Multiplicity of Solutions for an Elliptic Kirchhoff Equation

Abstract

In this paper we study the existence of positive solution to the Kirchhoff elliptic problem -1+γG′‖∇u‖L2(Ω)2Δu=λf(u)inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle -\\left( 1+\\gamma G'\\left( \\Vert \ abla u\\Vert ^2_{L^2(\\Omega )}\\right) \\right) \\Delta u = \\lambda f(u) &{} \ ext{ in } \\; \\Omega ,\\\\ u = 0 &{} \ ext{ on } \\; \\partial \\Omega ,\\\\ \\end{array}\\right. } \\end{aligned}$$\\end{document}where Omega is an open, bounded subset of mathbb {R}^N (Nge 3), f is a locally Lipschitz continuous real function, f(0)ge 0, G'in C(mathbb {R}^+) and G'ge 0. We prove the existence of at least two solutions with L^infty (Omega ) norm between two consecutive zeroes of f for large lambda .