Multiple positive solutions to Kirchhoff equations with competing potential functions in mathbb{R}^{3}

Abstract

In this paper, we study the existence of multiple positive solutions to the following Kirchhoff equation with competing potential functions: \t\t\t{−(ε2a+εb∫R3|∇v|2)Δv+V(x)v=K(x)|v|p−1vin R3,v>0,v∈H1(R3),\\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}$$ \\textstyle\\begin{cases} -(\\varepsilon ^{2}a+\\varepsilon b{\\int _{\\mathbb{R}^{3}}}{ \\vert \\nabla v \\vert } ^{2})\\Delta v+V(x)v=K(x) \\vert v \\vert ^{p-1}v \\quad \\mbox{in }\\mathbb{R}^{3}, \\\\ v>0, \\quad v\\in H^{1}(\\mathbb{R}^{3}), \\end{cases} $$\\end{document} where varepsilon >0 is a small parameter, a,b>0 are constants, 3< p<5. We relate the number of solutions with the topology of the global minima set of the function {V^{frac{2}{p-1}-frac{1}{2}}(x)}/ {K^{frac{2}{p-1}}(x)}. The Nehari manifold and Ljusternik–Schnirelmann category theory are applied in our study.