also by Regione Friuli Venezia Giulia L.R. 3/98 (2002), progetto di ricerca “Equazioni differenziali in geometria ed in fisica matematica”. 2000 Mathematics Subject Classification: 53A10 (49J10). Keywords: H-systems, blow up, prescribed mean curvature, Rellich’s conjecture.

1

Introduction In this paper we deal with the Dirichlet problem for the prescribed mean curvature equation:  ∆u = 2H(u)ux ∧ uy in D2 (0.1) u=γ on ∂D2 where D2 is the two-dimensional open unit disc, H: R3 → R is a given bounded smooth function, and γ ∈ H 1 (D2 , R3 ) ∩ C 0 (∂D2 , R3 ) defines the boundary datum. Problem (0.1) is related to the Plateau problem concerning existence and multiplicity of disc-type parametric surfaces with prescribed mean curvature, spanning a given contour (see, e.g., [28]). Indeed, if u ∈ C 0 (D2 , R3 ) ∩ C 2 (D2 , R3 ) solves (0.1) and it is conformal and regular on D2 , then u parametrizes a surface M = range u in R3 such that ∂M = γ(∂D2 ) and the mean curvature of M at any p ∈ M equals H(p). Problem (0.1) is variational and its solutions can be formally found as critical points of the energy functional Z 1 EH (u) = |∇u|2 + 2VH (u) 2 D2

on the affine space Hγ1 = γ + H01 (D2 , R3 ). Here VH is H-volume functional, introduced by Steffen in [25]. VH extends to the whole space Hγ1 the H-volume integral defined by Hildebrandt in [19] for mappings in Hγ1 ∩ L∞ and it coincides with the classical volume functional studied by Wente [32] when H is constant. We point out that both the functional EH and the differential equation in (0.1) are invariant with respect to conformal transformations of the disc into itself. As a consequence, in general the functional EH does not satisfy good compactness properties, like weak lower semicontinuity, neither Palais Smale condition. An important property enjoyed by the H-volume functional is the isoperimetric inequality: 3/2 Z Z 2 2 for every u ∈ Hγ1 (0.2) |∇γ| |∇u| + |VH (u) − VH (γ)| ≤ CH D2

D2

where the constant CH depends only on kHk∞ (see Lemma 1.2). We point out that for u ∈ Hγ1 regular enough, the right hand side in the inequality (0.2) measures the H-weighted volume enclosed by the surfaces parametrized by γ and u. The inequality (0.2) implies that for γ small VH is essentially cubic in a neighborhood of γ and hence, for u ∈ Hγ1 close to γ, the leading term in the energy functional EH (u) is given by the Dirichlet integral. Under quite weak conditions on H (e.g., H is continuous and nonzero at infinity, or even, H is far from zero on a sufficiently large set), one can check that inf u∈Hγ1 EH (u) = −∞. Hence for γ small EH admits a nice mountain pass geometry at some positive level (see Theorem 4.1). In fact the energy functional EH is not of class C 1 , nevertheless it turns out to be sufficiently regular in order to apply the critical point theory. These observations suggest that problem (0.1) should admit a pair of solutions: a first solution (the so called “small” solution) as a local minimum point of EH close to γ, and a 2

second solution with higher energy (the “large” solution) at the mountain pass level. This is the so called Rellich’s conjecture. Let us give a more precise outline of the variational problem by starting with the case γ ≡ 0. Clearly u ≡ 0 is a solution to (0.1). In particular, thanks to (0.2), u ≡ 0 turns out to be a strict local minimum for EH . If H does not vanish at infinity, the functional EH possesses a mountain pass geometry on H01 . However the mountain pass level for EH in H01 fails to be critical, because of a uniqueness result by Wente [33]. Clearly enough, this means that every Palais Smale sequence at the mountain pass level is not relatively compact in H01 . Let us turn to the case of a nonconstant boundary datum γ. If γ is suitably close to zero, then problem (0.1) might be seen as a perturbation of the correspondent problem with vanishing boundary condition. As far as concerns the existence of the “small” solution, the problem has been solved firstly in case H constant by Hildebrandt [18] and Wente [32] (see also previous papers by Heinz [17], Werner [35], etc.) and then, for arbitrary bounded functions H by Hildebrandt again [19] and Steffen [25], [26] under different smallness conditions on γ, by minimizing the energy functional EH on certain subsets of Hγ1 suited to recover locally the weak lower semicontinuity. Now let us discuss the problem of the existence of the “large” solution. When H is a nonzero constant, the “perturbation” γ allows us to recover compactness at the mountain pass level. In this case, a multiplicity result was proved by Brezis and Coron in [8] and independently by Struwe in [29] (see also [27], [30] and [34]). In particular in [29] the Author pointed out the following result. Theorem 0.1 [Struwe] Let H be a nonzero constant. If the functional EH has a nonconstant local minimum u, then the mountain pass level for EH is a critical level, hence there exists a (weak) solution u to (0.1) different from u. Notice that Struwe’s Theorem describes a situation shared by a large class of variational problems with lack of compactness, characterized by the fact that, even if the (global) Palais Smale condition can fail, the presence of a strict local minimum point corresponding to a nonconstant solution of the problem is helpful in order to recover some compactness, ensuring the existence of a mountain pass type critical point (see, e.g., [2], [7], [23]). As remarked by some authors (see [5, pag. 602] and [31, pag. 642]), for a long time it was believed that this situation could occur also for the problem of H-surfaces with H noncontant and that the difficulty if finding such a result were due to the strong technical obstacles arising when H is variable. In this paper we show that in general an analogous version of Struwe’s Theorem in case H nonconstant is false, even if H is “very close” to a constant in any reasonable topology. The main result is given by Theorem 5.5. As a consequence of Theorem 5.5 we have, for example, that: Theorem 0.2 For every λ > 0 there exists δ > 0 and a curvature H ∈ C ∞ (R3 ) with kH − 1k∞ + k∇Hk∞ < λ 3

such that for every boundary datum γ satisfying kγkH 1 + kγkL∞ < δ problem (0.1) has a small solution u but no large solution at a mountain pass level. In fact we are able to exhibit quite a large class of curvatures H for which the nonexistence 2 result in Theorem 0.2 holds. An example of such an H’s is the map H(p) = 1 − λe−|p| for λ > 0 small enough. We notice that Theorem 0.1 can be proved by investigating in a quite precise way the behavior of the Palais Smale sequences of EH (see, e.g. [9]). Apart from the case H constant, up to now, for arbitrary variable curvatures H a complete description of the behavior of Palais Smale sequences of EH is not available in the literature. The difficulties occurring in the study of Palais Smale sequences in the general case of (bounded) variable curvatures are not just technical. Actually, for nonconstant functions H, Palais Smale sequences may have quite a wild behavior. In [14] we point out some remarks and we illustrate some examples on this subject. To obtain the nonexistence result stated in Theorem 0.2 (or the more general version given by Theorem 5.5), instead of studying the behavior of the Palais Smale sequences for EH in Hγ1 at the mountain pass level, we adopt a different viewpoint. More precisely, assuming that for γ small EH admits a critical point uγ ∈ Hγ1 at the mountain pass level, we study the behavior of uγ as γ shrinks to zero. More generally, we study the behavior of a bounded sequence (uγ ) ⊂ H 1 (D2 , R3 ) of solutions of (0.1) as kγkH 1 + kγkL∞ → 0 and we prove some results describing typical blow up phenomena. In order to state these results, some preliminary definitions are helpful. We call H-bubble a nonconstant, bounded solution of the problem  ∆ω = 2H(ω)ωx ∧ ωy on R2 R (0.3) |∇ω|2 < +∞ . R2 The energy of an H-bubble ω is given by Z Z 1 2 ˆ |∇ω| + 2 QH (ω) · ωx ∧ ωy . EH (ω) = 2 R2 R2

We also say that a sequence (un ) ⊂ H 1 (D2 , R3 ) blows up an H-bubble ω if there exist sequences (ζn ) ⊂ D2 and (εn ) ⊂ (0, 1) such that, for a subsequence, one has that εn → 0, 1 (1/εn )dist(ζn , ∂D2 ) → +∞ and un (εn z + ζn ) → ω(z) in Hloc (R2 , R3 ). Clearly a blow up corresponds to a lack of compactness according to a rather classical concentration phenomenon. Moreover, in the frame of the concentration-compactness principle, problem (0.3) constitutes a so called “problem at infinity” associated to the Dirichlet problem (0.1). We can summarize some of the results proved in Sections 2 and 3 in the next theorem. 2

Theorem 0.3 Let H ∈ C 1 (R3 ), let (γ n ) ⊂ H 1 ∩ C 0 (D , R3 ) be such that kγ n kH 1 + kγ n kL∞ → 0, and for every n ∈ N let un ∈ Hγ1n ∩ L∞ be a solution of the Dirichlet 4

problem (0.1) with boundary datum γ n . Assume also that C0 ≤ kun kH 1 ≤ C1 for some positive constants C0 and C1 . The following facts hold: (i) If H ∈ L∞ (R3 ), then the sequence (un ) blows up an H-bubble ω. (ii) If in addition supp∈R3 |∇H(p) · p p| =: MH < 1 then EˆH (ω) ≤ lim inf EH (un ). (iii) If MH < 1 and, furthermore, EH (un ) → EˆH (ω) then, up to a subsequence,

   

n 1

u − ω · − ζn − ω∞ → 0 and un (εn z + ζn ) → ω(z) in Cloc (R2 , R3 )

1

εn H

where ω∞ = lim|z|→+∞ ω(z), and (ζn ) ⊂ D2 and (εn ) ⊂ (0, 1) are suitable sequences satisfying εn → 0 and (1/εn )dist(ζn , ∂D2 ) → +∞.

The above theorem enables us to state a necessary condition in order that the Dirichlet problem (0.1) admits a large solution at a mountain pass level for γ small. More precisely, given H ∈ C 1 (R3 ) such that MH < 1 and kHk∞ = lim|p|→+∞ |H(p)|, it turns out that in order that the mountain pass level for EH in Hγ1n is a critical value for a sequence of boundary data γ n shrinking to zero in the H 1 ∩ C 0 topology, then a minimal H-bubble must exist (we refer to [10] for the discussion on the minimal H-bubbles). On the other hand, in section 5 we construct a class of (nonconstant) curvature functions for which there exists no minimal H bubble. Hence, in these cases, one obtains nonexistence of the large solution for (0.1) with small boundary datum. The main technical tools in our approach are some a priori estimates due to Bethuel and Rey [5] concerning the L∞ norm of solutions of the Dirichlet problem (0.1). We will also take advantage from some ε-regularity estimates, both in the spirit of the paper by Sacks and Uhlenbeck [24] (see also [10]), and in the direction followed by Bethuel and Rey in [5]. We point out that for a fixed small boundary datum γ, if H is suitably close to a constant, e.g., H(p) = 1 + εK(p) with K ∈ Cc∞ (R3 ) and |ε| small, Bethuel and Rey [5] and independently Jakobowsky [21] (see also [22]) proved the existence of a large solution for (0.1). Notice that our nonexistence result, stated by Theorem 0.2, is not in contradiction with the above mentioned results. In fact in the results by Bethuel and Rey and by Jakobowsky the perturbation parameter ε in the curvature function has to be taken smaller and smaller with γ. We finally quote also some very recent works by Chanillo and Malchiodi [15] and by Isobe [20] concerning larger multiplicity results for the Dirichlet problem (0.1) in case H nonzero constant and γ small.

1 1.1

Preliminaries Notation

For any domain Ω ⊆ Rk and every exponent σ ∈ [1, +∞], we denote by Lσ (Ω, Rn ) the usual space of Lσ integrable maps. When no confusion can arise, we shall often write Lσ instead 5

of Lσ (Ω, Rn ) and we denote with k · kσ the norm in Lσ . Let D2 be the unit disc in R2 . The usual Sobolev space H 1 (D2 , R3 ) will be denoted simply by H 1 . We shall denote the Dirichlet integral of a map u ∈ H 1 by Z 1 |∇u|2 . D(u) := 2 D2 For γ ∈ H 1 we set

Hγ1 := {u ∈ H 1 | u = γ on ∂D2 } = γ + H01 .

The space Hγ1 is an (affine) Hilbert space with respect to the Dirichlet norm. Notice that in dealing with the space Hγ1 we can always assume γ to be harmonic in D2 . If in addition γ is bounded, then the map z → |γ(z)|2 is subharmonic in D2 , hence kγkL∞ (D2 ,R3 ) = kγkL∞ (∂D2 ,R3 ) . Let us define kγk := k∇γk2 + kγk∞ . Let us introduce the spaces ˆ 1 := H 1 (S2 , R3 ) H ˆ s := Ls (S2 , R3 ) (1 ≤ s < +∞) . L ˆ 1 is a Hilbert space compactly embedded into L ˆ s for all It is known (see e.g. [1]) that H ∞ 2 3 1 ˆ . s ∈ [1, +∞) and that C (S , R ) is dense in H ˆ 1 with a corresponding map from In the following we will often identify any map U ∈ H 2 3 R into R obtained by composing U with the inverse of the stereographic projection of ˆ 1 turns out to be isomorphic to the Hilbert space S2 onto the R2 ∪ {∞}. In this way H R 1 {U ∈ Hloc (R2 , R3 ) | R2 (|∇U |2 + µ2 |U |2 ) < +∞} where µ(z) = 2/(1 + |z|2 ) for z ∈ R2 . ˆ 1 can be written as follows: Moreover the norm in H Z 2 (|∇U |2 + µ2 |U |2 ) . kU kHˆ 1 = R2

ˆ 1 we set For U ∈ H

ˆ ) := 1 D(U 2

Z

R2

|∇U |2 .

Finally, althrougout this work, Dr (z) and Dr (z) denote respectively the open and the closed disc in R2 centered at z ∈ R2 and with radius r > 0. In particular, if z = 0, we write Dr instead of Dr (0). Moreover, we denote Br (p) the open ball in R3 centered at p ∈ R3 and with radius r > 0. We also set Br = Br (0).

1.2

The Bononcini-Wente isoperimetric inequality

Let us introduce the classical Bononcini-Wente volume functional: Z 1 ˆ V(U ) := U · Ux ∧ Uy 3 R2 6

ˆ 1 ∩ L∞ . It is known that Vˆ extends to an analytic function on H ˆ 1, defined for every U ∈ H ˆ and that the following isoperimetric inequality holds: still denoted V, √ 3

ˆ )|2/3 ≤ D(U ˆ ) S|V(U

ˆ1 for every U ∈ H

(1.1)

where S = 36π is the isoperimetric constant (see [6], [32]). The constant S is the best one in the above inequality, namely inf

ˆ1 U ∈H ˆ )6=0 V(U

ˆ ) √ D(U 3 = 36π . ˆ )|2/3 |V(U

(1.2)

For future convenience, in the next result we make precise the characterization of minimizers and minimizing sequences for (1.2). Even if this result is known and essentially contained for instance in [9], for the sake of completeness we present here a self-contained and direct proof. Lemma 1.1 Any minimizer for (1.2) is a conformal parametrization of a sphere of degree ˆ 1 is a minimizing sequence for 1 or −1 (as a map from S2 into S2 ). Moreover, if (U n ) ⊂ H 2 (1.2), then there exist sequences (ρn ) ⊂ (0, +∞), (ζn ) ⊂ R and (pn ) ⊂ R3 such that ˜ n → ω strongly in H ˆ 1 , where U ˜ n (z) = U

1 U n (ρn z + ζn ) − pn k∇U n k2

and ω is a conformal parametrization of a sphere of degree 1 or −1. ˆ 1 with its corresponding mapping Proof. In this proof we shall often identify any map U ∈ H 1 2 3 in H (S , R ) obtained by composing U with the inverse of the standard stereographic projection of S2 onto R2 ∪ {∞}. The proof is accomplished in three steps: ˆ 1 is a minimizer for (1.2) then ω is a conformal parametrization of a sphere Step 1. If ω ∈ H of degree 1 or −1. ˆ 1 is a minimizing sequence for (1.2) such that U n → U weakly in H ˆ1 Step 2. if (U n ) ⊂ H n 1 ˆ and U is a minimizer for (1.2). and U is nonconstant, then U → U strongly in H ˆ 1 of (1.2) there exist sequences (ρn ) ⊂ Step 3. For every minimizing sequence (U n ) ⊂ H 2 3 ˜ n ) defined by (0, +∞), (ζn ) ⊂ R and (pn ) ⊂ R such that the sequence (U ˜ n (z) = U

1 U n (ρn z + ζn ) − pn k∇U n k2

ˆ 1 to some nonconstant function U ∈ H ˆ 1. admits a subsequence weakly converging in H ˆ 1 be a minimizer for (1.2). Given any ϕ ∈ Cc∞ (R2 , R3 ), the Proof of Step 1. Let ω ∈ H ˆ )/|V(U ˆ )|2/3 at ω along the direction ϕ is directional derivative of the functional U 7→ D(U zero, namely Z Z ϕ · ωx ∧ ωy = 0 ∇ω · ∇ϕ + 2λ R2

R2

7

for some λ ∈ R. Hence ω is a weak solution of ∆ω = 2λωx ∧ ωy on R2 . The case λ = 0 ˆ 1 , ω should be constant contrary to the fact that cannot occur because otherwise, since ω ∈ H ˆ V(ω) 6= 0. According to a result proved by Brezis and Coron [9] ω is a (possibly branched) conformal parametrization of a sphere of radius 1/|λ|. In particular, if d ∈ Z \ {0} denotes ˆ ˆ the degree of ω, then D(ω) = 4π|d|/λ2 and |V(ω)| = 4π|d|/(3|λ|3 ) (see [9]). Since ω is a minimizer for (1.2), we infer that |d| = 1. ˆ 1 be a minimizing sequence for (1.2) such that U n → U Proof of Step 2. Let (U n ) ⊂ H 1 ˆ weakly in H with U nonconstant. We notice that ˆ n) D(U ˆ n) V(U

ˆ ) + D(U ˆ n − U ) + o(1) = D(U ˆ ) + V(U ˆ n − U ) + o(1) . = V(U

(1.3) (1.4)

The expansion (1.4) has been proved by Wente [32] and by Brezis and Coron [9] for sequences ˆ 1 can be proved in the same way, by using the in H01 . The version (1.4) for sequences in H 3 ∞ 2 3 1 ˆ ˆ 1 . By (1.3), (1.4) and (1.2) density of R + Cc (R , R ) into H and the continuity of Vˆ in H we find that ˆ ) + D(U ˆ n − U) D(U

ˆ ) + V(U ˆ n − U )|2/3 + o(1) = S|V(U ˆ )|2/3 + S|V(U ˆ n − U )|2/3 + o(1) ≤ S|V(U

(1.5)

ˆ ) + D(U ˆ ≤ D(U − U ) + o(1). n

The inequality (1.5) follows from the fact that |1 + t|2/3 ≤ 1 + |t|2/3 for every t ∈ R. Hence in particular we obtain ˆ ) + V(U ˆ n − U )|2/3 = |V(U ˆ )|2/3 + |V(U ˆ n − U )|2/3 + o(1). |V(U ˆ ) = 0 or V(U ˆ n −U ) → 0. Since |1+t|2/3 = 1+|t|2/3 if and only if t = 0, we infer that either V(U ˆ )+ D(U ˆ n −U ) ≤ The first case cannot occur because otherwise from (1.5) it follows that D(U ˆ n − U ) + o(1) and then U is constant, contrary to the assumption. Hence it must be D(U ˆ ˆ n − U ) → 0. Therefore (1.5) yields D(U ˆ ) + D(U ˆ n − U ) ≤ D(U ˆ ) + o(1) V(U ) 6= 0 and V(U n n ˆ ˆ namely k∇(U − U )k2 → 0 and in particular D(U ) → D(U ), thanks to (1.3). Moreover ˆ n ) → V(U ˆ ). Thus U is a minimizer for (1.2). In addition, since U n → U (1.4) implies V(U ˆ 1 , k∇(U n − U )k2 → 0 and H ˆ 1 is compactly embedded into L ˆ 2 , it follows that weakly in H n 1 ˆ U → U strongly in H . ˆ 1 be a minimizing sequence for (1.2). Let αn = k∇U n k−1 Proof of Step 3. Let (U n ) ⊂ H 2 n ˆ n ) 6= 0). (notice that k∇U k2 6= 0 because of the isoperimetric inequality (1.1) and V(U Then, by homogeneity, (αn U n ) is again a minimizing sequence for (1.2) with k∇(αn U n )k2 = Sk 1. Fix k points z1 , . . . , zk ∈ ∂D2 such that ∂D2 ⊂ i=1 D1 (zi ). By a standard procedure, for every n ∈ N there exist ζn ∈ R2 and ρn > 0 such that Z Z 1 n 2 . |∇(αn U n )|2 = |∇(αn U )| = sup 2 k + 1 z∈R Dρn (z) Dρn (ζn ) 8

Let ˜ n (z) = αn U n (ρn z + ζn ) − pn U R ˜ n = 0. Since k∇U ˜ n k2 = k∇(αn U n )k2 = where pn ∈ R3 is taken in order to ensure that S2 U R n n ˜ = 0 for every n ∈ N, the sequence (U ˜ ) turns out to be bounded in H ˆ 1 and 1 and S2 U ˜ n ), weakly converging to some U ∈ H ˆ 1. therefore it admits a subsequence, still denoted (U We claim that U is nonconstant. Arguing by contradiction, assume that U is constant. Since R ˜ n = 0 and H ˆ 1 is compactly embedded into L ˆ 1 we infer that U = 0. We also observe U S2 ˆ U ˜ n ) = V(α ˆ n U n ) and then (U ˜ n ) is again a minimizing that, by conformal invariance, V( n ˜ sequence for (1.2) satisfying k∇U k2 = 1 and Z Z ˜ n |2 = 1 ˜ n |2 = sup (1.6) |∇U |∇U k+1 z∈R2 D1 (z) D2 for every n ∈ N. Let r > 1 be such that Dr ⊂ there exists rn ∈ [1, r] such that

Sk

D1 (zi ). We claim that for every n ∈ N

i=1

˜ n kH 1/2 (∂D ) → 0 . kU rn

(1.7)

˜ n k2 = 1 and, thanks to the compact embedding of H ˆ 1 into L ˆ2, U ˜n → 0 Indeed, since k∇U 2 ˆ , by Fubini’s Theorem we can find rn ∈ [1, r] such that strongly in L Z Z 2 n 2 ˜ ˜ n |2 → 0 . |∇U | ≤ 2 |U and r −1 ∂Drn ∂Drn Then the interpolation inequality kukH 1/2 (∂Drn ) ≤ kukH 1 (∂Drn ) kukL2 (∂Drn ) implies (1.7). Passing to a subsequence, if necessary, we may suppose that there exists Z ˜ n |2 = λ . lim |∇U n→+∞

There results

Indeed

Z

Dr n

Dr n

k 1 ≤λ≤ . k+1 k+1 Z ˜ n |2 ≥ ˜ n |2 = |∇U |∇U D2

(1.8) 1 k+1

by (1.6), and Z

Dr n

˜ n |2 ≤ |∇U

Z

Dr

˜ n |2 ≤ |∇U

k Z X i=1

D1 (zi )

˜ n |2 ≤ |∇U

k k+1

Sk ¯ n (z) = U ˜ n (rn z) and let by (1.6) again and since Dr ⊂ i=1 D1 (zi ). For every n ∈ N let U 2 n n 1 n ¯ kH 1/2 (∂D2 ) → 0 and ¯ 2 on D . We have that kU h ∈ H be the harmonic extension of U |∂D consequently Z D2

|∇hn |2 → 0 9

(1.9)

Now define un (z) =



¯ n (z) U as |z| < 1 hn (z/|z|2 ) as |z| ≥ 1

ˆ 1 and Notice that un , wn ∈ H Z |∇un |2 2 ZR |∇wn |2 R2

=

and wn (z) =

Z

D2

=

¯ n |2 + |∇U

Z

R2 \D2



¯ n (z/|z|2 ) U hn (z/|z|2 )

as |z| < 1 as |z| ≥ 1 .

Z

|∇hn |2 Z ¯ n |2 + |∇U |∇hn |2

(1.10)

D2

(1.11)

D2

having used the invariance of the Dirichlet integral with respect to the Kelvin transformation z 7→ z/|z|2 . Therefore (1.10) and (1.11) imply Z ˆ n ) + D(w ˆ n ) = D( ˆ U ¯ n) + D(u |∇hn |2 . (1.12) D2

¯ n ∈ L∞ , by direct computations we also obtain Moreover, assuming for a moment that U that Z Z Z n ¯n n n n n ¯ ¯ (1.13) hn · hnx ∧ hny U · Ux ∧ Uy − u · ux ∧ uy = D2 D2 R2 Z Z Z ¯n · U ¯n ∧ U ¯yn − U hn · hnx ∧ hny (1.14) wn · wxn ∧ wyn = − x R2 \D2

R2

D2

which yield ˆ n ) − V(w ˆ n ) = V( ˆ U ¯ n ) = V( ˆ U ˜ n) . V(u

(1.15)

¯n ∈ H ˆ 1 even without the condition U ¯ n ∈ L∞ and this In fact (1.15) holds true for any U 1 ∞ 1 ˆ ˆ ˆ 1 and can be obtained by using the density of H ∩ L into H , the continuity of Vˆ in H the formulas (1.13) and (1.14). Hence, by (1.1), (1.9), (1.12) and (1.15) we deduce that ˆ n ) + D(w ˆ n) D(u

ˆ U ˜ n ) + o(1) = D( ˆ U ˜ n )|2/3 + o(1) = S|V(

ˆ n ) − V(w ˆ n )|2/3 + o(1) = S|V(u ˆ n )|2/3 + S V(w ˆ n )|2/3 + o(1) ≤ S|V(u ˆ n ) + D(w ˆ n ) + o(1) ≤ D(u

where the first inequality has been obtained thanks to the fact that |1 + t|2/3 ≤ 1 + |t|2/3 for every t ∈ R. Hence we infer that ˆ n ) − V(w ˆ n )|2/3 = |V(u ˆ n )|2/3 + |V(w ˆ n )|2/3 + o(1) |V(u and, as in the proof of Step 2, we obtain that two alternative cases may occur: either ˆ n ) → 0 or V(w ˆ n ) → 0. In the first case, the previous chain of inequalities reduces to V(u 10

ˆ n ) + D(w ˆ n ) ≤ D(w ˆ n ) + o(1) and thus D(u ˆ n ) → 0. Similarly in the second case one D(u ˆ n ) → 0. Finally we show that in both the cases we get a contradiction. obtains that D(w ˆ Indeed, if D(un ) → 0, then (1.9) and (1.10) imply that Z Z ˜ n |2 = ¯ n |2 → 0 |∇U |∇U D2

Dr n

ˆ n ) → 0, then (1.9) and (1.11) imply that contrary to (1.8). If D(w Z ¯ n |2 → 0 |∇U R2 \D2

and then

Z

Dr n

˜ n |2 = |∇U

Z

D2

¯ n |2 = |∇U

Z

R2

¯ n |2 − |∇U

Z

R2 \D2

¯ n |2 → 1 , |∇U

again in contradiction with (1.8). Hence U cannot be constant. This ends the proof.

1.3

H-volume functional

The next step consists in defining a class of H-weighted volume functionals corresponding to variable curvature functions. A way to do this has been introduced and discussed by Steffen in [25] by means of the theory of currents. We just recall the main steps and those results which will be useful for our purposes. In [25] it is proved that for every pair u, w of maps in H 1 with u − w ∈ H01 , there exists a unique integer valued map iu,w ∈ L1 ∩ L3/2 (R3 ) such that Z Z 1 3/2 (D(u) + D(w)) (1.16) |iu,w (p)|3/2 dp ≤ √ |iu,w (p)| dp ≤ 3 3 36π R R and such that iu,w has compact support if u, w ∈ L∞ ([25], Theorem 2.10). Then, given H ∈ L∞ (R3 ), for every H ∈ L∞ (R3 ) one sets Z H(p)iu,w (p) dp . VH (u, w) := R3

Thus VH (u, w) is well defined, and from (1.16) and H¨older’s inequality it follows that √ 3 36π |VH (u, w)|2/3 ≤ D(u) + D(w) . 2/3 kHk∞ As noticed by Steffen [25], roughly speaking, the functional VH (u, w) measures the algebraic H-weighted volume enclosed by the surfaces parametrized by the maps u and w. Now, for every fixed boundary datum γ ∈ H 1 , let us set VH,γ (u) := VH (u, γ)

for every u ∈ Hγ1 .

In the next result, proved again in [25], we collect some properties of VH,γ . 11

Lemma 1.2 Let H ∈ L∞ (R3 ) and γ ∈ H 1 ∩ L∞ . Then (i) the functional VH,γ is continuous on Hγ1 ; (ii) if u, w ∈ Hγ1 then

√ 3

36π

2/3 kHk∞

|VH,γ (u) − VH,γ (w)|2/3 ≤ D(u) + D(w) ;

(1.17)

(iii) if H ∈ L∞ (R3 ) ∩ C 0 (R3 ) and Q: R3 → R3 is any vectorfield such that div Q = H (in the sense of distributions), then for every u ∈ Hγ1 ∩ L∞ it holds that Z Z Q(γ) · γx ∧ γy ; Q(u) · ux ∧ uy − VH,γ (u) = D2

D2

(iv) if H ∈ L∞ (R3 ) ∩ C 0 (R3 ), u ∈ Hγ1 and ϕ ∈ H01 ∩ L∞ , the directional derivative of VH at u in the direction of ϕ exists and it is given by Z ∂VH H(u)ϕ · ux ∧ uy . (u) = ∂ϕ D2 Given H ∈ L∞ (R3 ) let us set mH (p) :=

Z

0

1

H(sp)s2 ds for p ∈ R3 .

(1.18)

Notice that div(mH (p)p) = ∇mH (p) · p + 3mH (p) = H(p) and then QH (p) := mH (p)p can be chosen in (iii) of the previous lemma as a vectorfield whose divergence equals H. Moreover, for γ ∈ H 1 ∩ L∞ , let us define the functional VH : Hγ1 → R, by setting Z VH (u) := VH,γ (u) + mH (γ)γ · γx ∧ γy . D2

In particular, if in addition H ∈ C 0 (R3 ) then for every u ∈ Hγ1 ∩ L∞ we have Z mH (u)u · ux ∧ uy . VH (u) = D2

We also define the energy functional EH : Hγ1 → R as follows EH (u) := D(u) + 2VH (u) . This yields the variational setting suited to study the Dirichlet problem (0.1). More precisely, as a consequence of Lemma 1.2, we have that: 12

Lemma 1.3 Let H ∈ L∞ (R3 ) and γ ∈ H 1 ∩ L∞ . Then the functional EH is continuous on Hγ1 . If in addition H ∈ C 0 (R3 ), then (i) for every u ∈ Hγ1 ∩ L∞ it holds that Z Z 1 2 mH (u)u · ux ∧ uy ; |∇u| + 2 EH (u) = 2 D2 D2 (ii) if u ∈ Hγ1 and ϕ ∈ H01 ∩ L∞ , the directional derivative of EH at u in the direction of ϕ exists and it is given by Z Z ∂EH H(u)ϕ · ux ∧ uy . ∇u · ∇ϕ + 2 (u) = ∂ϕ D2 D2 In particular u ∈ Hγ1 is a critical point of EH (namely (∂EH /∂ϕ)(u) = 0 for all ϕ ∈ Cc∞ (D2 , R3 )) if and only if u is a weak solution of (0.1). Remark 1.4 The regularity theory for H-systems [17] yields that if u is a bounded weak solution of ∆u = 2H(u)ux ∧ uy on a domain Ω ⊆ R2 with H ∈ C 1 (R3 ), then u ∈ C 3 (Ω, R3 ). In particular if u ∈ H 1 is a bounded weak solution of (0.1) with H ∈ C 1 (R3 ) and γ ∈ H 1 ∩ C 0 (D2 , R3 ), then u ∈ C 3 (D2 , R3 ) ∩ C 0 (D2 , R3 ). The fact that a weak solution of (0.1) is bounded holds true under suitable conditions on H, e.g., when H ∈ C ∞ (R3 ) ∩ L∞ with supp∈R3 |∇H(p)|(1 + |p|) < +∞ (see [17]). Other regularity type results can be found for example in [4] and references therein. ˆ 1 . To every map Finally we introduce an H-weighted volume functional for maps in H 1 ′ 2 ˆ U ∈ H we can associate the pair of maps u, u on the disc D , defined by u = U|D2 and u′ = U (z/|z|2 ) for |z| < 1. Simple computations show that Z Z Z 2 2 |∇U | = |∇u| + |∇u′ |2 . (1.19) R2

D2

D2

Noticing that u − u′ ∈ H01 , we set iU := i

u,u′

and VˆH (U ) :=

Z

R3

H(p)iU (p) dp = VH (u, u′ ) .

(1.20)

ˆ ∩ L∞ , it results that If H ∈ L∞ ∩ C 0 (R3 ) and U ∈ H Z ˆ mH (U )U · Ux ∧ Uy VH (U ) = R2

with mH defined by (1.18). As before, VˆH (U ) represents the algebraic H-weighted volume of the bounded region enclosed by the surface parametrized by U . In particular, for H ≡ 1 the functional Vˆ1 coincides with the classical Bononcini-Wente volume Vˆ introduced at the beginning. 13

From (1.17) and (1.19) it follows that √ 3 36π ˆ ˆ ) |VH (U )|2/3 ≤ D(U 2/3 kHk∞

ˆ 1. for every U ∈ H

(1.21)

Finally, we set ˆ ) + 2VˆH (U ) EˆH (U ) := D(U

ˆ 1. for any U ∈ H ˆ 1 together with the energy functional EˆH constitute a variational setting The space H suited to study problem (0.3). ˆ 1 ∩ L∞ is a critical point of EˆH if and only if ω is a bounded, weak Observe that ω ∈ H solution of (0.3)). According to Remark 1.4, if H ∈ C 1 (R3 ), a bounded weak solution of (0.3) is actually a classical solution of the H-system on R2 and there exists lim|z|→∞ ω(z). In fact ω is of class C 3 as a map on S2 . In addition, it is known that a classical solution of (0.3) automatically satisfies the conformality conditions (see, e.g. [33], or also [11]). Thus an H-bubble (defined in the Introduction as a nonconstant, bounded solution of (0.3)) is a bounded, nonconstant critical point of EˆH , and viceversa. An H-bubble ω is said to be minimal if EˆH (ω) ≤ EˆH (U ) for every H-bubble U . Existence theorems for H-bubbles were proved in [10–13]. In particular we quote the paper [10] concerning existence of minimal H-bubbles, which contains some results which turn out to be useful in the sequel of the present work.

2

Blowing H-bubbles

In this Section we start to study the behavior of a sequence of solutions of  ∆un = 2H(un )unx ∧ uny in D2 un = γ n on ∂D2

(2.1)

as γ n shrinks to 0. Our first result is the following. Theorem 2.1 Let H ∈ C 1 ∩ L∞ (R3 ) and let γ n ∈ H 1 ∩ C 0 (D2 , R3 ) be a sequence of boundary data, with kγ n k → 0. Let un ∈ Hγ1n ∩ L∞ be a sequence of weak solutions to (2.1) with k∇un k2 bounded. Then, either un → 0 strongly in H 1 , and in this case EH (un ) → 0, or there exists an H-bubble ω with kωk∞ ≤ C(kHk∞ , supn k∇un k22 ). More precisely, if un does not converge to zero strongly in H 1 , then there exist sequences (ζn ) ⊂ D2 and (εn ) ⊂ (0, 1) such that εn → 0, (1/εn )dist(ζn , ∂D2 ) → +∞ and, for a subsequence, u ˜n → ω 1 2 3 n n in Cloc (R , R ), where u ˜ (z) = u (εn z + ζn ). A crucial result in the proof of Theorem 2.1 is the following estimate, that was proved in this generality by Bethuel and Rey (Theorem 4.8 in [5]).

14

Theorem 2.2 Let H ∈ C 1 ∩L∞ (R3 ), let γ ∈ H 1 ∩C 0 (D2 , R3 ) and let uγ ∈ Hγ1 ∩C 0 (D2 , R3 ) be a classical solution to (0.1). Then there exists a constant C > 0 (depending only on kHk∞ ) such that
kuγ k∞ ≤ kγk∞ + C k∇uγ k22 + 1 . (2.2)

In order to prove Theorem 2.1 first we notice that un ∈ C 2 (D2 , R3 ) (use Remark 1.4) and, by Theorem 2.2, sup kun k∞ =: R < +∞. (2.3) n

In the following we will strongly use some kind of ε-regularity estimates. The first ones are an adaptation of a similar result obtained by Sacks and Uhlenbeck in [24]. We refer to [10] for the proof. Lemma 2.3 Given H ∈ L∞ (R3 ) there exist ε¯ > 0 and, for every s ∈ (1, +∞) a constant Cs > 0 (depending only on kHk∞ ), such that if u is a weak solution of ∆u = 2H(u)ux ∧ uy 2,s on some domain Ω ⊆ R2 and u ∈ Wloc (Ω, R3 ), then k∇ukL2 (Dr (z)) ≤ ε¯ ⇒ k∇ukW 1,s (Dr/2 (z)) ≤ Cs r(2−2s)/s k∇ukL2 (Dr (z)) for every r ∈ (0, 1] and for every z ∈ R2 with Dr (z) ⊂ Ω. Other estimates will be needed in order to study the convergence of the sequence (un ) at the boundary of the domain, where we have no further information on the regularity of any un but the continuity and the H 1 regularity. The version of ε-regularity estimates in this case is essentially due to Bethuel and Rey [5], even if our setting is much simpler, since we deal with a sequence of solutions which are uniformly bounded in L∞ . We have that: Lemma 2.4 Let H ∈ C 1 (R3 ). For every R > 0 there exists εR > 0 (depending on R and on kHkC 1 (BR ) ) such that, given any sequence (un ) ⊂ H 1 satisfying: (i) ∆un = 2H(un )unx ∧ uny in D2 , (ii) un → 0 weakly in H 1 , (iii) kun k∞ ≤ R for all n ∈ N, it holds that lim sup k∇un kL2 (Dr (z)∩D2 ) ≤ εR ⇒ un → 0 strongly in H 1 (Dr/2 (z) ∩ D2 ) n→+∞

for every z ∈ D2 and for every r ∈ (0, 1]. We postpone the proof of Lemma 2.4. In fact in the sequel we shall discuss a slightly more general result (Lemma 2.10) which includes, as a special case, also the above lemma.

15

let

Now, having fixed H ∈ C 1 ∩ L∞ (R3 ), the sequence (un ), and consequently R as in (2.3), ε0 = min {¯ ε, εR }

where ε¯ is given by Lemma 2.3 and εR is given by Lemma 2.4 in correspondence of R. As a first step, obtained by using Lemma 2.3 and a uniqueness result by Wente [33], we have: 1 Lemma 2.5 un → 0 weakly in H 1 and strongly in Cloc (D2 \ N, R3 ) where N is a finite 2 subset of D .

Proof. Since the boundary data γ n are harmonic on D2 we first get that γ n → 0 in H 1 . Then set v n := un − γ n ∈ H01 . Thus, v n is bounded in H01 and there exists v ∈ H01 ∩ L∞ such that, for a subsequence, v n → v weakly in H01 . Moreover inf n k∇v n k2 > 0. Now we prove that v is a weak solution to the Dirichlet problem  ∆v = 2H(v)vx ∧ vy in D2 (2.4) v=0 on ∂D2 . Indeed, observe that for fixed ϕ ∈ Cc∞ (D2 , R3 ) Z Z n ∇v · ∇ϕ ∇v · ∇ϕ → 2 D2 ZD n n n n n 2 H(v + γ )ϕ · γ ∧ γ x y ≤ kHk∞ kϕk∞ k∇γ k2 = o(1) 2 ZD n n n n n n n n H(v + γ )ϕ · (v ∧ γ + γ ∧ v ) x y x y ≤ kHk∞ kϕk∞ k∇γ k2 k∇v k2 = o(1) D2

and thus, by (2.1)

Z

D2

∇v · ∇ϕ + 2

Z

D2

H(v n + γ n )ϕ · vxn ∧ vyn = o(1)

(2.5)

In order to estimate the second term in the left hand side of (2.5) we use Lemma 2.3 and 1,s we argue as in [24], Proposition 4.3 and Theorem 4.4. Notice that un ∈ Wloc (D2 , R3 ) for all s > 1 (in fact un is of class C 2 ). Let ε0 > 0 be given according to Lemma 2.3 and let b = sup k∇un k2 . Take a compact K ⊂ D2 and for every k ∈ N large enough take a finite open covering of K made by discs D2−k−1 (zi ) (i ∈ Ik ) and such that every point of K belongs to at most d discs D2−k (zi ). Then XZ |∇un |2 ≤ db for all k and for all n ∈ N. i∈Ik

D2−k (zi )

(n)

(n)

Consequently for every n ∈ N there exists a finite set Ik such that #Ik ≤ bd/ε20 R (n) and such that D −k (zi ) |∇un |2 > ε20 if and only if i ∈ Ik . By compactness, passing 2

16

to a subsequence, if necessary, we can find a set Iˆk with #Iˆk ≤ bd/ε20 and such that R |∇un |2 ≤ ε20 for every i ∈ Ik \ Iˆk . By Lemma 2.3 the sequence (∇un ) is bounded in D2−k (zi )   S W 1,s K \ i∈Iˆk D2−k−1 (zi ) . Since (un ) is bounded in L∞ , we infer that (un ) is bounded   S in W 2,s K \ i∈Iˆk D2−k−1 (zi ) . Hence, for fixed s > 2, by Rellich’s Theorem, un → v in   S C 1 K \ i∈Iˆk D2−k−1 (zi ) . Since #Iˆk ≤ bd/ε20 with a bound independent of k, taking a

sequence of compact sets Kj filling D2 as j → +∞, by a diagonal argument one can find a finite set F ⊂ D2 and a subsequence of (un ), always denoted (un ), such that un → v 1 1 in Cloc (D2 \ F ). Then v n = un − γ n → v strongly in Hloc (D2 \ F ) and pointwise almost everywhere. Then Z Z H(v n + γ n )ϕ · vxn ∧ vyn → H(v)ϕ · vx ∧ vy D2

D2

for every ϕ ∈ Cc∞ (D2 \ F ). Hence v is a weak solution of ∆v = 2H(v)vx ∧ vy in D2 \ F . In order to prove that v is a weak solution of the H-system in D2 , we can repeat the proof of Theorem 3.6 in [24]. Assume for simplicity that F = {0}. Let η ∈ C ∞ (R, [0, 1]) be such that η(s) = 0 for s ≤ 1 and η(s) = 1 for s ≥ 2, and set η k (s) = η(ks). Given ϕ ∈ Cc∞ (D2 , R3 ) we set ϕk (ζ) = η k (|ζ|)ϕ(ζ). Notice that ϕk can be used as test for v to get Z Z k H(v)ϕk · vx ∧ vy = 0 . (2.6) ∇v · ∇ϕ + 2 D2

D2

R R Now, since ϕk → ϕ weakly∗ in L∞ , we get D2 H(v)ϕk · vx ∧ vy → D2 H(v)ϕ · vx ∧ vy . Also, R R R ∇v·∇ϕk → D2 ∇v·∇ϕ, since, by H¨older inequality, D2 |∇v·∇η k ||ϕ| ≤ Ck∇vkL2 (D2/k ) = D2 o(1) as k → +∞. Therefore, (2.6) yields in the limit Z Z H(v)ϕ · vx ∧ vy = 0 ∇v · ∇ϕ + 2 D2

D2

for every test function ϕ ∈ Cc∞ (D2 , R3 ). Hence v is a weak solution to problem (2.4). Then, by a Heinz regularity result [17], v is smooth, and a nonexistence result by Wente [33] can be applied, to conclude that v ≡ 0. Therefore we have obtained that un → 0 weakly in 1 H 1 (D2 , R3 ) and strongly in Cloc (D2 \ F ). The argument shall go on with a (partial) blow up analysis, in the spirit of [9]. Let us introduce the set Z S := {z ∈ D2 | lim inf |∇un |2 > 0 ∀r ∈ (0, 1] } n→+∞

Dr (z)∩D2

which corresponds to the set of possible blow up points of the sequence (un ). Lemma 2.6 The set S is finite. Moreover S = Ø if and only if un → 0 strongly in H 1 .

17

Proof. As a preliminary result, let us prove that if ζ ∈ S then Z |∇un |2 > ε20 for every r ∈ (0, 1]. lim inf n→+∞

(2.7)

Dr (ζ)∩D2

Indeed, if, by contradiction, (2.7) is false, one can find r ∈ (0, 1] and a subsequence of (un ), still denoted (un ), such that k∇un kL2 (Dr (ζ)∩D2 ) ≤ ε0 for every n. Then Lemmata 2.4 and 2.5 imply that un → 0 in H 1 (Dr/2 (ζ) ∩ D2 ) contradicting the fact that ζ ∈ S. Assume now that S contains k points ζ1 , . . . , ζk and take r > 0 such that the discs Dr (ζi ) are pairwise disjoints. By (2.7) there exists n ¯ ∈ N such that k∇un kL2 (Dr (ζi )∩D2 ) ≥ ε0 for all i = 1, . . . , k and for all n ≥ n ¯ . Then Z

D2

|∇un |2 ≥

k Z X

Dr (ζi )∩D2

i=1

and then #S ≤

|∇un |2 ≥ kε20

1 sup k∇un k22 . ε20 n

Thus S is finite. Trivially, by definition of S, if un → 0 strongly then S = Ø. Let us prove the converse. Assuming that S = Ø, take a finite covering of D2 made by open discs D1/2 (zi ). Since lim inf k∇un kL2 (D1 (zi )∩D2 ) ≤ ε0 , by Lemmata 2.4 and 2.5 we infer that (for a subsequence) k∇un kL2 (D1/2 (zi )∩D2 ) → 0 and then k∇un kL2 (D2 ) → 0. This argument can be repeated for any subsequence of (un ) and this allows us to conclude that k∇un k2 → 0 for all the sequence. Finally the fact that un → 0 in H 1 follows by the Poincar´e inequality and by the assumption kγ n kH 1 → 0. For future convenience let us extend any function un on R2 in order to get a sequence ˆ 1 . More precisely for every n ∈ N set in H  n u (z) as |z| ≤ 1 n u ˆ (z) = n 2 γ (z/|z| ) as |z| > 1. Observe that Z

R2

|∇ˆ un |2 =

Z

D2

|∇un |2 +

Z

D2

|∇γ n |2 and kˆ un k∞ = kun k∞ .

Indeed notice that kˆ un kL∞ (R2 \D2 ) = kγ n kL∞ (D2 ) = kγ n kL∞ (∂D2 ) ≤ kun kL∞ (D2 ) . Thus sup k∇ˆ un k2 < +∞ and n

sup kˆ un k∞ = R < +∞

(2.8)

n

thanks to the boundedness assumption on the Dirichlet norm of (un ) and to (2.3). We shall go on with a rescaling argument and with the study of a suitably normalized sequence of solutions on expanding domains. To this aim the next result will be useful.

18

Lemma 2.7 For every ζ ∈ S there exist sequences (εn ) ⊂ (0, 1) and (ζn ) ⊂ D2 such that εn → 0, ζn → ζ and, for every n large enough, Z Z |∇un |2 = ε20 (2.9) |∇un |2 = sup z∈Dr (ζ)

Dεn (ζn )∩D2

Dεn (z)∩D2

where r ∈ (0, 1) is such that S ∩ Dr (ζ) = {ζ}. Proof. Let ζ ∈ S, let r ∈ (0, 1) as above and let δ > 0 be such that S ∩ Dr+δ (ζ) = {ζ}. Set Ω = Dr+δ (ζ) ∩ D2 . From (2.7) it follows that for every n ∈ N large enough Z |∇un |2 > ε20 . Ω

By standard arguments, for every n (large enough) one can find ζn ∈ Ω and εn ∈ (0, r + δ) such that Z Z |∇un |2 = ε20 . (2.10) |∇un |2 = sup Dεn (ζn )∩Ω

z∈Dr+δ (ζ)

Dεn (z)∩Ω

First we claim that εn → 0. Indeed, by contradiction assume that for a subsequence, inf n εn ≥ ε > 0. Then for every z ∈ Dr+δ (ζ) we have that Z |∇un |2 ≤ ε20 . Dε (z)∩Ω

In particular we can take z = ζ and, since ε ≤ r + δ, we have Z |∇un |2 ≤ ε20 Dε (ζ)∩D2

and this contradicts (2.7). Now we show that ζn → ζ. Assume that (for a subsequence) ζn → ζ ′ . For every r′ ∈ (0, 1], since εn → 0 and ζn → ζ ′ , there exists n′ ∈ N (depending on r′ ) such that Dεn (ζn ) ⊂ Dr′ (ζ ′ ) for every n ≥ n′ and then Z Z |∇un |2 = ε20 for every n ≥ n′ . |∇un |2 ≥ Dr′ (ζ ′ )∩D2

Dεn (ζn )∩Ω

By the arbitrariness of r′ > 0 it follows that ζ ′ ∈ S. But ζ ′ ∈ Dr+δ (ζ) and r and δ have been fixed in such a way that S ∩ Dr+δ (ζ) = {ζ}. Therefore ζ ′ = ζ. Finally notice that, since εn → 0 and ζn → ζ, we have that Dεn (ζn ) ∩ Ω = Dεn (ζn ) ∩ D2 . Moreover for n ∈ N large enough (so that εn < δ), for every z ∈ Dr (ζ) it holds that Dεn (z) ⊂ Dr+δ (ζ) and then (2.9) follows from (2.10). Given ζ ∈ S, and (εn ) and (ζn ) according to Lemma 2.7, for every n ∈ N define: u ˜n (z) = u ˆn (εn z + ζn ) .

19

(2.11)

Observe that, thanks to the conformal invariance, one has: Z Z |∇ˆ un |2 and k˜ un k∞ = kˆ un k∞ . |∇˜ un |2 = R2

R2

In particular, by (2.8), we have that sup k∇ˆ un k2 < +∞ and

sup kˆ un k∞ < +∞ .

n

(2.12)

n

Notice also that, always by the conformal invariance, for every n we have ∆˜ un = 2H(˜ un )˜ unx ∧ u ˜ny

in D1/εn (−ζn /εn ) .

Finally, observe that for every z ∈ R2 and n ∈ N one has Z Z Z |∇ˆ un |2 = |∇˜ un |2 = D1 (z)∩D1/εn (−ζn /εn )

In particular, by (2.9), Z

Dεn (εn z+ζn )∩D2

D2 ∩D1/εn (−ζn /εn )

|∇˜ un |2 =

Z

Dεn (ζn )∩D2

Dεn (εn z+ζn )∩D2

|∇un |2 = ε20 .

(2.13)

|∇un |2 .

(2.14)

(2.15)

As a next step, we will study the convergence property of the rescaled sequence (˜ un ) in correspondence of a blow up point ζ ∈ S. First we consider the case of a point inside the disc D2 . Here we will use the ε-regularity estimates stated in Lemma 2.3. Lemma 2.8 If S∩D2 6= Ø then there exists an H-bubble ω. More precisely, given ζ ∈ S∩D2 , 1 one has that, for a subsequence, u ˜n → ω in Cloc (R2 , R3 ), where (˜ un ) is defined by (2.11). Proof. Because of (2.12), the sequence (˜ un ) admits a subsequence, always denoted (˜ un ), 1 ˆ which converges weakly in H and pointwise almost everywhere to some function ω ∈ 1 ˆ 1 ∩ L∞ . First let us prove that u H ˜n → ω in Cloc (R2 , R3 ). Take a compact set K in R2 . Since ζn → ζ and |ζ| < 1, we have that K ⊂ D1/εn (−ζn /εn ) for n large enough. Fix a finite covering {D1/2 (zi )} of K. Since for n large Dεn (εn zi + ζn ) ⊂ D2 , from (2.14) and (2.9) it follows that Z D1 (zi )

|∇˜ un |2 ≤ ε20

for all i and for all n large enough. Since (2.13) holds and D1 (zi ) ⊂ D1/εn (−ζn /εn ), we can apply Lemma 2.3 with fixed s > 2 and then Rellich’s Theorem to deduce that u ˜n → ω in 1 1 C (D1/2 (zi )) for all i and hence in C (K). Therefore, passing to the limit in (2.13) (more precisely, in its weak formulation) we infer that ω is a weak solution of ∆ω = 2H(ω)ωx ∧ ωy in K. By the arbitrariness of K, we obtain that in fact ω is a weak solution of the H-system on R2 . Moreover, standard regularity theory implies that ω is a classical solution. Finally we show that ω is nonconstant. Indeed, by (2.15) we know that k∇˜ un kL2 (D2 ) = ε0 because 2 n 1 Dεn (ζn ) ⊂ D for n large. Since u ˜ → ω in C (D2 ) we conclude that k∇ωkL2 (D2 ) = ε0 and thus ω is nonconstant. This concludes the proof. 20

Notice that in the case considered in Lemma 2.8 the sequences (εn ) and (ζn ) given by Lemma 2.7 satisfy εn → 0 and (1/εn )dist(ζn , ∂D2 ) → +∞. Thus, in particular, ζn ∈ D2 for n large. Now let us consider the case of a blow up point on the boundary of D2 . Lemma 2.9 If S ∩∂D2 6= Ø then there exists an H-bubble ω. In particular, given ζ ∈ S ∩D2 and taking (εn ) and (ζn ) according to Lemma 2.7, one has that (1/εn )dist(ζn , ∂D2 ) → +∞ 1 and, up to a subsequence, u ˜n → ω in Cloc (R2 , R3 ), where (˜ un ) is defined by (2.11). Proof. Let ζ ∈ S ∩ ∂D2 and let (εn ) ⊂ (0, 1) and (ζn ) ⊂ D2 be given according to Lemma 2.7. Recall that εn → 0 and dist(ζn , ∂D2 ) → 0. Notice also that 0 ∈ D1/εn (−ζn /εn ) and that dist(0, ∂D1/εn (−ζn /εn )) = ε1n dist(ζn , ∂D2 ). We distinguish two cases: either

or, for a subsequence,

1 dist(ζn , ∂D2 ) → +∞ εn

(2.16)

1 dist(ζn , ∂D2 ) → d < +∞ . εn

(2.17)

If (2.16) holds then, in the limit, the sequence of discs D1/εn (−ζn /εn ) fills the whole plane R2 and in fact we are in the same situation of Lemma 2.8. Hence in this case the statement follows in the same way. The rest of the proof consists in showing that this is the only possible case, namely that (2.17) cannot occur. We argue by contradiction, assuming that (2.17) holds. Notice that in this case the sequence of discs D1/εn (−ζn /εn ) converges to a half-plane Ω with 0 ∈ Ω and dist(0, ∂Ω) = d. Taking into account of (2.12), let u be a weak limit of (˜ un ). We may also assume that u ˜n → u almost everywhere. Observe 1 3 2 that u ∈ H0 (Ω, R ). Indeed, for every z ∈ R \ Ω and for every n ∈ N one has that u ˜n (z) = γ n ((εn z + ζn )/|εn z + ζn |2 ). Since γ n → 0 uniformly on D2 , we infer that u = 0 on R2 \Ω. Arguing as in the proof of Lemma 2.8 we also obtain that for every compact set K ⊂ Ω we have u ˜n → u in C 1 (K). This property and (2.13) imply that u is a classical solution of the H-system in Ω. But since a half-plane is conformally equivalent to D2 and the H-system is invariant under conformal transformations, we find that u is conformally equivalent to a solution of the Dirichlet problem (2.4) in D2 with vanishing boundary condition. The Wente’s uniqueness result [33] already mentioned applies again and yields u ≡ 0. We will get a contradiction if we will prove that actually u 6= 0. To this aim we will apply the following result of ε-regularity estimates: Lemma 2.10 Let H ∈ C 1 (R3 ). For every R > 0 there exists εR > 0 (depending on R and on kHkC 1 (BR ) ) such that, given an arbitrary sequence of smooth domains Ωn having a limit ˆ 1 satisfying: domain Ω and any sequence (un ) ⊂ H (i) ∆un = 2H(un )unx ∧ uny in Ωn for all n ∈ N, ˆ 1, (ii) un → 0 weakly in H 21

(iii) kun k∞ ≤ R for all n ∈ N, it holds that lim sup k∇un kL2 (Dr (z)∩Ωn ) ≤ εR ⇒ lim n

n→+∞

Z

Dr/2 (z)∩Ωn

|∇un |2 = 0

for every z ∈ R2 and for every r ∈ (0, 1]. Let us complete the proof of Lemma 2.9. We apply Lemma 2.10 to the sequence (˜ un ), n with Ωn = D1/εn (−ζn /εn ). Observe that the sequence (˜ u ) satisfies the same uniform L∞ n bound as the sequence (u ) and then nothing changes in the choice of ε0 . In particular the next implication holds: Z Z lim sup |∇˜ un |2 = 0 |∇˜ un |2 ≤ ε20 ⇒ lim n→+∞

n

Dr (z)∩D1/εn (−ζn /εn )

Dr/2 (z)∩D1/εn (−ζn /εn )

for every z ∈ R2 and for every r ∈ (0, 1]. Therefore, taking a finite covering of D2 by open discs of radius 1/2, and using (2.9), (2.14) and the previous implication we find that Z |∇˜ un |2 → 0 D2 ∩D1/εn (−ζn /εn )

contradicting the fact that, by (2.15), for every n we have Z Z |∇un |2 = ε20 . |∇˜ un |2 = Dεn (ζn )∩D2

D2 ∩D1/εn (−ζn /εn )

This concludes the proof. Proof of Lemma 2.10. Assume that for every n ∈ N we have k∇un kL2 (Dr (z)∩Ωn ) ≤ ε

(2.18)

for some z ∈ R2 and r ∈ (0, 1] and for a suitable ε > 0 to be fixed later. Arguing as in the first step of the proof of Lemma A.1 in [5], we claim that for every n ∈ N there exists rn ∈ (r/2, r) such that kun kH 1/2 (∂An ) → 0

and kun kL∞ (∂An ) → 0

(2.19)

where An = Drn (z) ∩ Ωn . Indeed, using (2.18) and the fact that un → 0 strongly in L2 , by Fubini’s Theorem we can find rn ∈ (r/2, r) such that Z Z 2ε2 |un |2 → 0 . |∇un |2 ≤ 0 and r ∂An ∂An Then the interpolation inequality kukH 1/2 (∂An ) ≤ kukH 1 (∂An ) kukL2 (∂An ) and the compact embedding of H 1 (∂An ) into L∞ (∂An ) imply (2.19). Now let hn ∈ H 1 (An , R3 ) be the harmonic extension of un|∂An . From (2.19) it follows that khn kH 1 (An ) → 0

and kun kL∞ (An ) → 0 . 22

(2.20)

Testing ∆un = 2H(un )unx ∧ uny with un − hn on An we get that Z Z Z H(un )(hn − un ) · unx ∧ uny . ∇un · ∇hn + 2 |∇un |2 = −

(2.21)

An

An

An

Recall the Wente inequality: Z v · ux ∧ uy ≤ Ck∇vkL2 (A) k∇uk2 2 L (A)

(2.22)

A

which holds true for every v ∈ H01 (A, R3 ) and u ∈ H 1 (A, R3 ) with a universal constant C independent of the mappings v and u and also on the domain A (see, e.g., [4]). We apply (2.22) in order to estimate Z n n n n H(u )h · ux ∧ uy ≤ Ck∇(H(un )(hn − un ))kL2 (An ) k∇un k2L2 (An ) . (2.23) An

Observe that

|∇(H(un )(hn − un ))| = |(∇H)(un ) · (∇un )(hn − un ) + H(un )∇(hn − un )| ≤

|(∇H)(un )| (|hn | + |un |) |∇un | + |H(un )| (|∇hn | + |∇un |)

≤ (Rk∇HkL∞ (BR ) + kHkL∞ (BR ) + o(1))|∇un | +kHkL∞ (BR ) |∇hn |

where here and in the rest of the proof o(1) → 0 as n → +∞. Then, keeping into account of (2.20), from (2.23) it follows that Z n n n n (2.24) H(u )h · ux ∧ uy ≤ CH,R k∇un k3L2 (An ) + k∇un k2L2 (An ) o(1) An

where CH,R is a constant depending only on R and on the C 1 norm of H on BR . Hence by (2.21), (2.24) and (2.18) we obtain (1 + o(1))k∇un k2L2 (An ) ≤ o(1) + CH,R k∇un k3L2 (An ) ≤ o(1) + CH εk∇un k2L2 (An ) .

Therefore, if ε < 1/CH,R we infer that k∇un kL2 (An ) → 0 and finally, since Dr/2 (z) ∩ Ωn ⊂ An , also k∇un kL2 (Dr/2 (z)∩Ωn ) → 0.

We point out that Lemma 2.4 is a special case of Lemma 2.10, with Ωn = D2 for every n ∈ N. Proof of Theorem 2.1. If (un ) does not converge to zero strongly in H 1 , then, by Lemma 2.6, S 6= Ø and consequently there exists an H-bubble by Lemma 2.8 or by Lemma 2.9. If un → 0 strongly in H 1 then Z mH (un )un · unx ∧ uny |EH (un )| = o(1) + 2 D2

1 ≤ o(1) + kHk∞ kun k∞ k∇un k22 = o(1) 3

thanks to (2.3). 23

3

Blowing H-bubbles with low energy

In this Section we consider H ∈ C 1 (R3 ) and we make the crucial assumption MH := sup |(∇H(p) · p)p| < 1 .

(3.1)

p∈R3

Assumption (3.1) is far to be purely technical. Actually, it was essential in the paper [10] about the existence of minimal H-bubbles. Moreover, it plays a crucial role also in the behavior of Palais Smale sequences for problem (0.1), as we show in [14]. Our goal is to prove the following result. Theorem 3.1 Let H ∈ C 1 (R3 ) with MH < 1, and let (γ n ) ⊂ H 1 ∩C 0 (D2 , R3 ) be a sequence of boundary data, with kγ n k → 0. Assume that un ∈ Hγ1n ∩ L∞ is a sequence of solutions to (2.1) with EH (un ) bounded. Then the following facts hold true: (i) un → 0 strongly in H 1 if and only if EH (un ) → 0. (ii) If un does not converge to zero strongly in H 1 then there exists an H-bubble ω with kωk∞ ≤ C(kHk∞ , MH , supn EH (un )) and EˆH (ω) ≤ lim inf EH (un ) . n

(3.2)

Moreover, there exist sequences (ζn ) ⊂ D2 , and (εn ) ⊂ (0, 1) satisfying εn → 0, (1/εn )dist(ζn , ∂D2 ) → +∞ and such that, for a subsequence, un (εn z + ζn ) → ω(z)

1 (R2 , R3 ) . in Cloc

(iii) If case (ii) occurs and, in addition, EH (un ) → EˆH (ω), then

   

n

u − ω · − ζn − ω∞ → 0

1

εn H where ω∞ = lim|z|→+∞ ω(z).

Remark 3.2 As we will see in the next section (see also [10]), the condition MH < 1 yields a positive lower bound on the energies of the H-bubbles and precisely EˆH (ω) ≥ 4π/(3kHk2∞ ) for every H-bubble ω. Hence, if the sequence (un ) in the statement of Theorem 3.1 does not converge to zero strongly in H 1 , then lim inf EH (un ) ≥ 4π/(3kHk2∞ ). Let us point out some simple consequences of the condition (3.1) which will be helpful in the following. Remark 3.3 If H ∈ C 1 (R3 ) is such that MH < +∞ then (i) H, mH ∈ L∞ (R3 ) and kmH k∞ ≤ (1/3)kHk∞ ; 24

(ii) |∇mH (p) · p p| = |(3mH (p) − H(p))p| ≤ (1/2)MH for every p ∈ R3 ; ˜ ˜ ∈ C 0 (S2 ). (iii) for every p ∈ S2 there exists limt→+∞ H(tp) =: H(p) and H In order to prove Theorem 3.1 we point out a further consequence of assumption (3.1). Lemma 3.4 Let H ∈ C 1 (R3 ) be such that MH < +∞. (i) If γ ∈ H 1 ∩ L∞ and uγ ∈ Hγ1 ∩ L∞ is a weak solution of (0.1), then 1 (1 − MH − 2kγk∞ kHk∞ )k∇uγ k22 ≤ 3EH (uγ ) . 2

(3.3)

(ii) Assume MH < 1. Let (γ n ) ∈ H 1 ∩ C 0 (D2 , R3 ) be a sequence of boundary data with sup kγ n k∞ < (1−MH )/(2kHk∞ ), and for every n ∈ N let un ∈ Hγ1n ∩L∞ be a solution to (2.1). Then k∇un k2 is bounded if and only if the energies EH (un ) are bounded. If this is the case, then the L∞ norms of the sequence (un ) are bounded by a constant depending only on kHk∞ , MH and on supn EH (un ). Proof. (i) Since uγ − γ ∈ H01 ∩ L∞ we can evaluate the partial derivative of EH at uγ in the ′ direction of uγ − γ to get EH (uγ )(uγ − γ) = 0 and hence 3EH (uγ )

′ (uγ )(uγ − γ) 3EH (uγ ) − EH Z Z 1 (3mH (uγ ) − H(uγ ))uγ · uγx ∧ uγy ∇uγ · ∇γ + 2 k∇uγ k22 + = 2 D2 D2 Z H(uγ )γ · uγx ∧ uγy . +2 (3.4)

=

D2

From Remark 3.3 (ii) it follows that Z MH (3mH (uγ ) − H(uγ ))uγ · uγx ∧ uγy ≥ − 2 k∇uγ k22 . 2 2 D Moreover 2

Z

D2

H(uγ )γ · uγx ∧ uγy ≥ −kγk∞ kHk∞ k∇uγ k22

and then (3.4) yields (3.3). (ii) The proof of the if part immediately follows from (3.3). To prove the converse assume that supn k∇un k2 < +∞. Since un ∈ C 2 (D2 ) ∩ C 0 (D2 ), applying (2.2) we also have that supn kun k∞ < +∞. Then  Z Z 1 1 |mH (un )un · unx ∧ uny | ≤ |EH (un )| ≤ D(un ) + 2 |∇un |2 + kHk∞ sup kun k∞ 2 3 2 2 n D D (3.5) thanks to Remark 3.3 (i).

25

Proof of Theorem 3.1. Let (un ) be the sequence of solutions in Theorem 3.1. By Lemma 3.4 we get that (un ) is bounded in H 1 and in L∞ . Thus, we can assume that un → u weakly in H 1 , weakly∗ in L∞ and pointwise almost everywhere, for some u ∈ H01 ∩L∞ . Then Theorem 2.1 applies and in particular u = 0. The statement (i) follows from (3.3) and (3.5). Let us prove (ii). By Theorem 2.1, the sequence (un ) blows up an H-bubble ω. More precisely, there exist sequences (εn ) ⊂ (0, 1), with εn → 0, and (ζn ) ⊂ D2 such that 1 (1/εn )dist(ζn , ∂D2 ) → +∞ and, up to a subsequence, un (εn z + ζn ) → ω(z) in Cloc (R2 , R3 ). n n For every n ∈ N define Ωn := D1/εn (−ζn /εn ) and u ˜ (z) = u (εn z + ζn ) for z ∈ Ωn . We only have to show that (3.2) holds. Given a domain Ω in R2 and a mapping u ∈ H 1 (Ω, R3 ), denote Z Z 1 mH (u)u · ux ∧ uy . |∇u|2 + 2 EH (u, Ω) = 2 Ω Ω

Fix ε > 0 and take rε > 0 be such that

EH (ω, R2 \ Drε ) ≤ ε Z |∇ω|2 ≤ ε .

(3.6) (3.7)

R2 \Drε

1 By the strong convergence in Cloc (R2 , R3 ) and by (3.6) one has

EˆH (ω) ≤

EH (ω, Drε ) + ε

= EH (˜ un , Drε ) + ε + o(1)

= EH (˜ un , Ωn ) − EH (˜ un , Ωn \ Drε ) + ε + o(1)

= EH (un ) − EH (˜ un , Ωn \ Drε ) + ε + o(1)

(3.8)

with o(1) → 0 as n → +∞. Let γ˜ n : Ωn → R3 be defined by γ˜ n (z) = γ n (εn z + ζn ). Note that γ˜ n is the harmonic extension of u ˜n|∂Ωn on Ωn and Z Z n 2 |∇γ n |2 → 0 |∇˜ γ | = Ωn n

D2

k˜ γ kL∞ (Ωn ,R3 ) = kγ n kL∞ (D2 ,R3 ) → 0.

Since every u ˜n solves the H-system in Ωn , testing with u ˜n − γ˜ n , and using the divergence theorem one has Z Z Z Z ˜n n n ∂u n n n 2 H(˜ un )(˜ un −˜ γ n )·˜ unx ∧˜ uny . =2 (˜ u −˜ γ )· ∇˜ u ·∇˜ γ − |∇˜ u | + − ∂ν Ωn \Drε ∂Drε Ωn \Drε Ωn \Drε By Remark 3.3 (ii) one has that for every n ∈ N and for every z ∈ Ωn |(H(˜ un (z)) − 3mH (˜ un (z)))˜ un (z)| ≤ MH and then one can estimate EH (˜ un , Ωn \ Drε )

=

1 6

Z

Ωn \Drε

|∇˜ un |2 −

2 3

26

Z

Ωn \Drε

(H(˜ un ) − 3mH (˜ un ))˜ un · u ˜nx ∧ u ˜ny

+ −

2 3 1 3

Z

Ωn \Drε

Z

∂D

H(˜ un )˜ γn · u ˜nx ∧ u ˜ny +

(˜ un − γ˜ n ) ·

∂u ˜n ∂ν

1 3

Z

Ωn \Drε

∇˜ un · ∇˜ γn

rε Z Z 1 1 ≥ (1 − MH ) γ n k∞ |∇˜ un |2 |∇˜ un |2 − kHk∞ k˜ 6 3 Ωn \Drε Ωn \Drε Z Z 1 ∂ u ˜n 1 . (3.9) ∇˜ un · ∇˜ γn − (˜ un − γ˜ n ) · + 3 Ωn \Drε 3 ∂Drε ∂ν

Clearly k˜ γ n k∞

Z

Ωn \Drε

|∇˜ un |2 ≤ kγ n k∞ k∇un k22 = o(1)

where o(1) → 0 as n → +∞. Moreover Z n n un k2 k∇˜ γ n k2 = k∇un k2 k∇γ n k2 = o(1). ∇˜ u · ∇˜ γ ≤ k∇˜ Ωn \Drε 1 Furthermore, since u ˜n → ω in Cloc (R2 , R3 ), one has Z Z ∂u ˜n ∂ω n n (˜ u − γ˜ ) · ω· = + o(1) ∂ν ∂ν ∂Drε ∂Drε

and Z ∂ω ω· ∂Drε ∂ν

Z
2 ω · ∆ω + |∇ω| = R2 \Drε Z
2H(ω)ω · ωx ∧ ωy + |∇ω|2 = R2 \Drε Z |∇ω|2 ≤ (kωk∞ kHk∞ + 1) R2 \Drε

≤ (kωk∞ kHk∞ + 1) ε

thanks to (3.7). Hence from (3.9) it follows that Z 1 EH (˜ un , Ωn \ Drε ) ≥ (1 − MH ) |∇˜ un |2 − Cε + o(1) 6 Ωn \Drε for some positive constant C independent of ε and n. Then, by (3.8), one obtains Z 1 (1 − MH ) |∇˜ un |2 + EˆH (ω) ≤ EH (un ) + Cε + o(1). 6 Ωn \Drε

(3.10)

Hence EˆH (ω) ≤ lim inf EH (un ) because MH < 1 and ε > 0 can be taken small as we want. 27

Finally let us prove (iii). Setting n

ω ˜ (z) = ω



z − ζn εn



,

R observe that the sequence (un − ω ˜ n ) is bounded in L∞ and in H 1 because D2 |∇˜ ω n |2 = R 2 n n 1 |∇ω| . Up to a subsequence, u − ω ˜ converges weakly in H and pointwise almost Ωn everywhere to some function u ¯ ∈ H 1 ∩ L∞ . Let us prove that k∇(un − ω ˜ n )k2 → 0 .

(3.11)

Fixing ε > 0, let rε > 0 as in the part (ii). If EH (un ) → EˆH (ω), from (3.10) we infer that Z |∇˜ un |2 ≤ Cε + o(1) (3.12) Ωn \Drε

where the constant C depends only on H. We have that Z Z |∇(˜ un − ω)|2 |∇(un − ω n )|2 = 2 Ω D Z Z n |∇(˜ un − ω)|2 + 2 ≤

Ωn \Drε

Dr ε

≤ o(1) + Cε

|∇˜ un |2 + 2

Z

Ωn \Drε

|∇ω|2

thanks to (3.7), (3.12) and because u ˜n → ω in C 1 (Drε ). Hence (3.11) holds. Consequently, u ¯ is constant. According to Lemma A.2 in [9] ω n → ω∞ := lim|z|→+∞ ω(z) in H 1/2 (∂D2 , R3 ). We also know that un|∂D2 = γ n → 0 and then we infer that u ¯ = ω∞ and un − ω ˜ n → ω∞ strongly in H 1 . This concludes the proof.

4

Blowing minimal H-bubbles

It is known that for H ∈ L∞ (R3 ) and γ ∈ H 1 ∩ L∞ with kγk small enough, problem (0.1) admits a weak solution uγ ∈ H 1 characterized as a local minimum point for the energy EH in a suitable subset of Hγ1 (see, e.g., the results by Hildebrandt [19] and Steffen [25]). The solution uγ is often called “small” solution. In view of a multiplicity result for the Dirichlet problem (0.1), a good candidate to be a critical level for the energy functional EH in Hγ1 is the mountain pass level, whenever it can be defined. In the first part of this Section we investigate the mountain pass geometry for the energy functional EH in Hγ1 for a given bounded and regular curvature H: R3 → R and for γ ∈ H 1 ∩ L∞ small enough. Clearly the existence of a solution at the mountain pass level (namely a so called “large” solution) strongly depends on suitable compactness properties for the energy EH in Hγ1 and precisely on the validity of the Palais Smale (shortly PS) condition at the mountain pass 28

level. Here we do not study the PS condition for EH in Hγ1 but, in the second part of this section, we discuss some consequences of the possible existence of “large” solutions for the Dirichlet problem (0.1) with γ small. The main result of this section is Theorem 4.7. As a first result, we show that under rather weak assumptions on H (ensuring that the energy EH becomes negative somewhere) a mountain pass geometry always exists as soon as kγk is small enough. Moreover some uniform estimates on the mountain pass levels hold. Theorem 4.1 Let H ∈ L∞ (R3 ) ∩ C 0 (R3 ). If there exists v ∈ H01 such that EH (v) < 0, then for every γ ∈ H 1 ∩ L∞ with kγk small enough EH admits a mountain pass geometry in Hγ1 at level cH,γ := inf max EH (g(s) + γ) (4.1) g∈G s∈[0,1]

where G = {g ∈ C 0 ([0, 1], H01 ) | g(0) = 0, EH (g(1)) < −δ0 } for some δ0 > 0. In particular 4π + o(1) ≤ cH,γ ≤ cH,0 + o(1) 3kHk2∞

(4.2)

where o(1) → 0 as kγk → 0. Remark 4.2 If there exists v ∈ H01 such that EH (v) < 0, then inf v∈H01 EH (v) = −∞. Indeed, defining v n (z) = v(z n ) (in complex notation) one can see that for every n ∈ N one has v n ∈ H01 and EH (v n ) = nEH (v). Hence there is no restriction on the value of δ0 in the statement of Theorem 4.1. Remark 4.3 Under the assumptions of Theorem 4.1, if for kγk small enough the functional EH satisfies the PS condition in Hγ1 at the level cH,γ , then there exists a weak solution uγ to (0.1) with EH (uγ ) = cH,γ . This occurs in case H is a nonzero constant and γ is nonconstant (and small enough), as proved by Struwe in [29]. In order to prove Theorem 4.1, we start with the following preliminary result. Lemma 4.4 Let H ∈ L∞ (R3 ) ∩ C 0 (R3 ). Then lim EH (γ + v) = EH (v)

kγk→0

uniformly on BR := {v ∈ H01 ∩ L∞ | k∇vk2 + kvk∞ ≤ R } for every R > 0. Proof. First we notice that, since γ is harmonic, D(γ + v) = D(v) + D(γ) = D(v) + o(1) uniformly in v ∈ H01 . Fixing R > 0 and taking v ∈ BR , we have that VH (γ + v) − VH (γ) = J1γ (v) + J2γ (v) + J3γ (v) ,

29

where J1γ (v)

=

J2γ (v)

=

J3γ (v)

=

Z

ZD

ZD

2

mH (γ + v)γ · ( (γ + v)x ∧ (γ + v)y )

2

mH (γ + v)v · (γx ∧ γy + γx ∧ vy + vx ∧ γy )

D2

(mH (γ + v) − mH (v))v · vx ∧ vy .

Since k∇γk2 → 0 and kγk∞ → 0, we have that |J1γ (v)| ≤ kmH k∞ k∇(γ + v)k22 kγk∞ ≤ C(kHk∞ , R)kγk∞ = o(1) uniformly for v ∈ BR . Similarly |J2γ (v)| ≤ kmH k∞ kvk∞ (k∇γk2 + k∇vk2 )k∇γk2 ≤ C(kHk∞ , R)k∇γk2 = o(1) . For the last integral let us denote LR the Lipschitz constant of mH on B2R . Then for kγk small enough |mH (γ + v) − mH (v)| ≤ LR kvk∞ and therefore |J3γ (v)| ≤ LR kvk2∞ k∇vk22 kγk∞ = o(1) uniformly for v ∈ BR . The proof is complete. Remark 4.5 If H is constant the conclusion in Lemma 4.4 holds true uniformly for v in bounded sets of H01 . Indeed in case H constant J3γ ≡ 0 and we can estimate via Wente’s
inequalities [32], |VH (γ + v) − VH (v)| ≤ C kγk∞ k∇γk2 + k∇γk22 + k∇vk22 k∇γk2 with a constant C which depends only on H. Proof of Theorem 4.1. Using the fact that γ is harmonic and the isoperimetric inequality (1.17), for every v ∈ H01 we have that EH (v + γ) ≥ ≥ ≥

EH (γ) + D(v + γ) − D(γ) − 2 |VH (v + γ) − VH (γ)| kHk∞ 3/2 EH (γ) + D(v) − √ (D(v + γ) − D(γ)) 3 π kHk∞ 3/2 EH (γ) + D(v) − √ (D(v) + 2D(γ)) . 3 π

Therefore, by elementary estimates, we find that: EH (v + γ) ≥ EH (γ) −

kHk∞ √ k∇γk32 3 π

as

8π 2 − 2k∇γk22 kHk2∞

(4.3)

8π 2 − 2k∇γk22 . kHk2∞

(4.4)

k∇vk22 ≤

and EH (v + γ) ≥ EH (γ) − k∇γk22 +

4π 3kHk2∞ 30

as

k∇vk22 =

Let δ0 > 0 be such that {v ∈ H01 | EH (v) < −δ0 } 6= Ø. Hence the class G is nonempty. Moreover, by (4.3), the sets {v ∈ H01 | EH (v + γ) ≤ −δ0 /2} and {v ∈ H01 | k∇vk22 ≤ (8π 2 /kHk2∞ ) − 2k∇γk22 } are nonempty and separated. Consequently for kγk small enough EH admits a mountain pass geometry in Hγ1 at the level cH,γ defined in (4.1) and, by (4.4), cH,γ ≥

4π + o(1) 3kHk2∞

where o(1) → 0 as kγk → 0. Finally we check the upper estimate on the mountain pass levels cH,γ . Fixing ε > 0 let g ∈ G be such that maxs∈[0,1] EH (g(s)) ≤ cH,0 + ε. We claim that there exists g˜ ∈ G such that maxs∈[0,1] EH (˜ g (s)) ≤ cH,0 + 2ε and k∇˜ g (s)k2 + k˜ g (s)k∞ ≤ R for every s ∈ [0, 1] and for some R < +∞. If this claim is true, then one obtains that cH,γ

≤

s∈[0,1]

max EH (˜ g (s) + γ)

≤

s∈[0,1]

max EH (˜ g (s)) + max (EH (˜ g (s) + γ) − EH (˜ g (s))) s∈[0,1]

≤ cH,0 + 2ε + sup |EH (v + γ) − EH (v)| v∈BR

and the conclusion immediately follows from Lemma 4.4 and from the arbitrariness of ε. Hence it remains to prove the previous claim. First observe that by the uniform continuity of EH on the compact set range g there exists δ > 0 such that |EH (v) − EH (g(s))| < ε

for any v ∈ H01 with kv − g(s)kH01 < δ and for any s ∈ [0, 1] . (4.5)

Now take a partition 0 = s0 < s1 < . . . < sk = 1 such that kg(si ) − g(si−1 )kH01 < δ/2 for all i = 1, . . . , k. Since Cc∞ (D2 , R3 ) is dense in H01 , for every i = 0, . . . , k there exists vi ∈ Cc∞ (D2 , R3 ) such that kvi − g(si )kH01 < δ/2. In particular we can choose v0 = 0 and vk satisfying the additional condition |EH (vk ) − EH (g(1))|
0 such that k∇˜ g (s)k2 + k˜ g (s)k∞ ≤ R for every s ∈ [0, 1]. Finally, let s¯ ∈ [0, 1] be such that EH (˜ g (¯ s)) = max EH (˜ g (s)) . s∈[0,1]

31

(4.7)

There exists α ∈ (0, 1] and i ∈ {1, . . . , k} such that s¯ = (1 − α)si−1 + αsi . Then k˜ g (¯ s) − g(si )kH01 ≤ k˜ g (¯ s) − vi kH01 + kvi − g(si )kH01 < δ . Therefore, using (4.5), we obtain EH (˜ g (¯ s)) ≤ EH (g(si )) + ε ≤ max EH (g(s)) + ε ≤ cH,0 + 2ε s∈[0,1]

that, together with (4.7) completes the proof of the initial claim. In the rest of this section we focus on a class of curvatures H ∈ C 1 (R3 ) satisfying MH < 1, where MH is defined in (3.1). Another important parameter for our discussion is given by the value H∞ := lim sup |H(p)| |p|→+∞

which is finite under the previous condition on MH (see Remark 3.3). The next result states that if MH < 1 and H∞ > 0, then for γ small the energy EH again has a mountain pass geometry in Hγ1 with suitable uniform estimates on the corresponding mountain pass levels, in terms of kHk∞ and H∞ . Theorem 4.6 If H ∈ C 1 (R3 ) is such that MH < 1 and H∞ > 0, then the assumptions of Theorem 4.1 are satisfied and thus for every γ ∈ H 1 ∩ L∞ with kγk small enough EH admits a mountain pass geometry in Hγ1 at the level cH,γ defined in (4.1). Moreover 4π 4π + o(1) ≤ cH,γ ≤ + o(1) 2 2 3kHk∞ 3H∞

(4.8)

where o(1) → 0 as kγk → 0. Proof. Since MH is finite and H∞ > 0, by Remark 3.3 for every ε > 0 there exist s0 > 0 and a nonempty open set Σ in S2 such that |H(sp)| ≥ H∞ − ε > 0

for s ≥ s0 and p ∈ Σ .

(4.9)

This is enough to infer that there exists v ∈ H01 ∩ L∞ such that EH (sv) → −∞ as s → +∞ (see Lemma 2.5 in [10]). Hence, observing that H ∈ L∞ (R3 ) (see again Remark 3.3), the assumptions of Theorem 4.1 are fulfilled and the mountain pass levels cH,γ given by (4.1) are well defined and satisfy the estimate (4.2). In order to conclude the proof we shall show that 4π . (4.10) cH,0 ≤ 2 3H∞ To this purpose, let us introduce the value cˆH := inf 1 sup EH (sv) . v∈H0 s>0 v6=0

32

In [10] Lemma 2.9 we proved that cˆH ≥ cH,0 .

(4.11)

Moreover, using (4.9), as proved in [10], Lemma 2.12, we also have cˆH ≤

4π . 3(H∞ − ε)2

(4.12)

Thanks to the arbitrariness of ε > 0, (4.11) and (4.12) imply (4.10). We remark that the condition MH < 1 can be used to check that for kγk small enough every PS sequence for EH in Hγ1 is bounded in H 1 (an estimate similar to (3.3) with an additional negligible term holds true along a PS sequence). It is also known (see [3]) that the weak limit of a PS sequence for EH in Hγ1 is a weak solution of the Dirichlet problem (0.1). However, since in general the PS condition fails, one cannot exclude that the weak limit of a PS sequence at the mountain pass level is the “small” solution. The next result states that, under the assumptions of Theorem 4.1, if cH,γ is also a critical level for EH in Hγ1 and this holds for (a sequence of) boundary data γ with kγk → 0, then there exists a minimal H-bubble, with some upper and lower estimates on its energy. Theorem 4.7 Let H ∈ C 1 (R3 ) be such that MH < 1 and H∞ > 0 and assume that there exist a sequence (γ n ) ⊂ H 1 ∩ C 0 (D2 , R3 ) and a corresponding sequence (un ) ⊂ H 1 ∩ L∞ where for every n ∈ N the function un is a critical point of EH at level cH,γ n . Then there ˆ 1 ∩ L∞ with kωk∞ ≤ C(kHk∞ , MH , H∞ ) and exists a minimal H-bubble ω ∈ H 4π 4π ≤ EˆH (ω) ≤ . 2 3kHk2∞ 3H∞ Proof. Because of the uniform positive lower bound on EH (un ) = cH,γ n given by (4.8), the sequence un cannot converge to zero strongly in H 1 . Then, by Theorem 3.1, there exists 2 ˆ 1 ∩ L∞ and EˆH (ω) ≤ lim inf EH (un ) ≤ 4π/(3H∞ an H-bubble ω ∈ H ), thanks to the upper ˆ estimate in (4.8). In fact, by (4.2) and (4.11), we obtain that EH (ω) ≤ cH,0 ≤ cˆH . According to what proved in [10], Proposition 2.8, we get EˆH (ω) = cˆH = inf{EˆH (U ) | U H−bubble}

(4.13)

namely ω is a minimal H-bubble. It remains to prove that EˆH (ω) ≥ 4π/(3kHk2∞ ). Arguing as in the proof of Proposition 2.8 in [10], for every ε > 0 it is possible to find a map v ε ∈ H01 ∩ L∞ such that EH (v ε ) < 0 and max EH (sv ε ) ≤ EˆH (ω) + ε .

s∈[0,1]

By Lemma 4.4 we have EH (γ + sv ε ) → EH (sv ε )

33

uniformly in s ∈ [0, 1] as kγk → 0, and in particular EH (γ +v ε ) < 0 for kγk small. Therefore, the path s 7→ γ + sv ε can be used to estimate from above the level cH,γ , and using again Lemma 4.4 we get cH,γ ≤ max EH (γ + sv ε ) = o(1) + max EH (sv ε ) ≤ EˆH (ω) + ε s∈[0,1]

s∈[0,1]

where o(1) → 0 as kγk → 0. Using the lower estimate given in (4.8) and the arbitrariness of ε, the conclusion plainly follows. Remark 4.8 By (4.13), (4.11) and (4.2), if MH < 1 then EˆH (ω) ≥ (4π)/(3kHk2∞ ) for every H-bubble ω. We conclude this section with some remarks concerning the case of curvature functions H ∈ C 1 (R3 ) satisfying MH < 1 and H∞ = 0, namely H(p) → 0 as |p| → +∞. Under these conditions we can prove that EH is coercive in H01 and, more precisely, for every v ∈ H01 .

EH (v) ≥ (1 − MH )D(v)

(4.14)

Consequently, in this case, in general one cannot expect that EH admits a mountain pass geometry. In fact (4.14) together with Theorem 4.6 show that when MH < 1 a necessary and sufficient condition in order that EH admits a mountain pass geometry in H01 is that H∞ > 0. The proof of (4.14) is based on the following trivial estimate: Z 1 Z +∞ Z 1  2 2 ∇H(tp) · p p dt s ds H(sp)s p ds = |mH (p)p| = 0

0

≤ MH

Z

0

1

Z

+∞

s

dt t2



s

MH s2 ds = . 2

Therefore 2|VH (v)| ≤ MH D(v) for all v ∈ H01 ∩ L∞ . Then, by density and by the continuity of VH in H01 , (4.14) follows.

5

Nonexistence results

In this Section we collect some nonexistence results of H-bubbles and of large solutions to the Dirichlet problem (0.1) with small boundary data. We shall always consider curvature functions H ∈ C 1 (R3 ) satisfying MH < 1. We start with some simple facts concerning the case of curvatures vanishing at infinity. Theorem 5.1 Let H ∈ C 1 (R3 ) be such that MH < 1 and lim|p|→+∞ H(p) = 0. Then: (i) there is no H-bubble.

34

(ii) For every α, β > 0 there exists ε > 0 such that for every γ ∈ H 1 ∩ C 0 (D2 , R3 ) the Dirichlet problem (0.1) with kγk < ε admits no solution u ∈ Hγ1 ∩ L∞ with energy EH (u) ∈ [α, β]. Proof. First let us prove that |H(p)p| ≤ MH for all p ∈ R3 . Indeed, fixing p ∈ R3 \ {0}, since H(tp) → 0 as t → +∞ we have that Z +∞ Z +∞ |∇H(sp) · sp sp| |H(p)p| = ds ≤ MH ∇H(sp) · p p ds ≤ s2 1 1

ˆ 1 be a weak solution to (0.3). thanks to the definition (3.1) of MH . Now, let ω ∈ L∞ ∩ H Testing the equation in (0.3) with ω and using the previous estimate, we get Z Z Z 2 |∇ω| = 2 H(ω)ω · ωx ∧ ωy ≤ MH |∇ω|2 . R2

R2

R2

Therefore, ω is constant since MH < 1. Finally, part (ii) can be proved by contradiction using part (i) and Theorem 3.1. Now we consider curvatures H ∈ C 1 (R3 ) with MH < 1 and not vanishing at infinity. In particular, we investigate the case in which kHk∞ = H∞ := lim sup|p|→∞ |H(p)| > 0. We recall that in this case, by Theorem 4.6 for kγk small enough, the functional EH has a mountain pass geometry on Hγ1 and the corresponding mountain pass level cH,γ satisfies: cH,γ →

4π 3kHk2∞

askγk → 0 .

We look for conditions on H that guarantee the nonexistence of H-bubbles at the energy level 4π/(3kHk2∞ ). Therefore, according to Theorem 4.7, for kγk small the Dirichlet problem (0.1) will have no mountain pass type solution. We start with a preliminary result. For simplicity and without restrictions we limit ourselves to maps H with kHk∞ ≤ 1. ˆ 1 ∩ L∞ is an Lemma 5.2 Let H ∈ C 1 (R3 ) with MH < 1 and H ≤ 1 on R3 . If ω ∈ H H-bubble with iω ≤ 0 on R3 (iω defined in (1.20)) and EˆH (ω) ≤ 4π/3, then ω parametrizes a sphere of radius 1, and H ≡ 1 on a ball B1 (p) in R3 . Proof. By Remark 4.8, EˆH (ω) ≥ 4π/3, and therefore EˆH (ω) = 4π/3. Notice also that R Vˆ1 (ω) = R3 iω < 0, since iω ≤ 0, and iω < 0 on a set of positive measure. As already noticed in [10], the assumption MH < 1 implies that the map f : (0, +∞) → R, f (s) := EˆH (sω) attains it maximum at s = 1. This can be easily obtained by differentiating f twice (use ˆ 1 ∩ L∞ to check derivability). First, f ′ (1) = 0 since ω is an H-bubble. Secondly, also ω ∈ H ′ if f (s) = 0 at some point s > 0 then Z Z Z 2 ′′ |∇ω|2 < 0 . (∇H(sω) · sω)sω · ωx ∧ ωy ≤ (MH − 1) |∇ω| + 2 f (s) = − R2

R2

R2

35

Therefore, f do not have any local minima, that is, s = 1 is its global (and unique) maximum point. Setting ˆ D(ω) s˜ = − 3Vˆ1 (ω) we have that s˜ > 0 and

max Eˆ1 (sω) = Eˆ1 (˜ sω) = s>0

ˆ D(ω) ˆ 3|V1 (ω)|2/3

!3

≥

4π 3

by the isoperimetric inequality (1.2). Moreover 4π = EˆH (ω) = max EˆH (sω) ≥ EˆH (˜ sω) . s>0 3 Therefore, using (1.20) and the decomposition H = 1 + (H − 1), we obtain Z 4π 4π 4π 3 ˆ ˆ ˆ ≥ EH (˜ sω) = E1 (˜ sω) + 2VH−1 (˜ sω) ≥ + 2˜ s . (H(˜ sp) − 1)iω (p) dp ≥ 3 3 3 3 R ˆ ˆ In particular, we get that s˜ = 1 and consequently D(ω) = −3Vˆ1 (ω), D(ω)/| Vˆ1 (ω)|2/3 = √ 3 36π, and H ≡ 1 on the support of iω . Hence ω is a sphere of radius 1, by Lemma 1.1. Now we are in position to prove the following nonexistence result: Theorem 5.3 There exist M ∈ (0, 1) and δ > 0 such that for every H ∈ C 1 (R3 ) satisfying MH < M 0 < 1 − δ ≤ H(p) ≤ 1 inf

q∈B1 (p)

H(q) < 1

for all p ∈ R3 ,

for all p ∈ R3

ˆ 1 ∩ L∞ with energy EˆH (ω) ≤ 4π/3. there exists no H-bubble ω ∈ H Proof. By Remark 4.8, EˆH (ω) ≥ 4π/3 for every H-bubble ω. Thus we have to show that for δ, MH small enough and 1 − δ ≤ H ≤ 1 , H-bubbles with minimal energy 4π/3 do not exist, unless H is identically equal to 1 on a ball B1 (p) (otherwise, any conformal parametrization of ∂B1 (p) would be an H-bubble). We argue by contradiction assuming that there exists a sequence of functions Hn ∈ C 1 (R3 ) satisfying MHn → 0

0 < 1 − δn ≤ Hn (p) ≤ 1 inf

q∈B1 (p)

for all p ∈ R3

for all p ∈ R3 , n ≥ 1 ,

Hn (q) < 1

with δn → 0, and a corresponding sequence of Hn -bubbles ω n with energies EˆHn (ω n ) = 4π/3 for every n ∈ N. By a consequence of the a priori L∞ estimate proved by Gr¨ uter [16] (see the proof of Theorem 1.1 in [10]), one has that diam ω n ≤ ρ

for every n ∈ N 36

(5.1)

with a constant ρ > 0 that depends on supn MHn < 1 but is independent of n. Now we show that the sequence (ω n ) is a minimizing sequence for the Bononcini-Wente isoperimetric ˆ n ) → 4π. Indeed, since Eˆ′ (ω n )ω n = 0 we obtain inequality (1.2). Let us first show that D(ω Hn Z ˆ n) + 2 4π = 3EˆHn (ω n ) = D(ω (3mHn (ω n ) − Hn (ω n ))ω n · ωxn ∧ ωyn . R2

We can estimate the last integral by means of MHn (see Remark 3.3) getting Z n n n n n ˆ n ). 2 (3mHn (ω ) − Hn (ω ))ω · ωx ∧ ωy ≤ MHn D(ω R2

Hence

4π 4π ˆ n) ≤ ≤ D(ω . 1 + MHn 1 − MHn

ˆ n ) → 4π as n → +∞. Now we Since MHn → 0, passing to the limit, we obtain that D(ω write the curvature Hn as 1 + (Hn − 1) and we estimate the Hn − 1 volume with (1.21), to get kHn − 1k∞ ˆ n 3/2 δn ˆ n 3/2 4π ) = Eˆ1 (ω n ) + o(1) . = EˆHn (ω n ) ≥ Eˆ1 (ω n ) − 2 √ D(ω ) ≥ Eˆ1 (ω n ) − D(ω 3 3 36π We deduce also that ˆ n ) ≤ − 8π + o(1) , 2Vˆ1 (ω n ) = Eˆ1 (ω n ) − D(ω 3 and the conclusion readily follows, since √ 3

36π ≤

ˆ n) D(ω

|Vˆ1

(ω n )|2/3

≤

4π + o(1) 2/3

(4π/3 + o(1))

=

√ 3

36π + o(1) .

Thus we are in position to apply Lemma 1.1. Thanks to the invariance of H-systems with ˆ 1 , where ω is respect to the conformal group, we may assume that ω ˜ n := ω n − pn → ω in H R n a conformal parametrization of a sphere, and pn = S 2 ω . Notice that the sequence (˜ ω n ) is ∞ n ˜ n -bubble, where H ˜ n (p) = H(p + pn ), bounded in L , because of (5.1). Moreover, ω ˜ is a H 2 2 3 and it belongs to C (S , R ), thanks to regularity theory for H-systems. Since δn → 0, ˜ n → 1 uniformly on R3 and, taking advantage of the ε-regularity estimates (Lemma 2.3) H and arguing as in the proof of Lemma 2.5, we can infer that ω ˜ n → ω in C 1 (S2 , R3 ) (see also Lemma 2.5 in [13]). In particular this implies that for n large enough ω ˜ n parametrizes 3 2 a compact surface diffeomorphic to S , and iωn ≤ 0 on R . Therefore, by Lemma 5.2, for n large ω n parametrizes a sphere of radius 1, and Hn ≡ 1 on some ball of radius 1, a contradiction.

37

Theorem 5.4 For every R > 0 there exists δ > 0 such that for every H ∈ C 1 (R3 ) satisfying MH < 1 0 < 1 − δ ≤ H(p) ≤ 1 inf

q∈B1 (p)

for all p ∈ R3 ,

for all p ∈ R3

H(q) < 1

ˆ 1 ∩ L∞ with energy EˆH (ω) ≤ 4π/3 and such that kωk∞ ≤ R. there exists no H-bubble ω ∈ H Proof. The proof is similar to that of Theorem 5.3. We argue again by contradiction assuming that there exist R > 0 and a sequence of functions Hn ∈ C 1 (R3 ) satisfying MHn < 1 0 < 1 − δn ≤ Hn (p) ≤ 1

for all p ∈ R3

for all p ∈ R3 , n ≥ 1 ,

inf Hn (p) < 1

B1 (p)

ˆ 1 ∩ L∞ with kω n k∞ ≤ R with δn → 0, and a corresponding sequence of Hn -bubbles ω n ∈ H ˆ n ) → 4π we notice that and EˆHn (ω n ) = 4π/3 for every n ∈ N. In order to show that D(ω Z 1 |3mHn (p) − Hn (p)| ≤ 3 |Hn (sp) − Hn (p)|s2 ds 0

≤ 3 Hence

Z

0

1

|Hn (sp) − 1|s2 ds + 3

Z n n n n n 2 (3mHn (ω ) − Hn (ω ))ω · ωx ∧ ωy R

Thus we can conclude, since as before

ˆ n) + 2 4π = 3EˆHn (ω n ) = D(ω

Z

R2

Z

1

0

|1 − Hn (p)|s2 ds ≤ 2δn .

ˆ n) . ≤ 2δn RD(ω

(3mHn (ω n ) − Hn (ω n ))ω n · ωxn ∧ ωyn .

The proof can be completed as in Theorem 5.3. As a corollary of Theorems 4.7 and 5.4 we get the following nonexistence result of mountain pass solutions for the Dirichlet problem (0.1) with small boundary data. Theorem 5.5 For every M ∈ (0, 1) there exist δ > 0 such that for every H ∈ C 1 (R3 ) satisfying MH < M 0 < 1 − δ ≤ H(p) ≤ 1

for all p ∈ R3

for all p ∈ R3

inf H(p) < 1

B1 (p)

lim sup H(p) = 1 , |p|→+∞

there exists ε > 0 such that for every boundary datum γ ∈ H 1 ∩ C 0 (D2 , R3 ) with kγk < ε, problem (0.1) admits no solution uγ ∈ Hγ1 ∩ L∞ with energy EH (uγ ) = cH,γ . 38

Proof. The assumption lim sup|p|→+∞ H(p) = 1 = kHk∞ implies that cH,γ → 4π/3 as kγk → 0, by Theorem 4.7. Next, let us observe that Theorem 4.7 provides an a priori L∞ -bound C(kHk∞ , MH , H∞ ) for minimal H-bubbles that possibly blow up as a limit of a sequence of mountain pass solutions, as the boundary data go to 0. Now, fix M ∈ (0, 1), set R = C(1, M, 1) as in Theorem 4.7 and take δ > 0 according to Theorem 5.4. With this choice of δ, fix any curvature H as in the statement of the theorem, and argue by contradiction, using Theorems 4.7 and 5.4. 2

For example, one can take H(p) = 1 − λe−|p| with λ > 0 small, or also H(p) = 1 + λ(1 + tanh(p · ξ)), with ξ ∈ S2 fixed, and |λ| small, λ 6= 0. Considering for instance the Dirichlet problem  2 ∆u = 2(1 − λe−|u| ) ux ∧ uy in D2 (5.2) u=γ on ∂D2 one knows, by the results proved in [21] and [5], that if kγk is small enough, then there exists λ0 > 0, depending on γ, such that for λ ∈ [−λ0 , λ0 ] problem (5.2) has a small solution uγ , which is a local minimum for EH , and a large solution uγ at the mountain pass level cH,γ . As a consequence of Theorem 5.5 we get that λ0 → 0 as γ shrinks to 0.

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