Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

Abstract

We are interested in the following Dirichlet problem: -Δu+λu-μu|x|2-νudist(x,RN\\Ω)2=f(x,u)inΩu=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta u + \\lambda u - \\mu \\frac{u}{|x|^2} - \ u \\frac{u}{\ extrm{dist}(x,\\mathbb {R}^N \\setminus \\Omega )^2} = f(x,u) &{} \\quad \ ext{ in } \\Omega \\\\ u = 0 &{} \\quad \ ext{ on } \\partial \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}on a bounded domain Ω⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^N$$\\end{document} with 0∈Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\in \\Omega $$\\end{document}. We assume that the nonlinear part is superlinear on some closed subset K⊂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K \\subset \\Omega $$\\end{document} and asymptotically linear on Ω\\K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\setminus K$$\\end{document}. We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u. Moreover, we study also the multiplicity of solutions to the associated normalized problem.