Abstract:

We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity $\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in \mathbb{R} ^{N}, \end{eqnarray*}$ where $\varepsilon>0$ is a parameter, $s\in(0, 1)$, $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R}$ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$, and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.

Key words: Fractional Choquard equation, Variational method, Ljusternik-Schnirelmann theory, Positive solution, Concentrating phenomenon.