Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well

Abstract

<p style='text-indent:20px;'>In the present paper, we investigate a class of nonlinear Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V_\lambda(x)u+\mu \phi u = f(u) \quad \quad \ {\rm{in}} \ {\mathbb{R}}^{3},\\ -\Delta \phi = u^2 \quad \quad \quad \quad \ \ \quad \quad \quad \quad \quad \quad {\rm{in}} \ {\mathbb{R}}^{3}, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ V_\lambda(x) = \lambda V(x)+1 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>. Under some mild assumptions on <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, we prove the existence of ground state sign-changing solution for <inline-formula><tex-math id="M6">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> large enough by adopting the deformation lemma and constrained minimization arguments. Then, the least energy of sign-changing solutions is strictly large than two times the ground state energy. Additionally, the phenomenon of concentration for ground state sign-changing solutions is also analysed as <inline-formula><tex-math id="M7">\begin{document}$ \lambda\rightarrow \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mu\rightarrow0 $\end{document}</tex-math></inline-formula>.</p>