A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schr\"odinger equation in $\mathbb R^{N}$

TL;DR

This paper proves the existence and multiplicity of positive solutions for a fractional Schrödinger equation in rom variational methods, using topological tools like Ljusternick-Schnirelmann and Morse theory to relate solutions to the topology of the potential's minimum set.

Contribution

It introduces a novel application of Ljusternick-Schnirelmann and Morse theory to establish multiple solutions for fractional Schrödinger equations with variable potentials.

Findings

Multiple positive solutions exist as  approaches zero.

Solutions concentrate near the minimum set of the potential.

Topological complexity of the minimum set determines the number of solutions.

Abstract

In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-\Delta)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$ where $0<s<1$, $(-\Delta)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions and the nonlinearity $f:\mathbb R \to \mathbb R$ satisfy suitable assumptions; in particular it is assumed that $V$ achieves its positive minimum on some set $M.$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\varepsilon\to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set $M$.