Concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system involving critical exponent

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 $\varepsilon\rightarrow0$.