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

In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well

$$\begin{array}{r}\{\begin{array}{l}-(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^{2}dx)\mathrm{\Delta }u+\lambda V(x)u=f(x,u)+g(x)|u{|}^{q-2}u\text{in}{\mathbb{R}}^3,\\ \\ u\in H^1({\mathbb{R}}^3),\end{array}\end{array}$$

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 $\lambda \to \mathrm{\infty }$ is also explored.