Lower order term for the fractional Laplacian with a Hardy potential

Abstract

This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta {)}^{s}u+g|u{|}^{p-1}u=\lambda \frac{u}{|x{|}^{2s}}+f(x),$$ in a bounded domain $\Omega$ of ${\mathbb{R}}^N(N>2s)$, subject to the zero Dirichlet condition in ${\mathbb{R}}^N\setminus \Omega$, where $0<g\in L_{loc}^1(\Omega )$, $p>1$ and $f\in L_g^{(p+1)/p}(\Omega )$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda \in \mathbb{R}$. Moreover, the regularity of solution is also obtained.