Abstract
In this paper, we study the following nonlinear Kirchhoff type equation: −(a+b∫RN|∇u|2dx)△u+Vu=f(u)+h(x),x∈RN,\\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} $$\\begin{aligned} - \\biggl(a+b \\int_{\\mathbb{R}^{N}} \\vert \\nabla u \\vert ^{2}\\,dx \\biggr) \\triangle u+Vu=f(u)+h(x),\\quad x\\in\\mathbb{R}^{N}, \\end{aligned}$$ \\end{document} where a, b, V are positive constants, N=2 or 3. Under appropriate assumptions on f and h, we get that the equation has two positive solutions by using variational methods.
Highlights
Let us defineBy (3.1), the Hölder inequality and the Sobolev inequality, for all μ ∈ J and u ∈ E, one has
Introduction and main resultsWe consider the following nonlinear Kirchhoff type equation:– a + b |∇u|2 dx u + Vu = f (u) + h(x), x ∈ RN, (1.1)where a, b, V are positive constants, N = 2 or 3
The nonlinearity f satisfies 3-superlinear growth at infinity, which assures the boundedness of any Palais-Smale sequence or Cerami sequence
Summary
By (3.1), the Hölder inequality and the Sobolev inequality, for all μ ∈ J and u ∈ E, one has
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