Critical exponent Neumann problem with Hardy-Littlewood-Sobolev nonlinearity

Abstract

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation with Neumann boundary condition \begin{equation*} \begin{aligned} -Δu &= λα(x)u + \left(\int\limits_Ω\frac{u(y)^{2^*_μ}}{|x-y|^μ}\;dy\right)u^{2^*_μ-1}, \;\;\text{in} \; Ω,\\ \frac{\partial u}{\partial ν} &= 0\;\; \text{on} \; \partialΩ, \end{aligned} \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ $(N\ge 4)$, $\nu$ is the unit outer normal to $\partial \Omega$ and $\mu \in (0,N)$. According to the parameter $\lambda$, we prove necessary and sufficient conditions for the existence and non-existence of positive weak solutions to the problem. The proof is based on variational arguments.