On a class of N/s-fractional Hardy–Schrödinger equations with singular exponential nonlinearity in RNdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {R}}^{N}$$end{document}

摘要

The aim of this paper is to study the existence of weak solutions for a class of Hardy–Schr?dinger-type equations driven by the fractional N/s-Laplace operator. More precisely we consider (-Δ)N/ssu+V(x)|u|Ns-2u=λh(x)+|u|Ns-2u|x|σ(N-s)s+f(x,u)|x|γinRN,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} (-Delta )_{N/s}^{s}u+V(x)|u|^{frac{N}{s}-2}u=lambda frac{ h(x) +|u|^{frac{N}{s}-2}u}{|x|^{frac{sigma (N-s)}{s}}}+ frac{f(x,u)}{|x|^{gamma } };;;text {in};;{mathbb {R}}^{N}, end{aligned}$$end{document}where (-Δ)N/ssdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(-Delta )_{N/s}^{s}$$end{document} is the fractional N/s-Laplace operator, N≥2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Nge 2$$end{document}, 00,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha >0, $$end{document}V:RN→Rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$V :{mathbb {R}}^{N}rightarrow {mathbb {R}}$$end{document} is the potential function changing signs over RNdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{o