Multiplicity and concentration results for some nonlinear Schr\"odinger equations with the fractional $p$-Laplacian

TL;DR

This paper investigates the existence, multiplicity, and concentration of positive solutions to a class of nonlinear Schrödinger equations involving the fractional p-Laplacian, using variational methods and topological tools.

Contribution

It introduces new results on solution multiplicity and concentration for fractional p-Laplacian Schrödinger equations with variable potentials.

Findings

Existence of multiple positive solutions for small parameters.

Solutions concentrate around minima of the potential.

Results cover subcritical and critical nonlinear growth cases.

Abstract

We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.