Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

Solutions positives de problèmes semi-linéaires elliptiques dans des domaines non bornes ayant frontière non borné

Giovanna Cerami

, Riccardo Molle

, Donato Passaseo

aDipartimento di Matematica, Politecnico di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy bDipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n^◦1, 00133 Roma, Italy

cDipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy Received 15 March 2005; received in revised form 8 August 2005; accepted 16 September 2005

Available online 24 February 2006

Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the equation−u+u=u^p−1, 2< p <2^∗= 2N

N−2, with Dirichlet zero data, in an unbounded smooth domainΩ⊂R^Nhaving unbounded boundary. Under the assumptions:

(h₁) ∃τ₁, τ₂, . . . , τ_k∈R⁺\ {0}, 1kN−2, such that

(x₁, x₂, . . . , x_N)∈Ω ⇐⇒ (x₁, . . . , x_i−1, x_i+τ_i, . . . , x_N)∈Ω,∀i=1,2, . . . , k, (h₂) ∃R∈R⁺\ {0}such thatR^N\Ω⊂ {(x1, x₂, . . . , x_N)∈R^N:_N

j=k+1x_j²R²} the existence of at leastk+1 solutions is proved.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article on étudie l’existence et la multiplicité de solutions positives pour l’équation−u+u=u^p−1, 2< p <2^∗= 2N

N−2, avec la condition de Dirichletu=0 sur∂Ω. Le domaineΩ⊂R^Nest non borné et∂Ωest non borné aussi. En supposant que les conditions

(h₁) ∃τ1, τ₂, . . . , τ_k∈R⁺\ {0}, 1kN−2, tels que

(x₁, x₂, . . . , x_N)∈Ω ⇐⇒ (x₁, . . . , x_i−1, x_i+τ_i, . . . , x_N)∈Ω,∀i=1,2, . . . , k, (h₂) ∃R∈R⁺\ {0}tel queR^N\Ω⊂ {(x₁, x₂, . . . , x_N)∈R^N:_N

j=k+1x_j²R²}

✩ Work supported by national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

* Corresponding author.

E-mail addresses:cerami@poliba.it (G. Cerami), molle@mat.uniroma2.it (R. Molle).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2005.09.007

soient vérifiées, on démontre que le problème possède au moinsk+1 solutions.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35J20; 35J65

Keywords:Nonlinear elliptic problems; Unbounded domains; Unbounded boundary

1. Introduction and statement of the results

In this paper we are concerned with the existence and the multiplicity of solutions to (P )

⎧⎨

⎩

−u+u=u^p−1 inΩ,

u >0 inΩ,

u∈H₀¹(Ω)

where 2< p <2^∗=2N/(N−2)andΩ⊂R^N,N3, is an unbounded smooth domain with∂Ωunbounded.

Problem(P )has a variational structure: its solutions can be found looking for positive functions that are critical points of the functional

E(u)=

Ω

|∇u|²+u² dx constrained to lie on the manifold

V = u∈H₀¹(Ω): |u|L^p(Ω)=1 .

However, the usual variational techniques (minimization, minimax methods) cannot be applied straightly, because of the lack of compactness, due to the unboundedness ofΩ. Indeed, since the embeddingj:H₀¹(Ω) →L^p(Ω)is continuous, but not compact, the manifoldV is not closed for the weakH₀¹-topology and, moreover, the basic Palais–

Smale condition is not satisfied byEat every energy level. Furthermore, as we shall see, the situations, one has to face, are strongly depending on the shape of the domain in which(P )is considered, so the corresponding technical difficulties can be considerably different.

WhenR^N\Ω is a ball the existence of a solution to(P )can be, quite easily, proved (see [13]), by minimizing Eon the manifoldV_r= {u∈H_r(Ω): |u|L^p(Ω)=1},H_r(Ω)being the subspace ofH₀¹(Ω)consisting of spherically symmetric functions, that, as well known [21], embeds compactly inL^p(Ω). Analogous devices can be used when R^N\Ω is bounded and it is “nearly” spherical, in a suitable sense, or enjoys of some other kind of symmetry [10]

and, moreover, whenΩis a “strip-like” domain [12].

The question becomes more difficult when R^N \Ω has no symmetry properties. Indeed a classical, by now, result [13] states that, for a very large class of unbounded domains, those satisfying the condition: ∃ ¯x ∈R^N: (ν(x),x)¯ 0 ∀x∈∂Ω,(ν(x),x)¯ ≡0, (P )admits only the trivial solution u≡0. Moreover, even whenR^N\Ω is bounded, problem(P )is not easy to handle and cannot be solved by minimization: the infimum ofEonV equals the infimum ofu²_H1(R^N) on the manifold{u∈H¹(R^N): |u|L^p(R^N)=1}and is not achieved [4]. Nevertheless, in this case, a careful analysis of the Palais–Smale sequences behaviour [4] has made possible an estimate of the en- ergy levels in which the compactness is saved and to give some answers to the existence and multiplicity questions for(P ). The existence of a positive nonminimizing solution to(P )has been proved (see [4,2]) by minimax meth- ods, and furthermore multiplicity results have been obtained, by using subtle geometric and topological arguments, in [6–8,19].

When bothΩ andR^N\Ω are unbounded the compactness situation can be even more complex. Indeed, when R^N\Ω is bounded, the above mentioned result states that a Palais–Smale sequence either converges strongly to its weak limit or differs from it by a finite number of sequences that are noting but normalized solutions of the limit problem

(P_∞)

−u+u=u^p−1 inR^N, u∈H¹

R^N

“travelling to infinity” and infinitely far away each other. WhenΩ andR^N\Ω are unbounded and invariant with respect to a group of translations, it is not difficult to understand that, besides the above described behaviour, a non- compact Palais–Smale sequence can also look like a solution of(P )(normalized inL^p(Ω)) “travelling to infinity”, of course by means of translations that leaveΩ invariant. This happens, for instance, whenΩ is the complement of a cylinder:Ω=R^N\ {(x₁, . . . , x_N)∈R^N:_N

j=k+1x_j²R²}, where 1kN−2 andR >0; in fact, ifusolves(P ), thenu(x−yⁱ)/|u|L^p(Ω), withyⁱ=(x₁ⁱ,0, . . . ,0)andx₁ⁱ −→

i→+∞+ ∞, is a noncompact Palais–Smale sequence at the same energy level of the critical point, ofEonV,u/|u|L^p(Ω). Moreover, we remark that even worse phenomena can occur, in fact unbounded domains with unbounded boundary exist such that the Palais–Smale condition for the related energy functionalEcan fail at every energy level (see [17]).

The question of the existence of solutions of(P )whenΩis an unbounded domain having unbounded boundary is still very partially investigated. Most of the known existence results concern domains bounded in some co-ordinates (strip-like, cylinders) (see [23] and references therein), while only recently some existence results have been proved under suitable condition at infinity onΩand on∂Ω [18,17].

The research, we present here, deals with problem(P )when Ω is an unbounded “exterior” domain having un- bounded boundary. Precisely, we suppose thatΩ satisfies the following assumptions:

(h1) there existkpositive real numbersτ1, τ2, . . . , τk, 1kN−2, such that

(x₁, x₂, . . . , x_N)∈Ω ⇐⇒ (x₁, . . . , x_i−1, x_i+τ_i, . . . , x_N)∈Ω, ∀i=1,2, . . . , k, (h2) there existsR∈R,R >0, such that

R^N\Ω⊂

(x1, x2, . . . , xN)∈R^N: N j=k+1

x_j²R²

.

The main result we obtain is contained in the following

Theorem 1.1. Let Ω be a smooth domain verifying conditions (h1) and (h2). Then problem (P) has at least (k +1) solutions, u1, u2, . . . , uk+1, nonequivalent, in the sense that ∀i =j, i, j =1, . . . , k+1, does not exist (h1, h2, . . . , hk)∈Z^ksuch thatui(x1, . . . , xN)=uj(x1+h1τ1, x2+h2τ2, . . . , xk+hkτk, xk+1, . . . , xN).

We stress the fact that, dropping assumption (h1), Theorem 1.1 is not true and moreover(P ) could not have any solution, as one easily understands considering Ω= {(x₁, x₂, . . . , x_N)∈R^N: |(x₂, . . . , x_N)|> f (x₁)}, where f:R→Ris a bounded smooth function such that 0<inf_Rf <sup_Rf < H,H∈Randf(t ) >0∀t∈R.Ω is a domain satisfying the above mentioned nonexistence condition, our condition(h2), but not condition(h1). Moreover we point out that if in assumptions(h1),(h2)we havek=N−1 then Theorem 1.1 does not hold, in general. Consider, for example,D= {(x₁, x₂, . . . , x_N)∈R^N: |x₁|> R},R >0: since no solution exists on half-spaces, problem(P )has no solution onD. Indeed, the proof we carry on does not work whenk=N−1, because in such a case Lemma 4.7 is not true, withq=2.

On the contrary, a simple example of a domain to which Theorem 1.1 applies, giving the existence of at least two so- lutions, is obtained consideringΩ= {x∈R³: dist(x,E) > H},H∈R⁺\ {0}, withE= {(x₁,cosx₁,sinx₁): x₁∈R}.

We, also, must observe that, in domains verifying the assumptions of Theorem 1.1, but having richer symmetry properties (as the complement of a cylinder), solutions nonequivalent in the sense of Theorem 1.1 can be identified by a different kind of translation. Because of this we explicitly state the following corollary, for which, on the other hand, we have an independent proof, simpler than that of Theorem 1.1.

Theorem 1.2.Let Ω be a smooth domain verifying conditions(h1)and (h2). Then problem(P ) has at least one solution.

The paper is organized as follows: Sections 2 and 3 are devoted to build the variational framework for the study, namely, in Section 2, after some remarks, the necessary equivariant, Ljusternik–Schnirelmann type, theory is exposed, while in Section 3 the compactness question is studied. Section 4 contains some basic asymptotic estimates and in Section 5 the proofs of Theorems 1.1 and 1.2 are displayed.

2. Notations, preliminary remarks, equivariant theory recalls Throughout the paper we make use of the following notations:

• L^p(D), 1p <+∞,D⊆R^N, denotes a Lebesgue space; the norm inL^p(D)is denoted by| · |p,D;

• H₀¹(D), D ⊂R^N, and H¹(R^N)≡H₀¹(R^N) denote the Sobolev spaces obtained, respectively, as closure ofC₀^∞(D), andC^∞₀ (R^N), with respect to the norms

u_D=

D

|∇u|²+u² dx

1/2

, u_R^N=

R^N

|∇u|²+u² dx

1/2

;

• ifD1⊂D2⊆R^N andu∈H₀¹(D1), we denote also byuits extension toD2obtained settingu≡0 outsideD1. Hence for allu∈H₀¹(Ω)

E(u)=

Ω

|∇u|²+u² dx≡

R^N

|∇u|²+u² dx;

• the generic point x =(x₁, x₂, . . . , x_k, x_k+1, . . . , x_N)∈R^N =R^k ×R^N−k is denoted by (x₁, x₂, . . . , x_k, x) wherex=(xk+1, . . . , xN)∈R^N−k, k being the number that appears in assumption(h1), we put also |x| = (_N

j=k+1x_j²)^1/2;

• B(y, r)denotes the open ball ofR^N, having radiusrand centred aty. We set

m:=inf u²_RN: u∈H¹ R^N

,|u|p,R^N =1

. (2.1)

The infimum in (2.1) is achieved (see [21,5]) by a positive functionω, that is unique modulo translation [16] and radially symmetric about the origin, decreasing when the radial co-ordinate increases and such that

|x|→+∞lim D^sω(x)|x|^(N−1)/2e^|x|=d_s>0, s=0,1 (2.2) (see [15] and [5]).

The following proposition shows that, on the contrary,(P )cannot be solved by minimization.

Proposition 2.1.Setting m_Ω:=inf E(u): u∈V

(2.3) the relation

mΩ=m (2.4)

holds and the minimization problem(2.3)has no solution.

Proof. Since we may considerH₀¹(Ω)as a subspace ofH¹(R^N), m_Ωm.

To prove that the equality holds, let us consider the sequence(ω_yn)_n∈Ndefined by ωy_n(x):= ϕ(x)ω(x−y_n)

|ϕ(x)ω(x−y_n)|p,Ω

, x∈Ω,

where,∀n∈N,y_n=((y_n)₁, (y_n)₂, . . . , (y_n)_k, y_n)∈Ω, limn→+∞|y_n| = +∞,ωis the function realizing (2.1) and ϕ∈C^∞(R^N,[0,1])is a cut-off function defined byϕ(x₁, x₂, . . . , x_k, x)= ˜ϕ(|x|),ϕ˜:R⁺→ [0,1]being aC^∞non- decreasing function such thatϕ(s)˜ =0∀sR,ϕ(s)˜ =1∀sR+1 (where R is the number defined in assump- tion(h2)). Using (2.2) it is not difficult to verify that

(a) limn→+∞ϕ(x)ω(x−y_n)−ω(x−y_n)

p,R^N=0, (b) limn→+∞ϕ(x)ω(x−y_n)−ω(x−y_n)

R^N=0 (2.5)

hence

n→+∞lim E(ω_yn)=m. (2.6)

Let us now assumeu^∗∈V exists so thatE(u^∗)=m, then by the uniqueness of the family of functions realizing (2.1) u^∗(x)=ω(x−y^∗) for somey^∗∈R^N.

This is impossible becauseω(x) >0∀x∈R^NandR^N\ Ω= ∅. 2

From the above result we can deduce, also, some useful estimates on theL^p norm of a critical point and a lower bound for the energy of a changing sign critical point ofEonV.

Corollary 2.2.Letu¯be a nontrivial solution of (P_μ)

−u+u=μ|u|^p−2u inD, u∈H₀¹(D)

with eitherD=Ω orD=R^N. Then

| ¯u|p,D m

μ

1/(p−2)

. (2.7)

Proof. By (2.1), (2.3) and (2.4) we have m| ¯u|²_p,D ¯u²_D,

and beingu¯solution of(P_μ) ¯u²_D=μ| ¯u|^p_p,D. Thus

| ¯u|^p_p,⁻_D²m μ and (2.7) follows. 2

Corollary 2.3.Letu¯be a critical point ofEonV. IfE(u)¯ ∈(m,2^1−2/pm)thenu¯does not change sign.

Proof. Let us assumeE(u)¯ =μ,| ¯u|p,Ω =1, u¯= ¯u⁺− ¯u⁻ andu¯⁺≡0, u¯⁻≡0. Then, taking into account thatu¯ solves(Pμ)inΩand using (2.3),(2.4), we obtain

mu¯^±2

p,Ω u¯^±2

Ω=μu¯^±p

p,Ω

hence

1= | ¯u|^p_p,Ω =u¯^+p

p,Ω+u¯^−p

p,Ω2 m

μ

p/(p−2)

that implies

μ2^1−2/pm. 2

Remark 2.4. Obviously, the same conclusion holds true for every changing sign critical point of u²_RN on {u∈H¹(R^N): |u|p,R^N=1}and every normalized changing sign solution of(P_μ)inΩor inR^N.

We recall, now, some facts about equivariant critical points theory, that is the needful topological framework for our research.

LetXbe a normed space andGa topological group. Theaction ofGonXis a continuous map G×X−→^{[ ]} X, [g, u] →gu

verifying the conditions

(i) 1·u=u, (ii) (gh)u=g(hu), (iii) u→guis linear.

Theaction ofGonXissaidisometricif gu = u ∀g∈G∀u∈X.

AsetA⊆XisG-invariantifgA=Afor allg∈G.

Two elementsw, z∈XareG-equivalentifg(w)=zfor someg∈G. We denote by[w]theorbitofw, i.e. the subspace{gw: g∈G}, and byX/Gtheorbit space, i.e. the quotient space obtained by identifying each orbit to a point.

Two sequences(wn)n∈N,(zn)n∈N,wn, zn∈XareG-equivalentif, for alln∈N,wnisG-equivalent tozn, in other words, a sequence(g_n)_n∈Nexists so thatg_n∈Gandg_nw_n=z_n.

Amapf:Y→T,Y ⊆X G-invariant set,T normed space, isG-invariantif f◦g(y)=f (y) ∀g∈G∀y∈Y.

Amapf:Y→X,Y ⊆X G-invariant set, isG-equivariantif g◦f =f◦g ∀g∈G.

Definition 2.5.LetA, B, Y,B⊂A⊆Y be closedG-invariant subsets of a normed spaceXon which a topological groupGacts. TheG-equivariant category ofAinY, relative toB, denoted by cat^G_Y(A, B), is the least integerlsuch that there exist(l+1)closedG-invariant subsets ofY,C0, C1, . . . , Cland(l+1)mapshj∈C(Cj× [0,1], Y ), such that

(a)A⊆ l j=0

Cj, B⊆C0;

(b)

⎧⎪

⎨

⎪⎩

(i)h_j(·, t )isG-equivariant∀t∈ [0,1], j=0,1, . . . , l; (ii)h_j(c,0)=c ∀c∈C_j, j=0,1, . . . , l;

(iii)h₀(c,1)∈B ∀c∈C₀; h₀(B, t )⊂B ∀t∈ [0,1];

(iv)∀j=1,2, . . . , l∃w_j∈Ysuch thath_j(c,1)∈ [w_j] ∀c∈C_j.

If such a number does not exist, we say that theG-equivariant category ofAinY relative toBis+∞.

Definition 2.6.LetMbe aC¹,G-invariant manifold embedded in an Hilbert spaceHon which the topological group G acts isometrically. LetF ∈C¹(M,R)a G-invariant functional. The functional F satisfies the G-Palais–Smale condition, briefly (PS)^G, at the levelcif for every sequence(u_n)_n,u_n∈M, such that

F (u_n)_n−→_→+∞c, ∇F (u_n)_n→+∞−→ 0

there exists a sequence(v_n)_n,G-equivalent to(u_n)_n, relatively compact.

The following Ljusternik–Schnirelmann type theorem provides a lower bound for the number of critical points of an invariant functional in suitable ranges of its values.

Theorem 2.7.LetH be an Hilbert space on which the topological group Gacts isometrically. Let be M⊆H a G-invariantC^1,1- manifold andF∈C^1,1(M,R)aG-invariant functional. Put for anyc∈R

F^c= u∈M: F (u)c ,

K^c= {u∈M: F (u)=c, (∇F )(u)=0 .

Consider−∞< a < b <+∞, and assumeK^a= ∅ =K^band thatF satisfies the(PS)^G for allc∈ [a, b]. ThenF has at leastcat^G_Fb(F^b, F^a)critical points that are notG-equivalent and to which there correspond critical levels lying in(a, b).

Proof. SinceF isG-invariant, we have for allu∈M, ∀g∈G F(gu)[v] =lim

t→0

F (u+tg⁻¹v)−F (u) t

=F(u)[g⁻¹v].

Thus, being the action ofGisometric, we obtain ∇F (gu), v

=

∇F (u), g⁻¹v

=

g∇F (u), v

∀g∈G,∀u∈M

and we deduce that∇F:M→H is an equivariant map. Furthermore, because of the (PS)^Gcondition, ifF has not critical values in the interval[α, β] ⊆ [a, b], there exists a positive numberδ >0 for which

∇F (u)> δ ∀u∈F⁻¹ [α, β]

. (2.8)

Indeed, if (2.8) were false, a sequence(u_n)_n,u_n∈M, would exist so that F (u_n)_n→+∞−→ c∈ [α, β],

∇F (u_n)_n−→_→+∞0.

Thus, a sequence(g_n)_n,g_n∈G, would exist so that, passing eventually to a subsequence,g_nu_nn−→_→+∞v∈M. Hence we infer

F (v)= lim

n→+∞F (gnun)= lim

n→+∞F (un)=c, ∇F (v)= lim

n→+∞∇F (g_nu_n)= lim

n→+∞g_n∇F (u_n)= lim

n→+∞∇F (u_n)=0 contradicting the nonexistence of critical values in[α, β].

Therefore, using well known methods (see e.g. [24] Lemmas 1.14 and 3.1), a number ε >0 and a continuous deformationη∈C([0,1] ×M, M)can be constructed so that

(i)η(t,·) ∀t∈ [0,1]is aG-equivariant homeomorphism ofM; (ii)η(0, u)=u, ∀u∈M;

(iii)η(t, u)=u, ∀t∈ [0,1]ifu /∈F⁻¹[α−ε, β+ε];

(iv)η(1, F^β)⊂F^α; (v)F

η(·, u)

is nonincreasing∀u∈M.

Then, the conclusion follows, by applying classical arguments of the generalized Ljusternik–Schnirelmann theory (see [24] Theorem 5.19). 2

Remark 2.8.The same result could be also proved supposingH Banach, instead of Hilbert, space and under weaker regularity assumptions onF.

We end this section pointing out that, in our setting,a noncompact group of translations,G, acting onR^Nand, in turn, onH¹(R^N)andH₀¹(Ω)is considered. Namely, for allh≡(h1, h2, . . . , hk)∈Z^k, we defineTh:R^N→R^N, by

Th(x)=Th(x1, x2, . . . , xk, x):=(x1+τ1h1, x2+τ2h2, . . . , xk+τkhk, x)

hence we say that x, y∈R^N are equivalent if and only if there exists (h₁, h₂, . . . , h_k)∈Z^k such that y=(x₁+ τ₁h₁, x₂+τ₂h₂, . . . , x_k+τ_kh_k, x).

ClearlyΩ is invariant under the action ofG.

Analogously, for allh≡(h1, h2, . . . , h_k)∈Z^k, we defineTh:H¹(R^N)→H¹(R^N), by Th(u)(x):=u

Th(x)

=u(x1+τ1h1, x2+τ2h2, . . . , xk+τkhk, x)

and we say thatu, v∈H¹(R^N)are equivalentif and only if there exists(h1, h2, . . . , hk)∈Z^k such that v(x₁, x₂, . . . , x_k, x)=u(x₁+τ₁h₁, x₂+τ₂h₂, . . . , x_k+τ_kh_k, x).

We remark that the action onH¹(R^N)of the groupG:= {Th: h∈Z^k}is isometric, thatH₀¹(Ω)can be seen as an invariant subspace ofH¹(R^N),V is an invariant manifold inH₀¹(Ω)and the functionalEis invariant.

3. A compactness result

The purpose of this section is to show that there exists an energy interval in which the compactness of the functional Eis saved.

The result we prove is stated in the following

Proposition 3.1.The functionalEsatisfies theG-Palais–Smale condition onV at every levelc∈(m,2^1−2/pm).

Proof. Let(un)nbe a Palais–Smale sequence forEconstrained onV, e.g.

⎧⎨

⎩

(a)|u_n|p,Ω=1,

(b) limn→+∞E(u_n)=c, (c) limn→+∞∇E_|V(u_n)=0

(3.1) and assume

c∈

m,2^1−2/pm

. (3.2)

By definition ofE, (3.1)(b) implies that(un)nis bounded inH₀¹(Ω), so there existsu0∈H₀¹(Ω)such that, up to a subsequence,

(a)u_n u₀ weakly inH₀¹(Ω)and inL^p(Ω),

(b)u_n(x)→u₀(x) a.e. inΩ. (3.3)

By (3.1)(c), there exists a sequence(μ_n)_n,μ_n∈R, such that ∇E_|V(u_n), w

=

Ω

(∇u_n,∇w)+u_nw

dx−μ_n

Ω

|u_n|^p−2u_nw=o(1)wΩ ∀w∈H₀¹(Ω) (3.4) and, in view of (3.1)(a), (b), setting in (3.4)w=u_n, we deduce

n→+∞lim μ_n=c. (3.5)

Henceu₀solves

−u+u=c|u|^p−2u inΩ,

u∈H₀¹(Ω). (3.6)

Set now v_n(x)=

(u_n−u₀)(x), x∈Ω,

0, x∈R^N\Ω.

Then, by (3.3)(a),

vn0 weakly inH¹ R^N

andL^p R^N

(3.7) and

v_n²_RN= u_n²_Ω− u₀²_Ω+o(1). (3.8)

Furthermore, by (3.3)(b), the Brezis–Lieb theorem can be applied and it gives

|v_n|^p_p,RN= |u_n|^p_p,Ω − |u₀|^p_p,Ω+o(1). (3.9)

Let us suppose, now,vn_R^N→0 strongly (otherwise we are done). So, up to a subsequence,vn_R^Nk0>0

∀n∈N, for somek₀∈R. Then, using (3.4), (3.6), (3.8), (3.9), we deduce thatk₁∈Rexists such that|v_n|^p_p,RN k₁>0.

Let us decompose, now,R^N intoN-dimensional hypercubesQ_l, having unitary sides and vertices with integer co-ordinates, and put for alln∈N

d_n=max

l∈N|v_n|p,Q_l.

We claim thatγ∈R,γ >0, exists such that (up to a subsequence)

d_nγ >0 ∀n∈N. (3.10)

Indeed

0< k₁|v_n|^p_p,RN=

l∈N

|v_n|^p_p,Ql

max

l∈N |vn|^p_p,Q⁻²_l

l∈N

|vn|²p,Q_l

d_n^p−2k₂

l∈N

v_n²_Ql d_n^p−2k₂v_n²_RN,

k₂∈R⁺\ {0}independent ofi. Thus, in view of (3.1) and (3.8), (3.10) follows.

Let us call, for alln∈N,y_nthe centre of an hypercubeQ_nin which|v_n|p,Qn=d_n.

If(y_n)_n were bounded, then, passing eventually to a subsequence, we could assume that they_n, for alln, belong to the same cubeQ, and, hence, that they coincide. Thus, in Qwe would have, for all n,|v_n|_p,Qγ >0 and, on the other hand,v_nQv_nR^N k₃; as a consequence, by the Rellich Theorem,(v_n)_nwould converge strongly in L^p(Q)to a nonzero function, contradicting (3.7). Therefore

|yn| −→_n→+∞+ ∞.

Let us, now, call v˜₀the weak limit, inH¹(R^N), of the sequencev˜_n(x):=v_n(x+y_n). Arguing as before in the hypercubeQcentred at the origin and having unitary sides, we conclude thatv˜₀≡0. Moreover, as a consequence of (3.4), (3.5),v˜0is a weak solution, on its domainD, of−u+u=c|u|^p−2uand, since|y_n| → +∞andΩ satis- fies(h2), we deduceD=R^N, when dist(yn,R^N\Ω)_n→+∞−→ + ∞,D=Ω(up to a translation) when dist(yn,R^N\Ω) is bounded.

Now, we claim that

⎧⎨

⎩

(a)u0=0,

(b)v˜0does not change sign, (c)v˜n_n−→_→+∞v˜0strongly inH₀¹(D).

(3.11) Equality (3.11)(a) follows by observing that (3.8) implies

u_n²Ωu₀²Ω+ ˜v₀²_D+o(1), (3.12)

thus, ifu0were not zero, taking into account thatv˜0≡0 and that Corollary 2.2 applies to bothu0andv˜0, we would infer

E(un)= un²Ω2m· m

c

2/(p−2)

+o(1) and, then

c2^1−2/pm contradicting (3.2).

Assertion (3.11)(b) is a direct consequence of Remark 2.4 and of the arguments of Corollary 2.3.

Let us prove, then, (3.11)(c). Let us assume, by contradiction, thatv˜_nv˜₀strongly. Then, settingw_n(x):=(v˜_n−

˜

v₀)(x),w_n(x) 0 weakly inH¹(R^N)and inL^p(R^N), andw_n(x)0 strongly inH¹(R^N). So, we can repeat step by step the argument before applied to(v_n)_n, concluding that a sequence of points(z_n)_n,z_n∈R^N,|z_n| −→_n→+∞+ ∞, and a nonzero function,w₀, exist such that

w_n(x):=w_n(x+z_n) w₀(x) weakly inH₀¹D ,

Dbeing eitherR^NorΩ, andw₀being a solution of−u+u=c|u|^p−2uinD. Furthermore the inequality w_n²_RN= w_n²_RN= u_n²_Ω− ˜v₀²_D+o(1)

holds, thus

u_n²Ω˜v₀²_D+ w₀²_D+o(1) and, then,

c2^1−2/pm

follows, contradicting (3.2) and giving (3.11)(c). Now, (3.11) and (3.9) imply|˜v0|p,D=1 and, by (3.11)(b), we can supposev˜₀0 onD. Hence, ifD=R^N, the uniqueness of the positive regular solutions to−u+u=c|u|^p−2u in R^N implies c=E(v˜₀)=m, contradicting (3.2). Therefore, D=Ω and 0<dist(yn,R^N \Ω) < H for some H∈R⁺\ {0}. Let us consider the sequence(h_n)_n=(h_n,1, h_n,2, . . . , h_n,k)∈Z^ksuch that

τ_ih_n,iy_n,i< τ_i(h_n,i+1), i=1,2, . . . , k, n∈N, and define

u^∗_n(x1, x2, . . . , xk, x):=un(x1+τ1hn,1, . . . , xk+τkhn,k, x).

The sequence(u^∗_n)_nisG-equivalent to(u_n)_n and converges strongly inH₀¹(Ω)to a functionu^∗that is a nontrivial critical point ofEonV. 2

4. Useful tools and basic estimates

For what follows we need to introduce a barycenter type function. For allu∈L^p(R^N)we set

˜

u(x)= 1

|B(x,1)|

B(x,1)

u(y)dy ∀x∈R^N,

|B(x,1)|denoting the Lebesgue measure ofB(x,1), and ˆ

u(x)=

˜ u(x)−1

2max

R^N u(x)˜ ₊

∀x∈R^N; we, then, defineβ:L^p(R^N)\ {0} →R^Nby

β(u)= 1

| ˆu|^p_p,RN

R^N

x ˆ u(x)p

dx. (4.1)

We point out that β is well defined for all u∈L^p(R^N)\ {0}, because uˆ≡0 and has compact support, thatβ is continuous and

(a)β

u(x−y)

=β u(x)

+y ∀u∈L^p R^N

\ {0}, ∀y∈Rⁿ, (b)β

ω(x)

=0. (4.2)

Remark 4.1.We stress the fact that the above barycenter map has been introduced some years ago by the first and the third author ([9], pages 265–266).

This map has been useful in many situations; in fact, it has been used also in [6,17,18] and recently, with a very slight modification, in [3] (actually, in [3] the definition of barycenter map is introduced as a new one and the definition given in [9] is not quoted).

We set, for allr∈R⁺,

D_r:= (x1, x2, . . . , x_k, x)∈R^N: |x|< r and

Br :=inf E(u): u∈V , β(u)∈D_r

, (4.3)

so, in particular,

D₀:= (x₁, x₂, . . . , x_k, x)∈R^N: x=0

and

B0:=inf E(u): u∈V , β(u)

= (β(u)

k+1, β(u)

k+2, . . . , β(u)

N)=0

. (4.4)

We remark that

B0Br ∀r >0. (4.5)

In what follows, for everyy=(y₁, y₂, . . . , y_k, y)∈R^N, we set z_y=(y₁, y₂, . . . , y_k,1,0, . . . ,0)∈R^N

and we denote byS,ΣandΛ, respectively, the sets S:= (x₁, x₂, . . . , x_k, x)∈R^N: |x| =2

, (4.6)

Σ:=S+z₀, (4.7)

Λ:= σ y+(1−σ )z_y: y∈Σ, σ∈ [0,1]

. (4.8)

For everyρ >0 we define the operator Ψ_ρ:Σ× [0,1] −→V

by

Ψ_ρ[y, σ](x)= ϕ(x)[(1−σ )ω(x−ρy)+σ ω(x−ρz_y)]

|ϕ(x)[(1−σ )ω(x−ρy)+σ ω(x−ρzy)]|p,Ω

, (4.9)

whereϕ∈C^∞(R^N,[0,1])is the cut-off function introduced in Proposition 2.1.

We note that

Ψ_ρ[y,0](x)= ϕ(x)ω(x−ρy)

|ϕ(x)ω(x−ρy)|p,Ω =ω_ρy(x). (4.10)

For everyρ >0 we consider, also, the map ξ_ρ:Σ× [0,1] −→ρΛ

defined by

ξ_ρ[y, σ] =ρ

(1−σ )y+σ z_y

. (4.11)

Proposition 4.2.LetBr be the numbers defined in(4.3). Then for allr∈R⁺, there existsμr∈R, such that

Brμr> m. (4.12)

Proof. Clearly, for allr0,Brm; to prove (4.12) we argue by contradiction and we assume thatBr_ˆ=mfor some ˆ

r0. Hence, a sequence(un)nmust exist such thatun∈V and (i)β(u_n)∈D_rˆ ∀n∈N,

(ii) limn→+∞E(u_n)=m. (4.13)

Then, by the uniqueness of the minimizers family of (2.1), a sequence of points(y_n)_n,y_n∈R^N, and a sequence of functions(χ_n)_n,χ_n∈H¹(R^N)exist so that, passing eventually to a subsequence, still denoted by(u_n)_n,

(i)un(x)=ω(x−yn)+χn(x) ∀x∈R^N, (ii) limn→+∞χn(x)=0 inH¹

R^N

and inL^p

R^N (4.14)

(see also [4] Lemma 3.1). Therefore, by (4.2), (4.14)(i)and the continuity ofβ, the relation β(un)−yn_n−→_→+∞0

holds and, together with (4.13)(i), implies that the sequence(y_n)_n is bounded. Hence, either the sequence(y_n)_n is bounded, or it is unbounded, but, in view of the assumption(h1), it can be replaced by an equivalent sequence, still

denoted by(yn)n, contained in a bounded set; so, passing eventually to a subsequence, we conclude thatyn_n→+∞−→ y.¯ Thus, either(u_n)_nor aG-equivalent sequence, still denoted by(u_n)_n, satisfies

n→+∞lim u_n(x)=ω(x− ¯y).

Then, by (4.13)(ii), m= lim

n→+∞E(u_n)=

Ω

∇ω(x− ¯y)²+

ω(x− ¯y)2 dx

that is impossible, becauseω(x)realizes (2.1),ω >0 inR^NandR^N\ Ω= ∅, so the statement follows. 2

Lemma 4.3.LetΣ, Ψ_ρ,B0be as defined, respectively, in(4.7), (4.9), (4.4). Then there existsρ˜∈Rsuch that for all ρρ˜

B0 max

Σ×[0,1]E

Ψ_ρ[y, σ]

. (4.15)

Proof. Taking into account (2.2), (4.2), (4.10), (2.5)(a), it is not difficult to verify that

ρ→+∞lim β◦Ψ_ρ[y,0] −ρy

R^N=0 ∀y∈Σ. (4.16)

Thus, forρ large enough, β◦Ψ_ρ(Σ× {0})is homotopically equivalent inR^N\D₀toρΣ and, then, there exists (y,ˆ σ )ˆ ∈Σ× [0,1]such that(β◦Ψ_ρ)[ ˆy,σˆ] ∈D₀, so

B0E

Ψ_ρ[ ˆy,σˆ]

max

Σ×[0,1]E

Ψ_ρ[y, σ] , as desired. 2

Corollary 4.4.LetΣ, Ψ_ρandρ˜as in Lemma4.3. LetBr,r∈R⁺, as defined(4.3). Then for allρρ˜ Br max

Σ×[0,1]E

Ψρ[y, σ]

. (4.17)

Proof. Inequality (4.17) is an immediate consequence of (4.15) and (4.5). 2

Next step is to establish some, crucial, asymptotic estimates on the energy ofΨ_ρ(Σ× {0})and ofΨ_ρ(Σ× [0,1]).

To this end, we need, first, to recall some known results:

Lemma 4.5.For alla, b∈R⁺, for allp2, the relation (a+b)^pa^p+b^p+(p−1)

a^p−1b+ab^p−1 holds true.

Lemma 4.6.Letg∈C(R^N)∩L^∞(R^N)andh∈C(R^N)be radially symmetric functions satisfying for someα0, b0,γ∈R

|x|→+∞lim g(x)exp α|x|

|x|^b=γ ,

R^N

h(x)exp α|x|

1+ |x|^b

dx <+∞. Then

|y|→+∞lim

R^N

g(x+y)h(x)dx

exp α|y|

|y|^b=γ

R^N

h(x)exp(−αx1)dx holds.