Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities

TL;DR

This paper proves the existence of multiple positive solutions for a fractional elliptic problem involving Hardy potential and nonlinearities, using variational methods and Nehari manifold decomposition.

Contribution

It introduces a variational framework to establish multiple positive solutions for a nonlocal elliptic problem with Hardy potential and concave-convex nonlinearities.

Findings

At least two positive solutions exist for small parameter mbda.

Solutions are obtained under specific conditions on parameters and functions.

The approach extends variational methods to fractional problems with Hardy potential.

Abstract

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}= \lambda f(x) |u|^{q - 2} u + g(x) {\frac{|u|^{p-2}u}{|x|^s}} \ \text{ in } {\Omega,} \quad \text{ with Dirichlet boundary condition } u = 0 \ \text{ in } \mathbb{R}^n \setminus \Omega,$$ where $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain in $\mathbb{R}^n$ containing $0$ in its interior, and $f,g \in C(\overline{\Omega})$ with $f^+,g^+ \not\equiv 0$ which may change sign in $\overline{\Omega}.$ We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for $\lambda$ sufficiently small. The variational approach requires that $0 < \alpha <2,$ $ 0 <s < \alpha <n,$ $ 1<q<2<p \le 2_{\alpha}^*(s):= \frac{2(n-s)}{n-\alpha},$ and $ \gamma < \gamma_H(\alpha) ,$ the latter being the best fractional Hardy constant on $\mathbb{R}^n.$