Type

Publication

Commun. Contemp. Math. 21, no. 6, 1850027, 46 pp

We are concerned with the existence and concentration behavior of ground state solutions of the fractional Schrödinger-Poisson system with critical nonlinearity \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=\lambda|u|^{p-2}u+|u|^{2^*_s-2}u, \end{equation*} \begin{equation*} \varepsilon^{2t}(-\Delta)^t \phi=u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $\lambda>0$, $\frac{4s+2t}{s+t}<p<2^*_s=\frac{6}{3-2s}$, $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha=s,t\in(0,1)$ and satisfies $2t+2s>3$. The potential $V$ is continuous and positive, and has a local minimum. We obtain a positive ground state solution for $\varepsilon>0$ small, and we show that these ground state solutions concentrate around a local minimum of $V$ as $\varepsilon\rightarrow0$.