On Palais-Smale sequences for H-systems. examples [Part 1]

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On Palais-Smale sequences for H-systems Part one: examples∗ 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]

1

Introduction

In this paper we are concerned with the Dirichlet problem  ∆u = 2H(u)ux ∧ uy in D2 u=0 on ∂D2

(1.1)

where H ∈ C 1 ∩ L∞ (R3 ) is a given map and D2 is the two-dimensional open unit disc. The equation appearing in (1.1) is the prescribed mean curvature equation and it is known as H-system. As proved by Wente in [16], problem (1.1) admits only the trivial solution u ≡ 0. The purpose of this work is to throw some light on Palais-Smale sequences for (1.1). Problem (1.1) is variational in nature and its energy EH : H01 (D2 , R3 ) =: H01 → R is defined by: Z 1 |∇u|2 + 2VH (u) , EH (u) = 2 D2 where VH (u) is the so called H-volume functional (see Section 2 and [15]) which, for u regular enough, can be defined as the H-weighted algebraic volume enclosed by the surface parametrized by u. ∗ Work

supported by Ministero dell’Istruzione, dell’Universit` a e della Ricerca, Progetto di Ricerca “Metodi Variazionali ed Equazioni Differenziali Nonlineari”

1

One of the main features in the study of a variational problem is the compactness property and, more precisely, the behavior of PS sequences. As regards problem (1.1), a PS sequence for EH in H01 is a sequence (un ) ⊂ H01 such that −∆un + 2H(un )unx ∧ uny → 0

in H −1

supn∈N |EH (un )| < +∞ .

The invariance of problem (1.1) with respect to the conformal group of the disc is one of the responsibles of possible lack of compactness phenomena. In many noncompact variational problems the lack of compactness is well understood and a complete description of the behaviour of PS sequences is available. This holds also for problem (1.1) when H is constant, as studied in 1985 by Brezis-Coron [4], who proved the following result. Theorem 1.1 [H. Brezis-J.M. Coron] Let H be a nonzero constant, and let (un ) ⊂ H01 be a PS sequence for EH . Then (un ) is bounded in H01 and either un → 0 strongly in H01 or there exist a subsequence of (un ), still denoted (un ), (i) a finite number of nonconstant solutions U 1 , . . . , U r of  ∆U = 2HUx ∧ Uy on R2 U (z) → 0 as |z| → +∞ ,

(1.2)

(ii) corresponding sequences (ε1n ), . . . , (εrn ) in (0, +∞) and (a1n ), . . . , (arn ) in D2 with 1 dist(ain , ∂D2 ) → +∞ εin such that

  r

n X i · − ain U

u −

εin i=1

as n → +∞ (i = 1, . . . , r)

H 1 (D2 ,R3 )

−→ 0

as n → +∞ .

(1.3)

Notice that (1.3) describes a typical lack of compactness by concentration. For next convenience, let us introduce some terminology. We say that, given a function K: R3 → R, the problem  ∆U = 2K(U )Ux ∧ Uy on R2 (1.4) U (z) → 0 as |z| → +∞ defines a problem at infinity for (1.1) if there exists a PS sequence (un ) for EH in H01 and corresponding sequences (an ) ⊂ D2 , (εn ) ⊂ (0, +∞), with ε1n dist(an , ∂D2 ) → 0, such that 1 the rescaled sequence u ˆn (z) = un (εn z + an ) converges in Hloc (R2 , R3 ) to some nonconstant solution U of (1.4) with finite Dirichlet integral. We also say that the PS sequence (un )

2

blows U . Moreover we call K-bubble any nonconstant solution of ∆U = 2K(U )Ux ∧ Uy on R2 with finite Dirichlet integral. Hence, according to the above definitions, Theorem 1.1 states that, for H nonzero constant, problem (1.2) is the only problem at infinity for (1.1) and any PS sequence either converges to zero strongly in H01 or it admits a limiting configuration made by finitely many H-bubbles passing through the origin. A natural question is to understand if a similar result holds true also in case H nonconstant. A first attempt in this direction has been done by Bethuel and Rey in 1994. In [3], Theorem 5.6, the authors claimed that the picture described by Theorem 1.1 can be drawn in a similar way also for a class of variable curvature functions H ∈ C 1 (R3 ) ∩ L∞ . More precisely, according to Bethuel and Rey, any bounded PS sequence blows finitely many nonconstant solutions to  ∆U = 2H(U )Ux ∧ Uy on R2 (1.5) U (z) → 0 as |z| → +∞ . In particular problem (1.5) would be the only problem at infinity for (1.1). As already observed in [3], [5], [6], [9], [13], H-systems with variable H exhibit relevant qualitative changes with respect to the case H constant. From the technical viewpoint, one observes that the integral representation of the H-volume functional VH (u) involves a term which, in general, is not a determinant homogeneous in u as in case H constant. As a consequence, several nice properties fail and one needs a more sophisticated machinery. As far as concerns the behavior of PS sequences for problem (1.1), we will see that the previous statement by Bethuel and Rey is incorrect: for H nonconstant, PS sequences may have quite a wild behaviour and there might exist lots of hugely different problems at infinity. Firstly, we can show that if there exists an H-bubble (anywhere placed in R3 ), then one can construct a PS sequence which converges to zero weakly in H01 and has a unique blow up on U . Notice that in this case a problem at infinity for (1.1) is given by (1.4) with K = H(· + p) and p ∈ range U . Actually the H-bubble problem (1.5) defines a problem at infinity for (1.1) if and only if there exists an H-bubble passing through 0 and this fact is far to be always true (see some nonexistence examples of H-bubbles in [9]). Of course one can plainly construct examples in which the lack of compactness occurs on finitely many nonconstant solutions U 1 , . . . , U r of (1.4) with K = H(· + p1 ), . . . , K = H(· + pr ), respectively, for possible different p1 , . . . , pr ∈ R3 . Hence, in general, problems at infinity for (1.1) arise in correspondence of any mapping obtained by composition of H with any translation in R3 . The matter becomes much worse if there exists an H-bubble with null energy (the definition of energy of an H-bubble is given in Section 2.2) or, more generally, a sequence of H-bubbles whose corresponding sequence of energies defines a convergent series. Indeed in this case we can construct unbounded PS sequences which blow infinitely many bubbles and

3

have infinitely many concentration points. An example of this kind is exhibited in Section 3. A further different situation can occur when H(p) → H∞ ∈ R \ {0} as |p| → +∞. If this holds, a PS sequence can lose compactness also by blow up on an H∞ -bubble passing through the origin (see Section 4). According to the situations described in Sections 3 and 4 we infer the following remark: differently from the case of H constant, in general for H variable (even very close to a nonzero constant in any reasonable topology), if (un ) ⊂ H01 is a PS sequence for EH and (θn ) ⊂ H01 converges to zero strongly in H01 , then (un + θn ) is not necessarily a PS sequence for EH but — at least in our examples — it is a PS sequence for a problem at infinity, blowing a bubble passing through the origin. We point out that in our constructive examples, a PS sequence blowing infinitely many bubbles can be unbounded in H01 . Motivated by this situation, we also discuss the problem of boundedness in H01 of PS sequences (Section 6). Notice that in case H constant, PS sequences for EH in H01 are always bounded. In case of H variable, as an useful tool for our purposes, let us introduce the value MH := sup |∇H(p) · p| |p| .

(1.6)

p∈R3

We can show that the inequality MH < 1 yields a sufficient and (essentially) necessary condition in order that every PS sequence for EH in H01 is bounded. More precisely we can distinguish two situations: • If MH < 1 and H(p) → 0 as |p| → +∞ then every PS sequence converges to zero strongly in H01 . ¯ 6= 0 for every p in some nonempty open subset • If MH < 1 and limt→+∞ H(tp) = H 2 of S , then every PS sequence converges to zero weakly in H01 but it can have finitely many blow up. When MH ≥ 1 in general we cannot provide a precise classification but many different situations can occur. In particular: • There exists H ∈ C 1 (R3 ) with MH = 1 and H(p) → 0 as |p| → +∞ for which there is an unbounded PS sequence with no blow up. • For every M > 1 there exists H ∈ C 1 ∩ L∞ (R3 ) with MH = M for which there are PS sequences (un ) ⊂ H01 which are not locally bounded, namely k∇un kL2 (Ω) → +∞ for every domain Ω ⊂ D2 and, in addition, one of the following cases holds: - (un ) is bounded in L∞ and it has a countable, dense subset of blow up points; - kun kL∞ (Ω) → +∞ for every domain Ω ⊂ D2 and it has a countable, dense subset of blow up; 4

- kun kL∞ (Ω) → +∞ for every domain Ω ⊂ D2 and it has no blow up. In particular in the first two cases the limiting configuration can be made by infinitely many spheres. We point out that the parameter MH and the condition MH < 1 have been already used in a very strong way in previous works ([5], [7], [8], [9]) and it enter in many other aspects, like lower semicontinuity, uniform positive bound on the energy of H-bubbles, etc. In a work in preparation [10] we will develop the study of PS sequences by making a (partial) blow up analysis.

2

Preliminaries

2.1

Functional spaces

Given k, n ∈ N, for any Borel set Ω in Rk we denote by Lσ (Ω, Rn ) (for 1 ≤ σ ≤ +∞) and by H 1 (Ω, Rn ) and H01 (Ω, Rn ) the usual Lebesgue and Sobolev spaces of mappings taking value in Rn . Let D2 be the open unit disc in R2 . The Sobolev spaces H 1 (D2 , R3 ) and H01 (D2 , R3 ) will be denoted simply by H 1 and H01 respectively. We shall denote the Dirichlet integral of a map u ∈ H 1 by Z 1 |∇u|2 . D(u) := 2 D2 Let us also introduce the space ˆ 1 := H 1 (S2 , R3 ) . H ˆ 1 is a Hilbert space compactly embedded into Lσ (S2 , R3 ) It is known (see, e.g., [1]) that H ∞ 2 ˆ 1. for all σ ∈ [1, +∞) and that C (S , R3 ) is dense in H ˆ 1 with a corresponding map from In the following we will often identify any map U ∈ H R2 into R3 obtained by composing U with the inverse of the stereographic projection of S2 ˆ 1 turns out to be isomorphic to the onto the compactified plane R2 ∪ {∞}. In this way H R 1 Hilbert space {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 (|∇U |2 + µ2 |U |2 ) . kU k2Hˆ 1 = R2

ˆ 1 we set For U ∈ H

ˆ ) := 1 D(U 2

Z

R2

5

|∇U |2 .

(2.1)

ˆ 1. Notice that, by obvious extension, the space H01 can be considered as a subspace of H Throughout this work, Dr (z) and Dr (z) denote the open and the closed disc in R2 centered at z ∈ R2 and with radius r > 0, respectively. In particular, if z = 0 we write Dr instead of Dr (0). Given 0 < r < r′ we denote Ar,r′ the open annulus in R2 centered at 0 and with radii r and r′ . Finally, in many estimates we will denote C a real constant which does not depend on the varying parameters entering in the estimates.

2.2

H-volume and H-energy functionals

Given H ∈ C 0 ∩ L∞ (R3 ) let QH : R3 → R3 be such that div QH = H in the sense of distributions. We can take Z 1 H(sp)s2 ds . (2.2) QH (p) = mH (p)p with mH (p) = 0

2

1

For every Borel set Ω in R and for every u ∈ H (Ω, R3 ) ∩ L∞ let Z QH (u) · ux ∧ uy . VH (u, Ω) := Ω

This integral functional is well defined since QH ∈ L∞ loc . In particular the following estimate holds: Z 1 |∇u|2 for every u ∈ H 1 (Ω, R3 ) ∩ L∞ . (2.3) |VH (u, Ω)| ≤ kHkL∞ (R3 ) kukL∞ (Ω) 6 Ω Remark 2.1 If H ∈ C 0 (R3 ) is such that |H(p)p| ≤ CH < +∞ for every p ∈ R3 , then given any domain Ω in R2 the functional VH (·, Ω) is well defined in H 1 (Ω, R3 ) and Z CH |VH (u, Ω)| ≤ |∇u|2 for every u ∈ H 1 (Ω, R3 ). (2.4) 4 Ω This follows by the easy estimate |QH (p)| ≤ CH /2 for every p ∈ R3 . When Ω = D2 we set VH (u) = VH (u, D2 ) . We also define the energy functional Z 1 EH (u, Ω) := |∇u|2 + 2VH (u, Ω) 2 Ω

for u ∈ H 1 (Ω, R3 ) ∩ L∞ .

For next utility observe that, by (2.3), for any Borel set Ω in R2 and for any u ∈ H 1 (Ω, R3 )∩ L∞ one has that Z  1 1 |∇u|2 . (2.5) + kHkL∞ (R3 ) kukL∞ (Ω) |EH (u, Ω)| ≤ 2 3 Ω 6

If Ω = D2 we set EH (u) := EH (u, D2 ) = D(u) + 2VH (u)

for u ∈ H 1 ∩ L∞ .

The space H01 together with the energy functional EH provide the variational setting suited to study the Dirichlet problem (1.1). More precisely, we have the following result, due to K. Steffen [15], Proposition 3.3 and Lemma 5.2. Proposition 2.2 If H ∈ C 0 ∩ L∞ (R3 ) then the functional EH : H01 ∩ L∞ → R admits a unique continuous extension on H01 . Moreover for every u ∈ H01 and ϕ ∈ H01 ∩ L∞ , the directional derivative of EH at u in the direction of ϕ exists and it is given by Z Z H(u)ϕ · ux ∧ uy . ∇u · ∇ϕ + 2 ∂ϕ EH (u) = D2

D2

In particular u ∈ H01 is a critical point of EH (namely ∂ϕ EH (u) = 0 for all ϕ ∈ H01 ∩ L∞ ) if and only if u is a weak solution of (1.1). For u ∈ H01 we define EH′ (u): H01 ∩ L∞ → R by setting EH′ (u)ϕ := ∂ϕ EH (u)

for every ϕ ∈ H01 ∩ L∞ .

We also set kEH′ (u)k :=

sup ϕ∈H01 ∩L∞ ϕ6=0

|EH′ (u)ϕ| . kϕkH01

If kEH′ (u)k < +∞ then EH′ (u) extends in a unique way to a bounded linear functional of H01 , still denoted EH′ (u) and clearly kEH′ (u)kH −1 = kEH′ (u)k. A sequence (un ) ⊂ H01 is called Palais-Smale (shortly PS) sequence if kEH′ (un )k → 0 and |EH (un )| ≤ C for some constant C > 0 independent of n. If in particular EH (un ) → c then (un ) is called PS sequence at level c. Finally we introduce an H-weighted volume functional and an H-energy functional for ˆ 1 . If H ∈ C 0 ∩ L∞ (R3 ), for every U ∈ H ˆ 1 ∩ L∞ we set maps in H Z VˆH (U ) = QH (U ) · Ux ∧ Uy R2

where QH : R3 → R3 is as in (2.2). Let ω: R2 → R3 be the function defined by ω(z) = (µx, µy, 1 − µ)

(2.6)

where µ = µ(z) is defined by (2.1). Observe that ω is the inverse of the standard stereographic projection and it defines a conformal parametrization of S2 . In addition one has that ω(∞) = (0, 0, 1) and ∆ω = 2ωx ∧ ωy = −2µ2 ω

and |∇ω|2 = 2µ2 7

on R2 .

(2.7)

Moreover, by the divergence theorem, VˆH (rω + p) = −

Z

H(q) dq

(2.8)

Br (p)

for any r > 0 and p ∈ R3 , where Br (p) is the ball in R3 of center p and radius r. More generally, VˆH (U ) represents the algebraic H-weighted volume of the bounded region enclosed by the surface parametrized by U (see [15]). In particular, for H ≡ 1 the functional Vˆ1 coincides with the classical Bononcini-Wente volume functional. Finally, we set ˆ ) + 2VˆH (U ) EˆH (U ) := D(U

(2.9)

ˆ 1 ∩ L∞ . A result similar to Proposition 2.2 holds true also for Eˆ with respect for any U ∈ H ˆ 1 . In particular U ∈ H ˆ 1 ∩ L∞ is a critical point of EˆH if and only if U is a to the space H bounded, weak solution of  ∆U = 2H(U )Ux ∧ Uy on R2 R (2.10) |∇U |2 < +∞ . R2

2.3

H-bubbles

ˆ 1 ∩ L∞ of (2.10) will be called H-bubble. The energy of an Any nonconstant solution U ∈ H H-bubble U is given by (2.9). In the following we will always consider regular H-bubbles, more precisely, nonconstant solutions to (2.10) belonging to C 2 (S2 , R3 ). We point out that such a regularity for bounded weak solutions to (2.10) is guaranteed under some regularity conditions on the function H (see, e.g., [12], [2]). Given a regular H-bubble U , we set lim U (z) =: U (∞)

|z|→∞

and we note that |∇U (z)| = |dU (ω(z))|µ(z) ≤ kdU kL∞ (S2 ) µ(z)

for every z ∈ R2 ,

(2.11)

where d denotes differentiation in S2 . Notice that, since any regular H-bubble U is in fact conformal (use, for example, the ˆ ) measures the area of the surface parametrized by U . argument in [4], Lemma A.1), D(U The following uniform positive lower bound on the area of H-bubbles holds. Proposition 2.3 If H ∈ C 0 ∩ L∞ (R3 ) then there exists a constant AH > 0 such that for ˆ ) ≥ AH . every regular H-bubble U one has D(U 8

Proof. Let U be a regular H-bubble. Recall that U is conformal for free. Setting p¯ = U (∞), ¯ H(p) = H(p + p¯) for p ∈ R3 and ur (z) = U (rz) − p¯ for z ∈ D2 and r > 0, one has that for 2 ¯ every r > 0 the mapping ur is a classical, conformal solution of ∆u = 2H(u)u x ∧ uy in D . Hence one can use a Gr¨ uter estimate [11] and, more precisely the following version (see [3], Theorem 4.10)   ¯ L∞ (R3 ) k∇uk2 2 2 + k∇ukL2 (D2 ) , kukL∞ (D2 ) ≤ kukL∞ (∂D2 ) + C kHk L (D ) where C is a positive constant. In particular, for u = ur , one obtains   sup |U (z) − p¯| ≤ sup |U (z) − U (∞)| + C kHkL∞ (R3 ) k∇U k2L2 (R2 ) + k∇U k2L2 (R2 ) |z|rε

|∇U δε |2 → 0 .

ˆ 1. The proof of (3.5) and (3.6) plainly follows from: rε /δε → +∞, p = U (∞), and U ∈ H

Step 2. k∇uε kL2 (Arε ,1 ) → 0.

By direct computations one has that Z

Ar′ ,1 ε

and Z

ε 2

Arε ,r′

ε

2

|∇u | ≤ 2(diam U )

Z

|∇uε |2 =

2

|∇χ| +2

Arε ,r′

ε

Z

2π|p|2 log(rε′′ /rε′ )

C +2 |∇U | ≤ log(rε′ /rε ) δε 2

Arε ,r′

ε

Z

|z|>rε

|∇U δε |2

and then Step 2 follows from (3.4) and from (3.6). Step 3. EH (uε ) → EˆH (U ).

By conformal invariance one has that EˆH (U ) = EˆH (U δ ) for every δ > 0. Hence |EH (uε ) − EˆH (U )| = |EH (uε ) − EH (U δε , Drε ) − EH (U δε , R2 \ Drε )| ≤

|EH (uε , Arε ,1 )| + |EH (U δε , R2 \ Drε )|

and Step 3 follows from (3.6) and Step 2, using (2.5). Step 4. kEH′ (uε )kH −1 → 0.

For every ϕ ∈ H01 one has that Z Z ∇uε · ∇ϕ + 2H(uε )ϕ · uεx ∧ uεy + EH′ (uε )ϕ = |

Dr ε

{z

}

I1ε (ϕ)

Arε ,r′′

|

ε

∇uε · ∇ϕ + 2H(uε )ϕ · uεx ∧ uεy . {z

I2ε (ϕ)

Since uε = U δε in Drε and U δε solves the H-system, one has that Z ∂U δε . ϕ· I1ε (ϕ) = ∂n ∂Drε

}

By (2.11) one has that |∇U δε (z)| ≤ Cδε /rε2 for |z| = rε and then, by the embeddings 1 H 1 (Drε ) ֒→ H 2 ,2 (∂Drε ) ֒→ L1 (∂Drε ), one obtains that |I1ε (ϕ)| ≤ C

δε δε kϕkH 1 (Drε ) ≤ C 2 kϕkH01 (D2 ) . rε2 rε 12

In order to estimate I2ε (ϕ), we claim that kuεx ∧ uεy kL2 (Arε ,r′ ) → 0 .

(3.7)

ε

If (3.7) holds true, then one plainly infers that |I2ε (ϕ)|

≤ k∇uε kL2 (Arε ,r′′ ) k∇ϕkL2 (Arε ,r′′ ) + 2 ε

ε

≤



sup |q|≤kU k∞

|H(q)| kϕkL2 kuεx ∧ uεy kL2 (Arε ,r′ ) ε



k∇uε kL2 (Arε ,1 ) + Ckuεx ∧ uεy kL2 (Arε ,r′ ) kϕkH01 ε

and hence Step 4 is proved, because of Step 2 and (3.7). We have to check (3.7). One writes uεx ∧ uεy = χχx (U δε − p) ∧ Uyδε + χχy Uxδε ∧ (U δε − p) + χ2 Uxδε ∧ Uyδε . Thus one can estimate |uεx ∧ uεy | ≤ (diam U )|∇χ| |∇U δε | + 12 |∇U δε |2 and consequently kuεx ∧ uεy kL2 (Arε ,r′ ) ≤ Ck∇χkL∞ (Arε ,r′ ) k∇U δε kL2 (Arε ,r′ ) + Ck∇U δε k2L4 (Ar ε

ε

ε

′) ε ,rε

.

By (2.11) one has that |∇U δε (z)| ≤ Cδε /|z|2 for z ∈ Arε ,rε′ and then  σ Z δε |∇U δε |σ ≤ Crε2 2 r Ar ,r′ ε ε

ε

for every σ > 1. Moreover |∇χ| ≤ 1/(rε log(rε′ /rε )) on Arε ,rε′ . Hence kuεx ∧ uεy kL2 (Arε ,r′ ) ≤ C ε

1 δε + Crε ′ log(rε /rε ) rε2



δε rε2

2

.

Therefore (3.7) holds true, thanks to the assumptions (3.3)–(3.4). Step 5. kuε − (U δε − p)kH 1 → 0.

From (3.6) and from Step 2 it follows that k∇(uε − U δε )kL2 → 0. Moreover kuε − (U δε − p)kL2 (D2 )

kpkL2 (Drε ) + kuε kL2 (Arε ,r′′ ) + kU δε − pkL2 (Arε ,1 ) ε √ √ √ πrε |p| + πrε′′ kuε kL∞ (D2 ) + π sup |U δε (z) − p| ≤

≤

|z|>rε

and the conclusion follows from (3.3), (3.5) and since uε is uniformly bounded in L∞ . Remark 3.5 Using the same notation as in the previous proof, let us define   δε as z ∈ Drε′  as z ∈ Drε  U (z) − p p ′ ε ε δε v (z) = χ(z)(U (z) − p) as z ∈ Arε ,rε′ and θ (z) = χ (z)p as z ∈ Arε′ ,rε′′   0 0 as z ∈ Arε′′ ,1 . as z ∈ Arε′ ,1



One can check that v ε , θε ∈ H01 for every ε ∈ (0, 1), v ε + θε = uε , θε → 0 strongly in H01 , (uε ) is a PS sequence for EH and (v ε ) is a PS sequence for EH(·+p) . 13

Proof of Theorem 3.2. Fix a sequence (z k ) ⊂ D2 made by distinct points. For every n ∈ N let δn = min dist(zi , ∂D2 ) . 1≤i≤n

i

δεi

For every U (i = 1, . . . , r) let > 0 and uε,i ∈ H01 be given according to Lemma 3.4. Then for every n ∈ N let u ˜ε,i,n (z) = uε,i (δn (z − zi )) for i = 1, . . . , n. Since uε,i ∈ H01 (Dε , R3 ) one has that u ˜ε,i ∈ H01 (Dε/δn (zi ), R3 ). Since the points zi are distinct, for every n ∈ N there exists εn ∈ (0, 1) such that Dεn /δn (zi ) ⊂ D2 for every i = 1, . . . , n and Dεn /δn (zi ) ∩ Dεn /δn (zj ) = ∅ for all 1 ≤ i < j ≤ n.

(3.8)

Moreover, by Lemma 3.4, taking a smaller εn if necessary, we can also guarantee that for every i = 1, . . . , n one has: 1 |EH (uεn ,i ) − EˆH (U i )| ≤ 2 n 1 ′ εn ,i kEH (u )kH −1 ≤ 2 n i

(3.9) (3.10)

kuεn ,i − (U δεn ,i − U i (∞))kH 1 ≤ k∇U i kL2 (D1/δi

εn

Finally, for every n ∈ N set un =

)

1 n2

≥ k∇U i kL2 (R2 ) −

n X

1 . n2

(3.11) (3.12)

u ˜εn ,i,n .

i=1

Observe that by (3.8) one has  ε ,i,n u ˜ n (z) n u (z) = 0

as z ∈ Dεn /δn (zi ) and i = 1, . . . , n (3.13) otherwise. S In particular the support of un is contained in 1≤i≤n Dεn /δn (zi ). Hence un ∈ H01 . Moreover, by (3.13) and by the conformal invariance EH (un ) =

n X

EH (˜ uεn ,i,n , Dεn /δn (zi )) =

n X

EH (uεn ,i ) .

i=1

i=1

P∞ Hence, using (3.9) and the hypothesis that the series i=1 EˆH (U i ) converges to c we infer that EH (un ) → c as n → +∞ . 14

Given ϕ ∈ H01 , for every n ∈ N and for i = 1, . . . , n let   z + zi ϕ˜i,n (z) = ϕ δn so that ϕ˜i,n ∈ H01 (Dδn (zi ), R3 ). Noting that Dδn (−zi ) ⊆ D2 (by definition of δn ) and using again (3.13) and the conformal invariance, one has that n n X X ′ n ′ εn ,i,n ′ εn ,i i,n |EH (u )ϕ| = EH (˜ u )ϕ = EH (u )ϕ˜ i=1 i=1 ! n n X X kEH′ (uεn ,i )kH −1 kϕ˜i,n kH01 = ≤ kEH′ (uεn ,i )kH −1 kϕkH01 . i=1

i=1

Then, from (3.10) it follows that 1 for every n ∈ N. n Now for every n ∈ N and for i = 1, . . . , n set     i ˆ i,n (z) = U i z ˜ i,n (z) = U i z − zi − U i (∞) . − U (∞) and U U δεi n δεi n /δn kEH′ (un )kH −1 ≤

By using (3.13) one has that

n n

X

n X ˜ i,n ˜ i,n )kL2 (D2 ) + k˜ ˜ i,n kL2 (D2 ) U ≤ k∇(˜ uεn ,i,n − U uεn ,i,n − U

u −

1 2 i=1

i=1

H (D )

=

n X i=1

≤

n X i=1

ˆ i,n kL2 (D (−z )) ˆ i,n )kL2 (D (−z )) + δn kuεn ,i − U k∇(uεn ,i − U i i δn δn

ˆ i,n )kL2 (D2 ) + δn kuεn ,i − U ˆ i,n kL2 (D2 ) k∇(uεn ,i − U

and then (3.11) yields (3.1). Finally, given an open set Ω ⊆ D2 , for every n ∈ N let InΩ be the set of integers i ∈ {1, . . . , n} such that zi ∈ Ω. Since it is not restrictive to assume that εn /δn → 0, one has that Dεn /δn (zi ) ⊂ Ω for all i ∈ InΩ and then X X X k∇un k2L2 (Ω) ≥ k∇˜ uεn ,i,n k2L2 (Dε /δ (zi )) = k∇uεn ,i k2L2 (Dεn ) = kuεn ,i k2H 1 n

n

0

Ω i∈In

≥ ≥

X

Ω i∈In

Ω i∈In

ˆ i,n kH 1 − kuεn ,i − U ˆ i,n kH 1 kU 0

X

Ω i∈In

k∇U i kL2 (R2 ) −

2 n2

2

2

Ω i∈In

≥

X

Ω i∈In

k∇U i kL2 (D1/δi

εn

)−

1 n2

2

thanks to (3.11) and (3.12). In conclusion, since, by Proposition 2.3, inf i k∇U i kL2 (R2 ) > 0, one has that k∇un kL2 (Ω) → +∞ if the set {i ∈ N | zi ∈ Ω} is infinite.  15

Proof of Theorem 3.1. Fix distinct points z1 , . . . , zr ∈ D2 and set δ = min dist(zi , ∂D2 ) . 1≤i≤r

For every U i (i = 1, . . . , r) let δεi > 0 and uε,i ∈ H01 be given according to Lemma 3.4. Then, for every ε ∈ (0, 1) let r X uε,i (δ(z − zi )) . uε (z) = i=1

The rest of the proof goes on as the proof of Theorem 3.2, in fact with simpler and shorter computations and we leave it to the willing reader. 

4

PS sequences blowing H∞ -bubbles

In this section we exhibit a further phenomenon of lack of compactness for problem (1.1) when H: R3 → R is a continuous function such that H(p) → H∞ as |p| → +∞, for some H∞ ∈ R \ {0}. Our goal is to construct PS sequences for EH in H01 which blow H∞ -bubbles, namely spheres of radius |H∞ |−1 . Theorem 4.1 Let H ∈ C 0 ∩ L∞ (R3 ) satisfy:   1 as |p| → +∞ , for some a > 2 and H∞ ∈ R \ {0} . H(p) = H∞ + O |p|a

(4.1)

Then there exist continuous mappings ε 7→ δε ∈ (0, 1) and ε 7→ uε ∈ H01 defined for ε ∈ (0, 1), such that δε → 0 and EH (uε ) →

4π , 2 3H∞

kEH′ (uε )kH −1 → 0 ,

kuε − U δε kH 1 → 0

as ε → 0 ,

(4.2)

where U δ (z) = (1/H∞ )(ω (z/δ) − ω(∞)) and ω is defined in (2.6). In addition uε ∈ H01 (Dε , R3 ) ∩ L∞ for every ε ∈ (0, 1). Proof. Without loss of generality we may assume H∞ = 1. For every δ > 0 let U δ as in the statement of the theorem. Take arbitrary continuous mappings δε , rε , rε′ , rε′′ ∈ (0, 1) and pε ∈ R3 , defined for ε ∈ (0, 1) and such that rε < rε′ < rε′′

for every ε ∈ (0, 1)

For every ε ∈ (0, 1) set

rε′′ → 0 ,

and |pε | → +∞ ,

 δ U ε (z) + pε     χ(z)U δε (z) + p ε uε (z) = ′  χ (z)p ε    0 16

as as as as

z z z z

∈ Drε ∈ Arε ,rε′ ∈ Arε′ ,rε′′ ∈ Arε′′ ,1

δε → 0

as ε → 0 . (4.3)

where χ(z) =

log(|z|/rε′ ) log(rε /rε′ )

log(|z|/rε′′ ) . log(rε′ /rε′′ )

and χ′ (z) =

Notice that uε ∈ H01 ∩ L∞ . We show that (4.2) holds true provided that δε →0, rε2

rε′′ → +∞ , rε′

rε′ → +∞, rε

|pε |2 →0, log(rε′′ /rε′ )

| log δε | →0 |pε |a

as ε → 0 . (4.4)

Notice that conditions (4.4) are fulfilled taking, for instance, δε = ε7 , rε = ε3 , rε′ = ε2 , rε′′ = ε and |pε | = | log ε|b for some b ∈ ( a1 , 12 ), being a given by (4.1). This choice is possible because a > 2. Observe that in this way every uε belongs to H01 (Dε , R3 ). We point out that in the following several computations are the same as in the proof of Lemma 3.4. In this case we will omit the details. Step 1. k∇uε kL2 (Arε ,1 ) → 0. It follows from the estimates Z |∇uε |2 Ar′ ,1 ε

Z

Arε ,r′

ε

|∇uε |2

2π|pε |2 log(rε′′ /rε′ ) Z 4πkU k2∞ ≤ 2 |∇U |2 + log(rε′ /rε ) |z|>rε /δε =

and from (4.4). Step 2. EH (uε ) → 4π/3.

ˆ 1 , let us set Considering uε as a map in H ˆ 1 ∩ L∞ . θε = uε − (U δε + pε ) ∈ H Define K(p) = H(p) − 1. Observing that Eˆ1 (U δε + pε ) = Eˆ1 (ω) = 4π/3 and that U δε is a 1-bubble, one can write EH (uε ) −

4π 3

= Eˆ1 (uε ) − Eˆ1 (U δε + pε ) + 2VˆK (uε ) Z Z 1 |∇θε |2 + 2 uε · θxε ∧ θyε + 2VK (uε ) . = 2 R2 2 R

With similar estimates as in Step 1 one has that Z |∇θε |2 → 0 .

(4.6)

R2

Moreover, one has that Z

R2

Z uε · θxε ∧ θyε ≤ Ckuε kH01

R2

17

(4.5)

|∇θε |2

(see, e.g., Lemma A.2 in [6]). From Step 1 and since Z Z δε 2 |∇U |2 , |∇U | ≤ R2

Drε

ε

one has that ku kH01 ≤ C and thus, by (4.5) and (4.6), the statement of Step 2 holds true provided that one shows that VK (uε ) → 0 . (4.7) To this purpose, noticing that supp∈R3 |K(p)p| < +∞ and using (2.4) and (2.8), one can estimate |VK (uε )| = VK (uε , Arε ,1 ) + VˆK (U δε + pε ) − VK (U δε + pε , R2 \ Drε ) Z Z Z ε 2 K(p) dp + C |∇u | + ≤ C |∇U |2 B1 (pε −ω(∞)) Arε ,1 |z|>rε /δε and all these terms converge to zero, respectively by Step 1, since K(p) → 0 as |p| → +∞ and since rε /δε → +∞. Hence (4.7) holds true and Step 2 is proved. Step 3. kEH′ (uε )kH −1 → 0.

For every ϕ ∈ H01 one has that Z Z ′ ε δε δε δε EH (u )ϕ = ∇U · ∇ϕ + 2ϕ · Ux ∧ Uy + 2 |

Dr ε

+

{z

}

I1ε (ϕ)

Z

Arε ,r′′ ε

|

|

K(U δε + pε )ϕ · Uxδε ∧ Uyδε

Dr ε

{z

∇uε · ∇ϕ + 2H(uε )ϕ · uεx ∧ uεy . {z

}

I2ε (ϕ)

}

I3ε (ϕ)

One can estimate I1ε (ϕ) and I3ε (ϕ) as in the proof of Lemma 3.4. As far as concerns I2ε (ϕ) we use the fact that, by (2.7), 2Uxδε ∧ Uyδε = ∆U δε = −2µ2δε ω δε

(4.8)

where, for δ > 0, we define µδ (z) =

2δ 2 δ + |z|2

and ω δ (z) = ω

z 

.

δ

(4.9)

Recalling that |ω δ (z)| = 1 for all z ∈ R2 and δ > 0 and using (4.8) we estimate Z δε 2 δε ε K(U + pε )µδε ϕ · ω |I2 (ϕ)| = 2 Drε Z µ2δε |ϕ| ≤ 2kK(U δε + pε )kL∞ (R2 ) Dr ε

≤ 2kK(U

δε

+ pε )kL∞ (R2 )

Z

Drε

18

|ϕ|2 |z|α

!1/2

Z

Drε

|z|α µ4δε

!1/2

(4.10)

R where α is any value in (0, 2). Notice that for α ∈ (0, 2) the integral Dr |ϕ|2 /|z|α is finite ε thanks to the Hardy inequality (see, e.g., [14]): Z Z 4 |ϕ|2 ≤ |∇ϕ|2 for every ϕ ∈ H01 . (4.11) α (2 − α)2 D2 D2 |z| Moreover, by direct computation, one has that Z Z C 1 |z|α µ4 ≤ 2−α . |z|α µ4δε = 2−α δ δ ε ε Dr ε Drε /δε

(4.12)

Hence (4.10), (4.11) and (4.12) imply |I2ε (ϕ)| ≤

C (2−α)2

(2 − α)δε

sup |K(p − ω(∞) + pε )| kϕkH01

|p|=1

for every α ∈ (0, 2), with C positive constant independent of α and ϕ. Taking α = αε = 2 −

1 | log δε |

and using the assumption (4.1), we obtain that |I2ε (ϕ)| ≤ C

| log δε | kϕkH01 . |pε |a

Thanks to (4.4) we finally obtain that also I2ε (ϕ) → 0 as ε → 0 uniformly with respect to ϕ ∈ H01 . This completes the proof of Step 3. Step 4. kuε − U δε kH 1 → 0.

By Step 1 one has that k∇(uε − U δε )kL2 (D2 ) → 0. Moreover, observing that kuε k∞ ≤ |pε | + kU k∞ , one has that kuε − U δε k2L2 (D2 )

= kpε k2L2 (Drε ) + kuε − U δε k2L2 (Ar ≤ C(rε′′ |pε |)2 + π

sup |z|>rε′′ /δε

′′ ε ,rε

)

+ kU δε k2L2 (Ar′′ ,1 ) ε

|U (z)|2

and the conclusion follows using (4.3), (4.4) and the fact that U (z) → 0 as |z| → +∞. Remark 4.2 Using the same notation as in the previous proof, let us   δε  as z ∈ Drε  U (z)  pε ε ε δε v (z) = χ(z)U (z) as z ∈ Arε ,rε′ and θ (z) = χ′ (z)pε   0 0 as z ∈ Arε′ ,1



define as z ∈ Drε′ as z ∈ Arε′ ,rε′′ as z ∈ Arε′′ ,1 .

One can check that v ε , θε ∈ H01 for every ε ∈ (0, 1), v ε + θε = uε , θε → 0 strongly in H01 . Moreover (uε ) is a PS sequence both for EH and for E1 . However, in general (v ε ) is a PS sequence just for E1 but not for EH . Furthermore notice that (v ε ) is bounded both in H01 and in L∞ and it blows a 1-sphere passing through the origin. 19

Remark 4.3 From the technical point of view, the main difficulty in the previous proof occurs in the estimate of the term I2ε (ϕ) in Step 3 where we use in a rather sharp way the Hardy inequality (4.11) taking the exponent α closer and closer to the limiting value 2. The failure of the Hardy inequality in case of exponent α ≥ 2 is an obstruction in order to obtain the result stated by Theorem 4.1 just assuming that H(p) → H∞ as |p| → +∞, without the decay condition (4.1). This more general case is open. As one can recognize from the previous proof, instead of (4.1) one can ask for the following weaker condition: there exist p∞ ∈ S2 , H∞ ∈ R \ {0}, a > 2 such that   1 as t → +∞ . sup |H(p + tp∞ ) − H∞ | = O a t |p|≤1/|H∞ | According to this remark and similarly to what proved in Section 3, one can state a variant of Theorem 4.1 in which one builds PS sequences which blow finitely many spheres of possibly different radii. Theorem 4.4 Let H ∈ C 0 ∩ L∞ (R3 ), p1 , . . . , pr ∈ S2 , H1 , . . . , Hr ∈ R \ {0} and a > 2 be such that for every i = 1, . . . , r   1 as t → +∞ . sup |H(p + tpi ) − Hi | = O a t |p|≤1/|Hi | Then, given arbitrary different points z1 , . . . , zr ∈ D2 , there exist continuous mappings ε 7→ δε1 , . . . , ε 7→ δεr ∈ (0, 1) and ε 7→ uε ∈ H01 ∩ L∞ , defined for ε ∈ (0, 1), such that, as ε → 0, one has δεi → 0 for i = 1, . . . , r and

    r r

X X · − z 4π 1

i ′ ε ε −1 → 0, EH (uε ) → (u )k ω − ω(∞) u − , kE

→0, H H 2 i

1 3H H δ i ε i i=1 i=1 H

with ω defined by (2.6).

The proof of Theorem 4.4 can be accomplished as that one of Theorem 3.1.

5

Unbounded PS sequences blowing no bubble

Here we deal with a class of mean curvature continuous functions H: R3 → R vanishing at 1 when |p| → +∞, and we construct PS sequences for EH in H01 blowing no infinity as |p| bubble and which are unbounded both in H01 and in L∞ . Theorem 5.1 Let H ∈ C 0 ∩ L∞ (R3 ) be such that   1 1 H(p) = as |p| → +∞ , +O |p| |p|a 20

for some a > 4 .

Then there exist continuous mappings ε 7→ δε ∈ (0, 1), ε 7→ tε ∈ (0, +∞) and ε 7→ uε ∈ H01 defined for ε ∈ (0, 1), such that δε → 0, tε → +∞ and kuε − tε (ω δε − ω(∞))kH 1 → 0 , as ε → 0 , (5.1)  R 1 − H(p) dp. In addition where ω δ (z) = ω (z/δ), ω is defined in (2.6), and c = 2 R3 |p| EH (uε ) → c ,

kEH′ (uε )kH −1 → 0 ,

uε ∈ H01 (Dε , R3 ) and kuε kL∞ (D2 ) = tε for every ε ∈ (0, 1).

Proof. Take continuous mappings δε , rε , rε′ ∈ (0, 1) and tε ∈ (0, +∞) defined for ε ∈ (0, 1) and such that rε < rε′ for every ε ∈ (0, 1) and tε → +∞, rε′ → 0, δε → 0 as ε → 0. For every ε ∈ (0, 1) define  δε  as z ∈ Drε  tε ω (z) ε δε u (z) = χ(z)tε ω (z) as z ∈ Arε ,rε′  0 as z ∈ Arε′ ,1 where

χ(z) =

log(|z|/rε′ ) . log(rε /rε′ )

One can easily recognize that uε ∈ H01 and tε = kuε kL∞ (D2 ) = |uε (z)| for z ∈ Drε . We show that (5.1) holds true provided that t2ε →0, log(rε′ /rε )

δ ε tε →0, rε2

| log δε | →0 ta−2 ε

tε rε′ → 0 ,

as ε → 0 .

(5.2)

5 2 ′ Notice that conditions  (5.2) are  fulfilled taking, for instance, δε = ε , rε = ε , rε = ε and 1 tε = | log ε|b with b ∈ a−2 , 12 . Here the condition a > 4 makes possible this choice. Several computations in this proof are the same as in the proofs of Lemma 3.4 and of Theorem 4.1. In this case we will omit the details.

Step 1. k∇uε kL2 (Arε ,1 ) → 0.

Since |χ| ≤ 1 in Arε ,rε′ , |ω δε | = 1 and |∇ω δε |2 = 2µ2δε (see (2.7) and the definition of µδ given in (4.9)), one has that Z

Arε ,1

ε 2

|∇u | ≤

2t2ε

Z

δε 2

Arε ,r′

ε

|∇ω | +

2t2ε

Z

2

Arε ,r′

and the claim follows from (5.2).  R 1 − H(p) dp. Step 2. EH (uε ) → 2 R3 |p|

ε

|∇χ| ≤ 16π



δ ε tε rε

2

+ 4π

t2ε log(rε′ /rε )

By hypothesis, supp∈R3 |H(p)p| < +∞ for |p| large. Hence, from (2.4) and from Step 1 it follows that EH (uε , Arε ,1 ) → 0 .

21

Setting K(p) = H(p) − p1 , for every domain Ω in R2 and for every u ∈ H 1 (Ω, R3 ) ∩ L∞ with inf z∈Ω |u(z)| > 0 one has that Z 1 u · ux ∧ uy + VK (u, Ω) . VH (u, Ω) = 2 Ω |u| Since uε = tε ω δε in Drε and |∇ω δ |2 = 2µ2δ = −2ω δ · ωxδ ∧ ωyδ (because of (2.7)), one obtains that Z Z 1 uε ε ε 2 EH (u , Drε ) = · uεx ∧ uεy + 2VK (uε , Drε ) |∇u | + ε 2 Dr ε Drε |u | Z Z t2 = ε |∇ω δε |2 + t2ε ω δε · ωxδε ∧ ωyδε + 2VK (tε ω δε , Drε ) 2 Drε Dr ε 2VˆK (tε ω δε ) − 2VK (tε ω δε , R2 \ Drε ) .

= Now (2.4) and (2.7) yield

δε

2

|VK (tε ω , R \ Drε )| ≤

Ct2ε

Z

δε 2

|z|>rε

|∇ω | ≤ C



δ ε tε rε

2

.

Moreover, by conformal invariance of VˆK and by (2.8) one has that Z Z K(p) dp . K(p) dp → − VˆK (tε ω δε ) = VˆK (tε ω) = − R3

Btε

This completes the proof of Step 2. Step 3. kEH′ (uε )kH −1 → 0.

For every ϕ ∈ H01 one has that Z ′ ε ∇(tε ω δε ) · ∇ϕ + EH (u )ϕ = +2 |

|

Dr ε

2 ϕ · (tε ω δε )x ∧ (tε ω δε )y |tε ω δε | {z } I1ε (ϕ)

Z

Dr ε

K(tε ω δε )ϕ · (tε ω δε )x ∧ (tε ω δε )y + {z

}

I2ε (ϕ)

Z

Arε ,r′

|

ε

∇uε · ∇ϕ + 2H(uε )ϕ · uεx ∧ uεy . {z

I3ε (ϕ)

Arguing as in the proofs of Lemma 3.4 and of Theorem 4.1, one has that Z δ ε tε ∂ω δε ε |I1 (ϕ)| = tε ϕ· ≤ C 2 kϕkH01 ∂n rε ∂Drε

}

and then, by (5.2), I1ε (ϕ) → 0 as ε → 0 uniformly with respect to ϕ ∈ H01 . As far as concerns I2ε (ϕ), one argues as in the proof of Theorem 4.1, getting that |I2ε (ϕ)| ≤ C

| log δε | kϕkH01 . tεa−2 22

Hence also I1ε (ϕ) → 0 as ε → 0 uniformly with respect to ϕ ∈ H01 , by (5.2). Finally, one can handle I3ε (ϕ) as in the proof of Lemma 3.4, taking advantage of the following estimate ! Z Z Z Arε ,r′

ε

|uεx ∧ uεy |2

≤ Ct4ε

k∇χk2L∞ (Ar

≤ Ct4ε



′) ε ,rε

Arε ,r′

ε

δε 2 rε log(rε′ /rε )

2

+

δε4 rε6

|∇ω δε |2 +

!

Arε ,r′

ε

|∇ω δε |4

.

Step 4. kuε − tε (ω δε − ω(∞))kH 1 → 0.

One has that k∇(uε − tε ω δε )kL2 (D2 ) ≤ k∇uε kL2 (Arε ,1 ) + tε k∇ω δε kL2 (Arε ,1 ) . Then, using Step 1 and the estimate k∇ω δε kL2 (Arε ,1 ) ≤ Cδε /rε with (5.2) one deduces that k∇(uε − tε ω δε )kL2 (D2 ) → 0. Moreover kuε − tε (ω δε − ω(∞))k2L2 (D2 )

= ktε ω(∞)k2L2 (Drε ) + k(1 − χ)tε ω δε − tε ω(∞)k2L2 (Ar +ktε (ω ≤

δε

C(tε rε′ )2

≤ C

−

′) ε ,rε

ω(∞))k2L2 (Ar′ ,1 ) ε

+ t2ε sup |ω δε (z) − ω(∞)|2

(tε rε′ )2 +

|z|≥rε′



tε δ ε rε′

2 !

.

Hence one also has kuε − tε (ω δε − ω(∞))kL2 (D2 ) → 0 thanks to (5.2).



As one can observe from the proof (see, in particular, Step 2), the fact that H(p) decays exactly as 1/|p| at infinity is fundamental in the above construction. Notice that, since k∇ω δ kL2 (D2 ) = k∇ωkL2 (D1/δ ) → 8π as δ → 0, one can deduce that kuε kH 1 → +∞ as ε → 0. However, from the definition, one also has that uε → 0 strongly in H 1 (D2 \ Dr , R3 ) for any r ∈ (0, 1). Actually we can exhibit a situation with a PS sequence which is not even locally bounded either in H01 and in L∞ (compare with Remark 3.3).   1 Theorem 5.2 Let H ∈ C 0 ∩ L∞ (R3 ) be such that H(p) = |p| + O |p|1a as |p| → +∞, for  R 1 − H(p) dp = 0. Then there exists a PS sequence (un ) for EH in some a > 4 and R3 |p| H01 at level 0 and such that k∇un kL2 (Ω) → +∞ and kun kL∞ (Ω) → +∞ for every domain Ω ⊂ D2 .

Proof. Let N be a countable, dense subset of D2 . We can write N = {zk } where the zk ’s are distinct points of D2 . Using Theorem 5.1 and following the same procedure used in the proof of Theorem 3.2 one can build a sequence (un ) ⊂ H01 ∩ L∞ with the desired properties.  23

6

H 1 -boundedness of PS sequences (the role of MH )

In the previous sections we exhibited some cases of existence of PS sequences for EH in H01 which can be unbounded in H 1 and/or in L∞ . As one can see from Section 4, in general for a variable H: R3 → R, even smooth and close to a constant, it seems rather unlikely to find conditions on H ensuring the L∞ -boundedness of PS sequences. In this section we assume H ∈ C 1 ∩ L∞ (R3 ) and we focus on the problem of the H 1 boundedness of PS sequences. A crucial role is played by the quantity MH defined in (1.6). Proposition 6.1 If MH < 1 then every PS sequence for EH in H01 is bounded in H01 . If in addition H(p) → 0 as |p| → +∞, then every PS sequence for EH in H01 converges to zero strongly in H01 . Proof. For every u ∈ H01 ∩ L∞ we can write 3EH (u) − ∂u EH (u) = D(u) + 2

Z

D2

(3mH (u) − H(u))u · ux ∧ uy .

Then, using (2.14), we obtain (1 − MH )D(u) ≤ 3EH (u) + kEH′ (u)k kukH01

(6.1)

which, by continuous extension, holds true for every u ∈ H01 . In particular, when MH < 1, (6.1) ensures that every PS sequence for EH in H01 is bounded in H01 . Suppose now that H(p) → 0 as |p| → +∞. Firstly observe that, by definition of MH , for every p ∈ R3 one has Z +∞ |H(p)p| ≤ |∇H(sp) · p p| ds ≤ MH 1

and then it turns out that for every u ∈ H01 ∩ L∞ Z Z Z ′ 2 H(u)u · ux ∧ uy ≤ kEH (u)k kukH01 + MH |∇u| = ∂u EH (u) − 2 D2

D2

D2

|∇u|2

√ Hence 1 − MH kukH01 ≤ kEH′ (u)k for every u ∈ H01 and this implies the second part of the proposition.  Notice that Proposition 6.1 trivially applies when H is constant. In general, the condition MH < 1 is necessary in order to have H 1 -boundedness of PS sequences, as the following example shows. Example 6.2 For any R > 0 set ( R4 |p|3 − 2R3 |p|2 + 2R HR (p) = 1 |p|

24

if |p| ≤ R−1 . if |u| > R−1

Then HR ∈ C 1 (R3 ) with MHR = 1. Theorem 5.1 applies and one gets an unbounded PS sequence at level  Z  1 8π − HR (p) dp = . cR = 2 |p| 15R2 R3

We point out that for every ρ ≥ 1 the mapping R2 ∋ z 7→ U ρ (z) = ρω(z) with ω defined as in (2.6) is an HR -bubble with energy 8π/(15R2 ) (for the computation of VˆHR (U ρ ) use (2.8).

When a PS sequence is bounded in H 1 it can admit at most finitely many blow up points (see [3] or [10]). If MH > 1 one can construct PS sequences with a countable, dense set of blow up points, as discussed in the next example. Example 6.3 For R > 1 set ˜ R (p) = HR (p) + KR (p) , H where HR is defined in Example 6.2 and  −2 R (2|p|3 − 3|p|2 + 1) if |p| ≤ 1 KR (p) = 0 if |p| > 1 ˜ R ∈ C 1 (R3 ), H ˜ R (p) = 1/|p| for |p| > 1/R, and M ˜ ≤ 1 + C0 R−2 for some Notice that H HR constant C0 > 0 independent of R. In particular MH˜ R → 1 as R → ∞. Moreover  Z  1 ˜ dp = 0 (6.2) HR (p) − |p| R3 ˜ R -bubble with and for every ρ ≥ 1 the mapping U ρ = ρω introduced in Example 6.2 is an H null energy. Hence on can exhibit different kind of unbounded PS sequences for EH˜ R in H01 : applying Theorem 3.2 with U k = U ρ¯ for a fixed ρ¯ ≥ 1 and k ∈ N, one gets a PS sequence which is bounded in L∞ and it has a countable, dense set of blow up points. Again Theorem 3.2 with U k = U ρk and ρk → +∞ yields a PS sequence always having a countable, dense set of blow up points but which is unbounded in L∞ (Ω) for any domain Ω in D2 . Finally, thanks to (6.2), one can also apply Theorem 5.2 in order to construct a PS sequence which is unbounded both in H01 (Ω) and L∞ (Ω) for every domain Ω ⊆ D2 , and which blows no bubble. In the next example the curvature function does not vanish at infinity, as in the previous example, and in fact it equals a positive constant far away. Example 6.4 For R > 1 set  ˜ (p)  H   R 2   1 3 3 R − r + ¯ HR (p) = R3 2 4R      3 4R 25

if |p| ≤ R if R < |p| < if |p| ≥

3 R 2

3 R 2

˜ R is defined as in Example 6.3. One can check that H ¯ R ∈ C 1 ∩ L∞ (R3 ) and where H MH¯ R = MH˜ R = 1 + O(1/R2 ) (for R large). Moreover for every ρ ∈ [1, R] the mapping ¯ R -bubble with null energy. We can apply Theorem U ρ = ρω (with ω defined in (2.6)) is a H k ρk 3.2 with U = U , where (ρk ) is any sequence in [1, R] and we get a PS sequence for EH¯ R in H01 which is bounded in L∞ , it has a countable, dense set of concentration points and k∇un kL2 (Ω) → +∞ for every domain Ω in D2 . We conclude by recalling that the condition MH < 1 enters also in order to guarantee a positive lower bound for the energies of H-bubbles (Proposition 2.4). This result has been discussed in [5], Proposition 2.8. On the contrary, in the examples 6.3 and 6.4 we exhibit curvature functions H ∈ C 1 ∩ L∞ (R3 ) with MH close to 1 as we want, and for which there exist H-bubbles with null energy. Hence the condition MH < 1 is not just a decay condition at infinity but it provides also some essential bound on the behaviour of H on bounded sets of R3 in order to ensure both boundedness of PS sequences (Proposition 6.1) and a positive estimate from below on minimal energies of H-bubbles (Proposition 2.4). Remark 6.5 In Proposition 2.4 and in the first part of Proposition 6.1 we can replace the assumption MH < 1 with the weaker condition sup |(H(p) − 3mH (p))p| =: M H
2 |VˆKH (U )|

where KH (p) = ∇H(p) · p. Indeed one has MH ≥ M H and M H σH ≤ 1 (see [7]).

References [1] Aubin, Th., Nonlinear analysis on manifolds. Monge-Amp` ere equations, Grundlehren der Mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. [2] Bethuel, F., Ghidaglia, J.M., Improved regularity of solutions to elliptic equations involving Jacobians and applications, J. Math. Pures Appl. 72 (1993), 441–474. [3] Bethuel, F., Rey, O., Multiple solutions to the Plateau problem for nonconstant mean curvature, Duke Math. J. 73 (1994), 593–646. [4] Brezis, H., Coron, J.M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Rat. Mech. Anal. 89 (1985), 21–56.

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[5] Caldiroli, P., Musina, R., Existence of minimal H-bubbles, Commun. Contemp. Math. 4 (2002), 177–209. [6] Caldiroli, P., Musina, R., H-bubbles in a perturbative setting: the finite-dimensional reduction method, Duke Math. J. 122 (2004), 457–485. [7] Caldiroli, P., Musina, R., S2 -type parametric surfaces with prescribed mean curvature and minimal energy, in: Nonlinear equations: methods, models and applications (Bergamo, 2001), 61–77. Progr. Nonlinear Differential Equations Appl. 54. Birkh¨ auser, Basel, 2003. [8] Caldiroli, P., Musina, R., Existence of H-bubbles in a perturbative setting, Rev. Matem. Iberoamer. 20 (2004), 611–626. [9] Caldiroli, P., Musina, R., On the Dirichlet problem for H-systems with small boundary data: blow up phenomena and nonexistence results, Preprint SISSA 79/2004/M. [10] Caldiroli, P., Musina, R., On Palais-Smale sequences for H-systems. Part two: (partial) blow up analysis, in preparation.

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