Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions

TL;DR

This paper establishes the existence and multiplicity of positive solutions for a class of nonlinear fractional elliptic equations with sign-changing weights, using variational methods and topological tools.

Contribution

It introduces new multiplicity results for fractional elliptic equations involving sign-changing weights, employing Nehari manifold decomposition and Ljusternik-Schnirelmann category.

Findings

Proved existence of multiple positive solutions.

Developed a framework for handling sign-changing weights.

Applied variational methods to fractional elliptic equations.

Abstract

In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u= a_\lambda(x)|u|^{q-2}u+b(x)|u|^{2^*_\alpha-1}u &{\rm in}\,\,\Omega, u=0\,\,&{\rm in}\,\,\R^N\setminus\Omega, \end{array} \right. \end{eqnarray*} where $0<\alpha<1$, $ \Omega $ is a bounded domain with smooth boundary in $ \R^N $ with $N>2\alpha$ and $ 2^*_\alpha=2N/(N-2\alpha)$ is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Ljusternik-Schnirelmann category.