Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities

TL;DR

This paper develops a linking-type theorem to analyze strongly indefinite problems with sign-changing nonlinearities and applies it to a singular Schrödinger equation, establishing existence results for solutions.

Contribution

It introduces a new abstract linking-type theorem for indefinite problems and applies it to complex Schrödinger and curl-curl equations.

Findings

Established existence of solutions for the singular Schrödinger equation.

Extended the theory to nonlinear curl-curl problems.

Provided conditions under which solutions exist.

Abstract

We show the linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schr\"{o}dinger equation $$ -\Delta u + V(x)u + \frac{a}{r^2} u = f(u) - \lambda g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \ r = |y|, $$ where $$ 0 \not\in \sigma \left( -\Delta + \frac{a}{r^2} + V(x) \right). $$ As a consequence we obtain also the existence of solutions to the nonlinear curl-curl problem.