Multiplicity and concentration behavior of solutions of the critical Choquard equation

Abstract

In this paper, we study the following critical Choquard equation (1) $\begin{array}{rl}-{\epsilon }^2\mathrm{\Delta }u+u& ={\epsilon }^{\mu -3}\left({\int }_{{\mathbb{R}}^3}\frac{a\left(y\right)F\left(u\right(y\left)\right)+g\left(y\right)|u\left(y\right){|}^{6-\mu }}{|x-y{|}^{\mu }}\mathrm{d}y\right)\\ & \times \left(\frac{a\left(x\right)f\left(u\right)}{6-\mu }+g\left(x\right)|u{|}^{4-\mu }u\right)\mathrm{i}\mathrm{n}{\mathbb{R}}^3,\end{array}$(1) where $\epsilon >0$, $\mu \in (0,3)$, $F\left(u\right)={\int }_0^{u}f\left(s\right)\mathrm{d}s$, a, g are continuous functions. When $\epsilon >0$ is small, we obtain the multiplicity and concentration behavior of solutions of (1(1) $\begin{array}{rl}-{\epsilon }^2\mathrm{\Delta }u+u& ={\epsilon }^{\mu -3}\left({\int }_{{\mathbb{R}}^3}\frac{a\left(y\right)F\left(u\right(y\left)\right)+g\left(y\right)|u\left(y\right){|}^{6-\mu }}{|x-y{|}^{\mu }}\mathrm{d}y\right)\\ & \times \left(\frac{a\left(x\right)f\left(u\right)}{6-\mu }+g\left(x\right)|u{|}^{4-\mu }u\right)\mathrm{i}\mathrm{n}{\mathbb{R}}^3,\end{array}$(1) ).

COMMUNICATED BY:

• Alexander Pankov

Keywords:

• Choquard equation
• critical growth
• variational method

2010 Mathematics Subject Classifications:

• 35J60
• 35A15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project is supported by National Natural Science Foundation of China [grant nos. 11401583, 11571339] and the Fundamental Research Funds for the central Universities [grant no. 16CX02051A].