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

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.