The following critical Schr"odinger-Poisson system is considered: \begin{equation*} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u, \end{equation*} \begin{equation*} -\Delta \phi=u^2, \end{equation*} where $\lambda, \mu$ are two positive parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions.
In this paper, we study the following nonlinear Schrödinger-Poisson type equation \begin{equation*} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u), \end{equation*} \begin{equation*} -\varepsilon^2\Delta \phi=K(x)u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $V: \mathbb{R}^3\rightarrow \mathbb{R}$ is a continuous potential and $K: \mathbb{R}^3\rightarrow \mathbb{R}$ is used to describe the electron charge.
In this paper, we are concerned with positive solutions for the fractional Schrödinger-Poisson system : \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u), \end{equation*} \begin{equation*} \varepsilon^{2t}(-\Delta)^t\phi=u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha=s,t\in(\frac{3}{4},1)$, $V\in C(\mathbb{R}^3,\mathbb{R})$ is the potential function and $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and subcritical.
In this paper, we study the existence of infinitely many large energy solutions for the supercubic fractional Schr"{o}dinger-Poisson systems. We consider different superlinear growth assumptions on the nonlinearity, starting from the well-know Ambrosetti-Rabinowitz type condition.
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 $2t+2s>3$.
In this paper, we study the following critical fractional Schrödinger-Poisson system \begin{equation*} \varepsilon^{2s}(-\Delta)^s u +V(x)u+\phi u=P(x)f(u)+Q(x)|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, $s\in (\frac{3}{4},1),t\in(0,1)$ and $2s+2t>3$, $2_s^*:=\frac{6}{3-2s}$ is the fractional critical exponent for 3-dimension, $V(x)\in \mathcal{C}(\mathbb{R}^3)$ has a positive global minimum, and $P(x)$, $Q(x)\in \mathcal{C}(\mathbb{R}^3)$ are two positive and have global maximum.