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

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}u-\gamma \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}=\lambda f(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+g(x)\frac{\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^s}\text{ in }\mathrm{\Omega },\text{ with Dirichlet boundary condition }u=0\text{ in }{\mathbb{R}}^n\setminus \mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a smooth bounded domain in ${\mathbb{R}}^n$ containing $0$ in its interior, and $f,g\in C(\bar{\mathrm{\Omega }})$ with $f^{+},g^{+}\equiv ̸0$ which may change sign in $\bar{\mathrm{\Omega }}\mathrm{.}$ 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 $0<\alpha <2,$ $0<s<\alpha <n,$ $1<q<2<p\le 2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha },$ and $ \gamma < \gamma_H(\alpha) ,$thelatterbeingthebestfractionalHardyconstanton$$\mathbb{R}$^n.$