Existence and multiplicity results for a new Kirchhoff type problem with critical exponent and indefinite potential
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We study the following new Kirchhoff type problem \[\left\{\begin{array}{lll} -\left(a-b\di_{\Omega}|\n u|^2dx\right)\Delta u=\ld |u|^{q-2}u+Q(x)|u|^{2}u, & x\in\Omega,\\[10pt] u=0, & x\in\partial\Omega, \end{array}\right.\] where $a, b>0$, $1<q\le 2$, $\ld>0$ is a parameter, $\Omega$ is a smooth bounded domain in $\R^4$, the potential $Q(x)$ has indefinite sign and satisfies some assumptions. When $q=2$, the existence of a ground state solution is obtained. When $1<q<2$, we prove that the problem has two nontrivial solutions and one of them is a ground state solution. Moreover, some asymptotic behaviors of these ground state solutions are also discussed in this paper. In the proofs we use variational methods.
Abstract
We study the following new Kirchhoff type problem \[\left\{\begin{array}{lll} -\left(a-b\di_{\Omega}|\n u|^2dx\right)\Delta u=\ld |u|^{q-2}u+Q(x)|u|^{2}u, & x\in\Omega,\\[10pt] u=0, & x\in\partial\Omega, \end{array}\right.\] where $a, b>0$, $1<q\le 2$, $\ld>0$ is a parameter, $\Omega$ is a smooth bounded domain in $\R^4$, the potential $Q(x)$ has indefinite sign and satisfies some assumptions. When $q=2$, the existence of a ground state solution is obtained. When $1<q<2$, we prove that the problem has two nontrivial solutions and one of them is a ground state solution. Moreover, some asymptotic behaviors of these ground state solutions are also discussed in this paper. In the proofs we use variational methods.