High Energy Solutions for p-Kirchhoff Elliptic Problems with Hardy–Littlewood–Sobolev Nonlinearity

Abstract

This article deals with the study of the following Kirchhoff–Choquard problem: M∫RN|∇u|p(-Δp)u+V(x)|u|p-2u=∫RNF(u)(y)|x-y|μdyf(u),inRN,u>0,inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\begin{array}{cc} \\displaystyle M\\left( \\, \\int \\limits _{{\\mathbb {R}}^N}|\ abla u|^p\\right) (-\\Delta _p) u + V(x)|u|^{p-2}u = \\left( \\, \\int \\limits _{{\\mathbb {R}}^N}\\frac{F(u)(y)}{|x-y|^{\\mu }}\\,dy \\right) f(u), \\;\\;\ ext {in} \\; {\\mathbb {R}}^N,\\\\ u > 0, \\;\\; \ ext {in} \\; {\\mathbb {R}}^N, \\end{array} \\end{aligned}$$\\end{document}where M models Kirchhoff-type nonlinear term of the form M(t)=a+btθ-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(t) = a + bt^{\ heta -1}$$\\end{document}, where a,b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a, b > 0$$\\end{document} are given constants; 1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<N$$\\end{document}, Δp=div(|∇u|p-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _p = \ ext {div}(|\ abla u|^{p-2}\ abla u)$$\\end{document} is the p-Laplacian operator; potential V∈C2(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V \\in C^2({\\mathbb {R}}^N)$$\\end{document}; f is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for θ∈1,2N-μN-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in \\left[ 1, \\frac{2N-\\mu }{N-p}\\right) $$\\end{document} via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.