Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity

Abstract

We study the following class of elliptic equations: \begin{equation*} -\bigg(a+b\int_{{\mathbb R}^3}|\nabla u|^2\,dx\bigg)\Delta u+\lambda V(x)u=f(u), \quad x\in{\mathbb R}^3, \end{equation*} where $\lambda,a,b>0$, $V\in \mathcal{C}({\mathbb R}^3,{\mathbb R})$ and $V^{-1}(0)$ has nonempty interior. First, we obtain one ground state sign-changing solution $u_{b,\lambda}$ applying the non-Nehari manifold method. We show that the energy of $u_{b,\lambda}$ is strictly larger than twice that of the ground state solutions of Nehari-type. Next we establish the convergence property of $u_{b,\lambda}$ as $b\searrow0$. Finally, the concentration of $u_{b,\lambda}$ is explored on the set $V^{-1}(0)$ as $\lambda\rightarrow\infty$.