Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps

Cerami, Giovanna; Passaseo, Donato; Solimini, Sergio

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 23-40

In this paper we consider the equation

$$\left(E\right)-\Delta u+a\left(x\right)u={\left|u\right|}^{p-1}u\text{in}{ℝ}^N,$$

where $N⩾2$, $p>1$, $p<2^{⁎}-1=\frac{N+2}{N-2}$, if $N⩾3$. During last thirty years the question of the existence and multiplicity of solutions to (E) has been widely investigated mostly under symmetry assumptions on a. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in [6] admit a limit configuration and so (E) also admits a positive solution of infinite energy having infinitely many ‘bumps’.

Dans ce papier nous considérons l'équation

$$\left(E\right)-\Delta u+a\left(x\right)u={\left|u\right|}^{p-1}u\text{en}{ℝ}^N,$$

où $N⩾2$, $p>1$, $p<2^{⁎}-1=\frac{N+2}{N-2}$, si $N⩾3$. Pendant les trente dernières années la question de l'existence et de la multiplicité de solutions d'(E) a été largement examinée surtout conformément aux suppositions de symétrie sur a. Le but de ce papier est de montrer que, différemment de ceux trouvés conformément à la supposition de symétrie, les solutions trouvées dans [6] admettent une configuration de limite et donc (E) admet aussi une solution positive d'énergie infinie ayant une infinité de ‘bumps’.

Classification : 35J20, 35J60, 35Q55
Keywords: Variational methods, Solutions with infinitely many bumps, Schrödinger equation

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