A global compactness result for a critical nonlinear Choquard equation in RN $\mathbb{R} ^{N}$

• Main Authors: Wangcheng Huang, Wei Long, Aliang Xia, Xiongjun Zheng
• Format: Article
• Language: English
• Published: SpringerOpen 2019-06-01
• Series: Boundary Value Problems
• Subjects: Global compactness; (PS) condition; Choquard equation; Existence results
• Online Access: http://link.springer.com/article/10.1186/s13661-019-1223-z

Abstract This article is concerned with the following class of nonlinear Choquard equations: {−Δu+a(x)u=(|x|−μ∗|u|2μ∗)|u|2μ∗−2u+q(x)|u|p−1u,x∈RN,u∈H1(RN), $$\begin{aligned} \textstyle\begin{cases} -\Delta u+a(x)u= ( \vert x \vert ^{-\mu }* \vert u \vert ^{2^{*}_{\mu }} ) \vert u \vert ^{2^{*} _{\mu }-2}u+q(x) \vert u \vert ^{p-1}u, \quad x\in \mathbb{R} ^{N}, \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases}\displaystyle \end{aligned}$$ where 2μ∗=2N−μN−2 $2^{*}_{\mu }=\frac{2N-\mu }{N-2}$ is the critical exponent with N≥4 $N\ge 4$ and 0<μ<N $0<\mu <N$, 1<p<2∗−1=N+2N−2 $1< p<2^{*}-1=\frac{N+2}{N-2}$, a(x) $a(x)$ and q(x) $q(x)$ satisfy some assumptions. Through a compactness analysis of the functional corresponding to the above problem, we obtain the existence of weak solutions for this problem under certain assumptions on a(x) $a(x)$ and q(x) $q(x)$.