Combined effects of the Hardy potential and lower order terms in fractional Laplacian equations

Abstract

In this paper we consider the existence and regularity of solutions to the following nonlocal Dirichlet problems: \t\t\t{(−Δ)su−λu|x|2s+up=f(x),x∈Ω,u>0,x∈Ω,u=0,x∈RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta)^{s} u-\\lambda\\frac{u}{|x|^{2s}}+u^{p}=f(x), &x\\in\\Omega, \\\\ u>0, &x\\in\\Omega, \\\\ u=0, & x\\in\\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian operator, sin(0,1), Omegasubsetmathbb{R}^{N} is a bounded domain with Lipschitz boundary such that 0inOmega, f is a nonnegative function that belongs to a suitable Lebesgue space.