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

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 $\mathbb{R}^n.$