Local mountain-pass for a class of elliptic problems in RN involving critical growth

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Introduction

In this paper we shall be concerned with the existence and the concentration behavior of positive bound-state solutions (solutions with bounded energy) for the problem−ε2Δu+V(z)u=f(u)+u2−1,inRN,u∈C2(RN)∩H1(RN),u(z)>0forallz∈RN,where ε>0; 2=2N/(N−2), N≥3, is the critical Sobolev exponent; V:RNR is a locally Hölder continuous function satisfyingV(z)≥α>0forallz∈RN,andinfΩV<infΩVfor some bounded domain ΩRN; and the nonlinearity f:R+R is locally Lipschitz and such that
f(s)≥λsq1foralls>0andlims→∞f(s)sq2=0(when N=3, we need q1>2, otherwise we require a sufficiently large λ);0<θF(s)≤f(s)sforalls>0,where F(s)=0sf(t)dt;

Since we are interested in positive solutions, we define f(s)=0 for s<0.

Let us state our main result:

Theorem 1

Suppose that the potential V satisfies (V)(V∗∗) and f satisfies (f1)–(f4). Then there is an ε0>0 such that problem (Pε) possesses a positive bound state solution uε, for all 0<ε<ε0. Moreover, uε possesses at most one local (hence global) maximum zε in RN, which is inside Ω, such thatlimε→0+V(zε)=V0=infΩV.Besides, there are C and ζ, positive constants such thatuε(x)≤Cexp−ζx−zεεforallx∈RN.

We would like to remark that this kind of equation in (Pε) arises from the problem of obtaining standing waves solutions of the nonlinear Schrödinger equationiε∂ψ∂t=−ε2Δψ+(V(z)+E)ψ−|ψ|−1h(|ψ|)ψinRNwhere h(s)=f(s)+s2−1. A standing wave solution to problem (Sε) is one in the form ψ(x,t)=exp(−iε−1Et)u(x). In this case u is a solution of (Pε).

Some recent works have treated this problem in the subcritical case and we cite a couple of them.

Floer and Weinstein [7] have studied the problem (Sε) in the case N=1,h(s)=s3 and bounded potentials with nondegenerate critical point. They show that for small ε, this problem has a solution which concentrates around each nondegenerate point (see also [18] for a related work).

Roughly speaking, in this work, concentration behavior around the origin of a function means that it has the form ψ(εx) where ψ is a C2 function with exponential decay.

Under the potential conditioninfz∈RNV<liminf|z|→∞V(z),Rabinowitz [14] has proved the existence of positive ground-state solutions (solutions with minimal energy) for (Pε), in the case where h(s) behaves like sp,1<p<2−1, for small ε. The remaining concentration behavior result in this subcritical case was obtained by Wang [17].

Alves and Souto in [2] have established existence and concentration behavior of ground-state solutions when the nonlinearity has the critical form h(s)=λsq+s2−1,1<q<2−1, under condition (1).

In the very interesting article [5], del Pino and Felmer have obtained the complete treatment (existence and concentration behavior of solutions) with the potential under conditions , . They have obtained bound-state solutions but not ground-state solutions, and this is reasonable, because some problems under condition (V∗∗) do not admit any ground-state solution (see, for example, Theorem 4 of [2]). From this reason we cannot look for minimax critical points of the energy functional Jε:E→R,Jε(u)=12RN2|u|2+V(z)u2)dz−RNF(u+)dz−12RNu+2dz(where u+=max{u,0}), defined on the Hilbert spaceE=u∈H1(RN):RNV(z)u2dz<∞,associated to (Pε).

A way to solve this problem is to modify the nonlinearity into one more convenient in order to apply the mountain-pass theorem. Namely, we will consider the following Carathéodory function:g(z,s)=χΩ(f(s)+s2−1)+χDf̃(s)ifs≥0,0ifs<0,wheref̃(s)=f(s)+s2−1ifs≤a,αskifs>a,k>θ(θ−2)−1>1,a>0 is such that f(a)+a2−1=k−1aα,D=RNΩ̄ and χA denotes the characteristic function of subset A of RN. Similar modified nonlinearity has been used by del Pino and Felmer in [5] to study the subcritical case.

We have organized the paper as follows. In the second section, using the mountain-pass theorem, we shall prove that the functional associated to the modified problem possesses a critical point. In the following section we shall prove Theorem 1 by a couple of lemmas which show that for small ε this critical point of the modified functional has a concentration behavior and it is also critical point of functional Jε.

Section snippets

The modified functional

In this section we will consider the energy functional J:E→RN given byJ(u)=12||u||2RNG(z,u)dz,where||u||2=RN(|u|2+V(z)u2)dz.Here G(z,s)=χΩ(F(s)+12s2)+χDF̃(s) and F̃(s)=0sf̃(t)dt.

Notice that, using (f1)–(f4), it is easy to check that

(g1)g(z,s)=f(s)+s2−1=o(s), near the origin, uniformly in z∈RN;
(g2)g(z,s)≤f(s)+s2−1 for all s>0,z∈RN;
(g3)
0<θG(z,s)≤g(z,s)s,forallz∈Ω,s>0orz∈Dands≤a;and0≤2G(z,s)≤g(z,s)s≤1kV(z)s2forallz∈D,s>0,
(g4)the function s−1g(z,s) is increasing in s>0 for each z fixed.

Lemma 2

J

Proof of Theorem 1

In order to proof Theorem 1, let us fix some notations. First we suppose, without loss of generality that Ω is smooth and 0∈Ω. Furthermore,V(0)=V0=infΩV.We will denote by I0:H1(RN)→R the functional given byI0(u)=12RN(|u|2+V0u2)dx−RNF(u)+12(u+)2dx,associated to the autonomous problemΔu+V0u=f(u)+|u|2−2u,inRN.

It is known that under assumptions (f1)–(f4), (12) possesses a ground-state solution ω at the levelc0=I0(ω)=infv∈H1⧹{0}maxt≥0I0(tv)(see [1] for instance). Furthermore,c0<1NSN/2.

Remark 2

The

Acknowledgements

This work was done while the second author was visiting the Courant Institute of Mathematical Sciences. He thanks Professor Louis Nirenberg and all the faculty and staff of CIMS-NYU for their kind hospitality.

This work was done during the third author's visit to the Department of Mathematics at Rutgers University. He thanks Professor Yanyan Li for his kind hospitality.

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Partially supported by CNPq/Brazil.

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