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 $0 1$ and $f\in L^{(p+1)/p}_g(\Omega)$. 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.