Advances in Difference Equations (Oct 2018)

Existence of nontrivial solution for a nonlocal problem with subcritical nonlinearity

  • Jing Zhang,
  • Zhongyi Zhang

DOI
https://doi.org/10.1186/s13662-018-1823-4
Journal volume & issue
Vol. 2018, no. 1
pp. 1 – 8

Abstract

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Abstract In this paper, we consider the following new nonlocal Dirichlet boundary value problem: 0.1 {−(a−b∫Ω|∇u|2dx)Δu=λu+g(x,u),x∈Ω,u=0,x∈∂Ω, $$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=\lambda u+g(x,u),& x\in \Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$ where a and b are positive, λ is a positive parameter, 0≤λ<aλ1 $0\leq\lambda< a\lambda_{1}$, λ1 $\lambda_{1}$ is the first eigenvalue of operator −Δ. Under appropriate assumptions on the function g which is of subcritical growth, we obtain a nontrivial solution.

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