Local mountain-pass for a class of elliptic problems in involving critical growth
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 problemwhere ε>0; N≥3, is the critical Sobolev exponent; is a locally Hölder continuous function satisfyingandfor some bounded domain ; and the nonlinearity is locally Lipschitz and such that
(when N=3, we need q1>2, otherwise we require a sufficiently large λ);where ;
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 – 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 which is inside such thatBesides, there are C and ζ, positive constants such that
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 equationwhere . 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 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 conditionRabinowitz [14] has proved the existence of positive ground-state solutions (solutions with minimal energy) for (Pε), in the case where h(s) behaves like , 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 , 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 ,(where u+=max{u,0}), defined on the Hilbert spaceassociated 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:where is such that and χA denotes the characteristic function of subset A of . 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 given bywhereHere and .
Notice that, using (f1)–(f4), it is easy to check that Lemma 2 J, near the origin, uniformly in ; for all ; the function is increasing in for each z fixed.
Proof of Theorem 1
In order to proof Theorem 1, let us fix some notations. First we suppose, without loss of generality that is smooth and . Furthermore,We will denote by the functional given byassociated to the autonomous problem
It is known that under assumptions (f1)–(f4), (12) possesses a ground-state solution ω at the level(see [1] for instance). Furthermore, 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.
References (18)
- et al.
Nonspreading wave packets for the cubic Schrödinger equations with bounded potential
J. Funct. Anal.
(1986) The concentration–compactness principle in the calculus of variations, the locally compact case, Part II
Ann. Inst. H. Poincaré Anal. Non Linéaire
(1984)On a class of semilinear elliptic problem in with critical growth
Nonlinear Anal.
(1997)- et al.
Radially symmetric solutions for a class of critical exponent elliptic problems in
Electron. J. Differential Equations
(1996) - C.O. Alves, M.A. Souto, On existence and concentration behavior of ground state solutions for a class of problems with...
- et al.
Remarks on the Schrödinger operator with regular complex potentials
J. Math. Pures Appl.
(1979) - et al.
Homoclinic type solutions for a semilinear elliptic PDE on
Comm. Pure Appl. Math.
(1992) - et al.
Local Mountain Pass for semilinear elliptic problems in unbounded domains
Calc. Var.
(1996) - et al.
On the existence of positive entire solutions of a semilinear elliptic equation
Arch. Ration. Mech. Anal.
(1986)
Cited by (0)
- 1
Partially supported by CNPq/Brazil.