Abstract
In this paper, we consider the following new nonlocal problem: {−(a−b∫Ω|∇u|2dx)Δu=λf(x)|u|p−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}$$\\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} - (a-b\\int_{\\varOmega} \\vert \\nabla u \\vert ^{2}\\,dx )\\Delta u=\\lambda f(x) \\vert u \\vert ^{p-2}u, & x\\in\\varOmega,\\\\u=0, & x\\in\\partial\\varOmega, \\end{array}\\displaystyle \\right . $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{3}, a,b>0 are constants, 3< p<6, and the parameter lambda>0. Under some assumptions on the sign-changing function f, we obtain the existence of positive solutions via variational methods.
Highlights
Under some assumptions on the sign-changing function f, we obtain the existence of positive solutions via variational methods
1 Introduction and main resluts In this paper, we are concerned with the existence of positive solutions for the following new nonlocal problem:
U = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in R3, a, b > 0 are constants, 3 < p < 6, and the parameter λ > 0. f (x) is sign changing in Ω, which is the reason why we call it indefinite nonlinearity in the title
Summary
Under some assumptions on the sign-changing function f , we obtain the existence of positive solutions via variational methods. 1 Introduction and main resluts In this paper, we are concerned with the existence of positive solutions for the following new nonlocal problem: (2020) 2020:40 existence and multiplicity of positive solutions to the Kirchhoff problem with indefinite nonlinearity by using mountain pass theorem and minimization argument.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
