Communications in Analysis and Geometry

Volume 30 (2022)

Number 9

Existence and multiplicity of solutions for a class of indefinite variational problems

Pages: 1933 – 1954

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n9.a1

Authors

Claudianor O. Alves (Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, Campina Grande, PB, Brazil)

Minbo Yang (School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China)

Abstract

In this paper we study the existence and multiplicity of solutions for the following class of strongly indefinite problems\[(P)_k \qquad\begin{cases}-\Delta u + V(x)u=A(x/k)f(u) \; \textrm{in} \; \mathbb{R}^N, \\u ∈ H^1(\mathbb{R}^N),\end{cases}\]where $N \geq 1$, $k \in \mathbb{N}$ is a positive parameter, $f : \mathbb{R } \to \mathbb{R}$ is a continuous function with subcritical growth, and $V, A : \mathbb{R} \to \mathbb{R}$ are continuous functions verifying some technical conditions. Assuming that $V$ is a $\mathbb{Z}^N$-periodic function, $0 \notin \sigma (-\Delta+V)$ the spectrum of $(-\Delta+V)$, we show how the ”shape” of the graph of function $A$ affects the number of nontrivial solutions.

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C. O. Alves was partially supported by CNPq/Brazil CNPq/Brazil 307045/2021-8 and Projeto Universal FAPESQ-PB 3031/2021.

Minbo Yang was partially supported by NSFC(11971436, 12011530199) and ZJNSF(LZ22A010001, LD19A010001).

Received 27 January 2019

Accepted 15 April 2020

Published 17 August 2023