关键词: 分数阶Schr?dinger-Poisson系统, 凹凸非线性项, 变号解, 陡峭位势

Abstract:

In this paper, we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and steep potential well

\begin{equation}

\begin{cases}

(-\Delta)^s u+V_{\lambda} (x)u+\phi u=f(x)|u|^{q-2}u+|u|^{p-2}u, & ~\mathrm{in}~~\mathbb{R}^3, \\

(-\Delta)^t \phi=u^2, & ~\mathrm{in}~~\mathbb{R}^3,

\end{cases}

\nonumber

\end{equation}

where $s\in(\frac{3}{4},1), t\in(0,1)$, $q\in(1,2)$, $p\in(4,2_s^*)$, $2_s^*:=\frac{6}{3-2s}$ is the fractional critical exponent in dimension 3, $V_{\lambda}(x)$ = $\lambda V(x)+1$ with $\lambda>0$. Under the case of steep potential well, we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma. Furthermore, we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution. Our results improve the recent results in the literature.

Key words: Fractional Schr?dinger-Poisson system, Concave-convex nonlinearity, Sign-changing solution, Steep potential well