Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well

Abstract

In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well $ \begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*} $ where $ a $ and $ b $ are positive constants, $ \lambda > 0 $ is a parameter, $ 1 < q < 2 $, the potential $ V\in C(\mathbb{R}^3, \mathbb{R}) $ and $ V^{-1}(0) $ has a nonempty interior. The functions $ f $ and $ g $ are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as $ \lambda\rightarrow \infty $ is also explored.

Keywords

• kirchhoff-type problem
• steep potential well
• variational methods