An Improved Fountain Theorem and its Application

TL;DR

This paper introduces an enhanced fountain theorem that relaxes previous assumptions, enabling the analysis of more complex elliptic problems with indefinite and sign-changing nonlinearities, leading to new solutions for Schrödinger equations.

Contribution

The paper presents a new fountain theorem without the $ au$-upper semicontinuity condition, broadening the scope of variational methods for indefinite elliptic problems.

Findings

Established infinitely many solutions for a semilinear Schrödinger equation

Extended the applicability of fountain theorems to more general problems

Provided a new approach for problems with sign-changing nonlinearities

Abstract

The main aim of the paper is to prove a fountain theorem without assuming the $\tau$-upper semicontinuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schr\"odinger equation with strongly indefinite structure and sign-changing nonlinearity.