Spectral Theory Approach for a Class of Radial Indefinite Variational Problems

TL;DR

This paper develops a spectral theory approach to find nontrivial radial solutions for indefinite nonlinear Schrödinger equations using variational methods and spectral properties of associated operators.

Contribution

It introduces a spectral theory framework combined with variational methods to address indefinite radial Schrödinger problems, leveraging the properties of radially symmetric functions.

Findings

Established existence of nontrivial radial solutions.

Applied spectral analysis to indefinite variational problems.

Utilized compactness properties of radial functions for critical point theory.

Abstract

Considering the radial nonlinear Schrodinger equation - \Delta u + V(x)u = g(x,u) in R^N, N \geq 3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring this problem is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating to the problem an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator A:= - \Delta + V(x) restricted to H^1_{rad}(R^N), the space of radially symmetric functions in H^1(R^N), for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.