Ground state sign-changing solutions for a Schr\"{o}dinger-Poisson system with critical growth
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In this paper, we study the existence and asymptotic behavior of ground state sign-changing solutions for the following Schr\"{o}dinger-Poisson system with critical growth $$\left\{ \aligned &-\Delta u+V_\lambda(x)u+\phi u=f(x,u)+\mu h(u)\ \ \ \ \ &{\rm in} \ \mathbb{R}^{3}, \\ &-\Delta \phi =u^2\ \ \ \ \ &{\rm in} \ \mathbb{R}^{3},\\ \endaligned \right.$$ where $V_\lambda(x)=\lambda V(x)+1$ with $\lambda>0$, $f$, $h$ are continuous and $\mu>0$. Under some mild assumptions on $V$, $f$ and $h$, we first show the existence of ground state sign-changing solutions for each $\lambda>0$ large enough via Nehari manifold methods. Then we prove the ground state sign-changing solution has precisely two nodal domains and characterize the relationship between its energy and the energy of the least energy solution of the above system. Finally, we give the asymptotic behavior of sign-changing solutions as $\lambda\rightarrow\infty$.
Abstract
In this paper, we study the existence and asymptotic behavior of ground state sign-changing solutions for the following Schr\"{o}dinger-Poisson system with critical growth $$\left\{ \aligned &-\Delta u+V_\lambda(x)u+\phi u=f(x,u)+\mu h(u)\ \ \ \ \ &{\rm in} \ \mathbb{R}^{3}, \\ &-\Delta \phi =u^2\ \ \ \ \ &{\rm in} \ \mathbb{R}^{3},\\ \endaligned \right.$$ where $V_\lambda(x)=\lambda V(x)+1$ with $\lambda>0$, $f$, $h$ are continuous and $\mu>0$. Under some mild assumptions on $V$, $f$ and $h$, we first show the existence of ground state sign-changing solutions for each $\lambda>0$ large enough via Nehari manifold methods. Then we prove the ground state sign-changing solution has precisely two nodal domains and characterize the relationship between its energy and the energy of the least energy solution of the above system. Finally, we give the asymptotic behavior of sign-changing solutions as $\lambda\rightarrow\infty$.