Positive solutions for the fractional Schr\"{o}dinger equations with logarithmic and critical nonlinearities

TL;DR

This paper investigates positive solutions for fractional Schrödinger equations with logarithmic and critical nonlinearities, demonstrating existence, multiplicity, and the influence of weight potentials using variational methods.

Contribution

It introduces a novel analysis of how weight potentials affect the number of positive solutions in fractional Schrödinger equations with complex nonlinearities.

Findings

Existence of at least one positive ground state solution.

The energy associated with solutions can be positive or negative.

The number of solutions relates to the category of certain sets influenced by the potential.

Abstract

In this paper, we study a class of fractional Schr\"{o}dinger equations involving logarithmic and critical nonlinearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik-Schnirelmann category, we deeply investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.