Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

Claudianor O. Alves; Geilson F. Germano

doi:10.3934/cpaa.2020126

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2020, Volume 19, Issue 5: 2887-2906. Doi: 10.3934/cpaa.2020126

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Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

Claudianor O. Alves^, and
Geilson F. Germano

Universidade Federal de Campina Grande, Unidade Acdêmica de Matemática, CEP: 58429-900, Campina Grande - Pb, Brazil

^* Corresponding author
^* Corresponding author

Received: June 30, 2019

Revised: October 31, 2019

Published: March 2020

Claudianor O. Alves was partially supported by CNPq/Brazil 304804/2017-7

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Abstract

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like

$ \begin{cases} -\Delta u+V(x)u = A(\epsilon x)f(u) \quad \mbox{in} \quad \mathbb{R}^{N}, \\ u\in H^{1}( \mathbb{R}^{N}), \end{cases} \qquad\qquad\qquad{(P)_{\epsilon}} $

where $ N \geq 1 $, $ \epsilon $ is a positive parameter, $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function with subcritical growth and $ V,A: \mathbb{R}^{N} \to \mathbb{R} $ are continuous functions verifying some technical conditions. Here $ V $ is a $ \mathbb{Z}^N $-periodic function, $ 0 \not\in \sigma(-\Delta + V) $, the spectrum of $ -\Delta +V $, and

$ 0 < \inf\limits_{x \in \mathbb{R}^{N}}A(x)\leq \lim\limits_{|x|\rightarrow+\infty}A(x)<\sup\limits_{x \in \mathbb{R}^{N}}A(x). $

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Mathematics Subject Classification: Primary: 35B40, 35J20; Secondary: 47A10.

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