New multiplicity of positive solutions for some class of nonlocal problems

Abstract

In this paper, we study the following nonlocal problem: {−(a−b∫Ω|∇u|2dx)Δu=λ|u|q−2u,x∈Ω,u=0,x∈∂Ω,\\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}$$ \\textstyle\\begin{cases} - (a-b \\int _{\\Omega } \\vert \\nabla u \\vert ^{2}\\,dx ) \\Delta u= \\lambda \\vert u \\vert ^{q-2}u, & x\\in \\Omega , \\\\ u=0, & x\\in \\partial \\Omega , \\end{cases} $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{N} with Nge 3, a,b>0, 1< q<2 and lambda >0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as bsearrow 0.