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Ground state solutions of the Kirchhoff equation with linearly bounded nonlinearities
  • Yanyan Liu,
  • Leiga Zhao
Yanyan Liu
Beijing Technology and Business University
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Leiga Zhao
Beijing Technology and Business University

Corresponding Author:[email protected]

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Abstract

In this paper, we consider the following Schr\”{o}dinger-Kirchhoff equation $$ -\left(a+b \di_{\R^N} |\nabla u|^2 \ d x \right)\Delta u+V(x)u=f(x,u), \ \hbox{in}\ \mathbb{R}^N, $$ where $N\geq 3$, $a$ and $b$ are positive parameters. Under suitable assumptions on $V$ and $f$ which is linearly bounded at infinity, the existence of ground state solutions and their asymptotic behavior as $b\to 0$ are established via variational methods. The nonexistence of nontrivial solutions is also obtained for large $b$.