Existence of positive solutions for Kirchhoff-type problem in exterior domains

Abstract

We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$ : (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align} where a > 0, $b\geq0$ , and λ > 0 are constants, $\partial\Omega\neq\emptyset$ , $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$ , and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*} has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.