Choquard equation

Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

In this paper, we study the singularly perturbed fractional Choquard equation \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u=\varepsilon^{\mu-3}(\int_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u))\ \ \text{in}\ \mathbb{R}^3, \end{equation*} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes the fractional Laplacian of order $s\in(0,1)$, $0<\mu<3$, $2^*_{\mu,s}=\frac{6-\mu}{3-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.

Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

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.