Existence of solution for a class of indefinite variational problems with discontinuous nonlinearity

TL;DR

This paper proves the existence of nontrivial solutions for a class of indefinite variational problems involving discontinuous nonlinearities, using variational methods and generalized gradients.

Contribution

It establishes existence results for solutions to indefinite variational problems with discontinuous nonlinearities, extending previous work to include generalized gradients and periodic potentials.

Findings

Existence of solutions under certain spectral conditions.

Application of variational methods to discontinuous nonlinearities.

Extension to problems with periodic potentials.

Abstract

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a $\mathbb{Z}^{N}$-periodic Caratheodory function and $\lambda=0$ does not belong to the spectrum of $-\Delta+V$. Here, $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$.