Schr\"odinger-Maxwell equations driven by mixed local-nonlocal operators

TL;DR

This paper proves the existence of solutions to Schrödinger-Maxwell systems involving mixed local and nonlocal operators, including models with nonpositive definite nonlocal parts, and identifies parameter ranges for solitary wave solutions.

Contribution

It introduces new existence results for Schrödinger-Maxwell equations with mixed operators, allowing nonpositive definite nonlocal components and analyzing parameter conditions for solitary waves.

Findings

Existence of solutions for mixed local-nonlocal Schrödinger-Maxwell systems.

Identification of parameter ranges for solitary standing waves.

Application of Mountain Pass theorem to obtain critical points.

Abstract

In this paper we prove existence of solutions to Schr\"odinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schr\"odinger-Maxwell equations and Schr\"odinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals.