Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth
Xiaoping Chen, Chun‐Lei Tang
Abstract
<p style='text-indent:20px;'>In this paper, we investigate the existence and asymptotic behavior of least energy sign-changing solutions for the following Schrödinger-Poisson system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda\phi(x)u = |u|^4u+ f(u),\ \ \ &\ x \in \mathbb{R}^{3},\\ -\Delta \phi = u^2, \ \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is a parameter. Under some suitable conditions on <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ V $\end{document}</tex-math></inline-formula>, we get a least energy sign-changing solution <inline-formula><tex-math id="M4">\begin{document}$ u_\lambda $\end{document}</tex-math></inline-formula> via variational method and its energy is strictly larger than twice that of least energy solutions. Moreover, the asymptotic behavior of <inline-formula><tex-math id="M5">\begin{document}$ u_\lambda $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M6">\begin{document}$ \lambda\rightarrow 0^+ $\end{document}</tex-math></inline-formula> is also analyzed.