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# 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

#### 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.

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