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

Advances in Nonlinear Analysis, Volume 10, Issue 1, 732–774

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. $F$ is the primitive of $f$ which is a continuous subcritical term. Under a local condition imposed on the potential $V$, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.