A Semilinear Elliptic Problem with Critical Exponent and Potential Terms

This paper addresses the following problem. \begin{equation} \left{ \begin{array}{lr} -Δu=λI_α*_Ωu+|u|^{2^-2}u\mbox{ in }Ω,\nonumber u\in H_0^1(Ω).\nonumber \end{array} \right. \end{equation} Here, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$, $\lambda \in \mathbb{R}$, $\lambda \in (0,N)$, $I_{\alpha }$ is the Riesz potential and \begin{align} I_α_Ωu(x):=\int_Ω\frac{Γ(\frac{N-α}{2})}{Γ(\fracα{2})π^\frac{N}{2}2^α|x-y|^{N-α}} u(y)dy. \nonumber \end{align} We study the non-existence, existence and multiplicity results. Our argument combines Brezis-Nirenberg’s method with the regularity results involving potential terms. Especially, we study the following nonlocal eigenvalue problem. \begin{equation} \left{ \begin{array}{lr} -Δu=λI_α*_Ωu\mbox{ in }Ω,\nonumber λ\in$\mathbb{R}$,,u\in H_0^1(Ω).\nonumber \end{array} \right. \end{equation}