Multiplicity and concentration of solutions for a Choquard equation with critical exponential growth in $\mathbb {R}^N$

Abstract

In this paper, we consider the following Choquard equation

$$\begin{aligned} -\varepsilon ^{N}\Delta _{N}u+V(x)|u|^{N-2}u=\varepsilon ^{\mu -N}\left( I_\mu *F(u)\right) f(u) \quad {\text{ in }\quad \mathbb {R}^N}, \end{aligned}$$

where $N\ge 3$, $I_\mu =|x|^{-\mu }$ with $0<\mu <N$, $\Delta _{N}u=\textrm{div}(|\nabla u|^{N-2}\nabla u)$ denotes the N-Laplacian operator, V(x) is a continuous real function on $\mathbb {R}^N$, F(s) is the primitive of f(s) and $\varepsilon$ is a positive parameter. Assuming that the nonlinearity f(s) has critical exponential growth in the sense of Trudinger–Moser inequality, we establish the existence, multiplicity and concentration of solutions by variational methods and Ljusternik–Schnirelmann theory, which extends the works of Alves and Figueiredo (J Differ Equ 246:1288–1311, 2009) to the problem with Choquard nonlinearity, Alves et al. (J Differ Equ 261:1933–1972, 2016) to higher dimension.

A Several technical estimates

Lemma A.1

By the definitions of $\delta _{n}$ and $b_\rho$ in (2.9) and (2.10), it holds that $\delta _{n}=O(\frac{1}{\log n})>0$ and $b_\rho =\frac{W_\rho \rho ^N(N-1)!}{N^N}$.

Proof

From the definition of ${\overline{w}}_{n}$ in (2.8), we have

$$\begin{aligned} \int _{B_{\rho }}|{\overline{w}}_{n}|^{N}\textrm{d}x&=\int _{|x|\le \rho /n}|{\overline{w}}_{n}|^{N}\textrm{d}x +\int _{\rho /n\le |x|\le \rho }|{\overline{w}}_{n}|^{N}\textrm{d}x\\&=\int _{|x|\le \rho /n}\left[ \frac{1}{\omega ^{1/N}_{N-1}}(\log n)^{\frac{N-1}{N}}\right] ^{N}\textrm{d}x+\int _{\rho /n\le |x|\le \rho } \left| \frac{1}{\omega ^{1/ N}_{N-1}}\cdot \frac{\log (\rho /|x|)}{{\log n}^{1/ N}}\right| ^{N}\textrm{d}x\\ {}&=\frac{(\log n)^{N-1}}{\omega _{N-1}}\int _{|x|\le \rho /n}\textrm{d}x +\frac{1}{\omega _{N-1}\log n}\int _{\rho /n\le |x|\le \rho }\left| \log \frac{\rho }{|x|}\right| ^{N}\textrm{d}x\\ {}&=(\log n)^{N-1}\cdot \frac{1}{N}\cdot (\frac{\rho }{n})^{N} +\frac{1}{\log n}\int ^{\rho }_{\frac{\rho }{n}}\left| \log \frac{\rho }{r}\right| ^{N}r^{N-1}\textrm{d}r\\ {}&=\frac{(\log n)^{N-1}}{N}(\frac{\rho }{n})^{N} +\frac{1}{\log n}\int ^{\log n}_{0}t^{N}(\rho e^{-t})^{N}\textrm{d}t\\ {}&=\frac{(\log n)^{N-1}}{N}(\frac{\rho }{n})^{N} +\frac{1}{\log n}\cdot \rho ^{N}\int ^{\log n}_{0}e^{-Nt}t^{N}\textrm{d}t\\&=\frac{\rho ^N}{\log n}\int ^{\log n}_{0}e^{-Nt}t^{N-1}\textrm{d}t. \end{aligned}$$

Next, calculate

$$\begin{aligned}&\int ^{\log n}_{0}e^{-Nt}t^{N-1}\textrm{d}t\\&=-\frac{1}{N}t^{N-1}e^{-Nt}\Big |^{\log n}_{0}+\frac{N-1}{N}\int ^{\log n}_{0}e^{-Nt}t^{N-2}\textrm{d}t\\&=-\frac{1}{N}(\log n)^{N-1}\cdot \frac{1}{n^{N}}-\frac{N-1}{N^{2}}\int ^{\log n}_{0}t^{N-2}\textrm{d}(e^{-Nt})\\ {}&=-\frac{1}{N}(\log n)^{N-1}\cdot \frac{1}{n^{N}}-\frac{N-1}{N^{2}}t^{N-2}e^{-Nt}\Big |^{\log n}_{0}\\ {}&\quad +\frac{(N-1)(N-2)}{N^{2}}\int ^{\log n}_{0}t^{N-3}e^{-Nt}\textrm{d}t\\ {}&=-\frac{1}{N}(\log n)^{N-1}\cdot \frac{1}{n^{N}}-\frac{N-1}{N^{2}}(\log n)^{N-2}\cdot \frac{1}{n^{N}}\\ {}&\quad -\frac{(N-1)(N-2)}{N^{3}}\int ^{\log n}_{0}t^{N-3}\textrm{d}(e^{-Nt})\\&=-\sum ^{N-1}_{i=1}\frac{(N-1)!}{(N-i)!N^i}\cdot \frac{(\log n)^{N-i}}{n^{N}}+\frac{(N-1)!}{N^{N-1}}\int ^{\log n}_{0}e^{-Nt}\textrm{d}t\\&=-\sum ^N_{i=1}\frac{(N-1)!}{(N-i)!N^i}\cdot \frac{(\log n)^{N-i}}{n^{N}}-\frac{(N-1)!}{N^{N}}\cdot \frac{1}{n^{N}}+\frac{(N-1)!}{N^{N}}, \end{aligned}$$

then we have

$$\begin{aligned} b_\rho =\lim \limits _{n\rightarrow \infty }\delta _{n}\log n=\frac{W_\rho \rho ^N(N-1)!}{N^N} \ \ \ \textrm{and}\ \ \ \ \delta _{n}=W_\rho \int _{B_{\rho }}|{\overline{w}}_{n}|^{N}\textrm{d}x =O\left( \frac{1}{\log n}\right) >0. \end{aligned}$$

$\square$

Lemma A.2

For $N\ge 3$, we have

$$\begin{aligned} \int ^{\frac{\rho }{n}}_{0} (\frac{\rho }{n}-r)^{N-\mu }r^{N-1}\textrm{d}r =\frac{\mathop {\prod }\limits ^{N-2}_{j=1}(N-j)}{\mathop {\prod }\limits ^{2N}_{i=N+1}(i-\mu )}(\frac{\rho }{n})^{2N-\mu }. \end{aligned}$$

Proof

For $N\ge 3$, by a direct calculation, we obtain

$$\begin{aligned} \int ^{\frac{\rho }{n}}_{0} (\frac{\rho }{n}-r)^{N-\mu }r^{N-1}\text {d}r =&-\frac{1}{N-\mu +1}\int ^{\frac{\rho }{n}}_{0} r^{N-1}\text {d}(\frac{\rho }{n}-r)^{N-\mu +1} \\ =&-\frac{1}{N-\mu +1} \bigg [ r^{N-1}(\frac{\rho }{n}-r)^{N-\mu +1}\Big |^{\frac{\rho }{n}}_0 \\ {}&\quad -(N-1) \int ^{\frac{\rho }{n}}_{0}(\frac{\rho }{n}-r)^{N-\mu +1} r^{N-2}\text {d}r \bigg ] \\ =&\frac{N-1}{N-\mu +1} \int ^{\frac{\rho }{n}}_{0}(\frac{\rho }{n}-r)^{N-\mu +1} r^{N-2}\text {d}r. \end{aligned}$$

Then repeat the above steps to get

$$\begin{aligned} \int ^{\frac{\rho }{n}}_{0} (\frac{\rho }{n}-r)^{N-\mu }r^{N-1}\textrm{d}r =&\left( \mathop {\prod }\limits ^{N-2}_{i=1} \frac{N-i}{N-\mu +i}\right) \int ^{\frac{\rho }{n}}_{0}(\frac{\rho }{n}-r)^{2N-\mu -2} r\textrm{d}r \\ =&\left( \mathop {\prod }\limits ^{N-2}_{i=1}\frac{N-i}{N-\mu +i}\right) \left[ \int ^{\frac{\rho }{n}}_{0}t^{2N-\mu -2} (\frac{\rho }{n}-t)\textrm{d}t\right] \\ =&\left( \mathop {\prod }\limits ^{N-2}_{i=1}\frac{N-i}{N-\mu +i}\right) \left[ \frac{(\frac{\rho }{n})^{2N-\mu }}{(2N-\mu -1)(2N-\mu )}\right] \\ =&\frac{\mathop {\prod }\limits ^{N-2}_{j=1}(N-j)}{\mathop {\prod }\limits ^{2N}_{i=N+1}(i-\mu )}(\frac{\rho }{n})^{2N-\mu }. \end{aligned}$$

Particularly, for $N=2$,

$$\begin{aligned} \int ^{\frac{\rho }{n}}_{0} (\frac{\rho }{n}-r)^{2-\mu }r\textrm{d}r =\frac{(\frac{\rho }{n})^{4-\mu }}{(3-\mu )(4-\mu )}. \end{aligned}$$

The proof is completed. $\square$