Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

Abstract

In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system $ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $ where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.

Keywords

• fractional schrödinger-poisson system
• variational method
• sign-changing solution
• steep potential well