In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<\mu<\min{2s,N-2s}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution.
In this paper, we are concerned with the following general nonlocal problem \begin{equation*} -\mathcal{L}_K u=\lambda_1u+f(x,u)\ \ \text{in}\ \Omega,
u=0\ \ \text{in}\ \mathbb{R}^N\backslash\Omega, \end{equation*} where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded and with continuous boundary.
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.
In this paper, we study the following fractional Schrödinger equation
\begin{equation*} (-\Delta)^su+V(x)u=K(x)f(u)+\lambda W(x)|u|^{p-2}u,\ \ x\in\mathbb{R}^N, \end{equation*}
where $\lambda>0$ is a parameter, $(-\Delta)^s$ denotes the fractional Laplacian of order $s\in(0,1)$, $N>2s$, $W\in L^{\frac{2}{2-p}}(\mathbb{R}^N,\mathbb{R}^+)$, $1< p <2$, $V,K$ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth.