Electronic Journal of Differential Equations (Nov 2017)
Least energy sign-changing solutions for the nonlinear Schrodinger-Poisson system
Abstract
This article concerns the existence of the least energy sign-changing solutions for the Schrodinger-Poisson system $$\displaylines{ -\Delta u+V(x)u+\lambda\phi(x)u=f(u),\quad \text{in } \mathbb{R}^3,\cr -\Delta\phi=u^2,\quad \text{in } \mathbb{R}^3. }$$ Because the so-called nonlocal term $\lambda\phi(x)u$ is involved in the system, the variational functional of the above system has totally different properties from the case of $\lambda=0$. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $\lambda>0$, we show that the energy of a sign-changing solution is strictly larger than twice of the ground state energy. Finally, we consider $\lambda$ as a parameter and study the convergence property of the least energy sign-changing solutions as $\lambda\searrow 0$.
