A Semilinear Elliptic Problem with Critical Exponent and Potential Terms

This paper addresses the following problem. {equation} \{ {array}{lr} -{Δ}u=λ I_α*_Ω u+|u|2^*-2u in Ω , u∈ H_0^1(Ω). {array} . {equation} Here, Ω is a bounded domain in RN with N≥3, 2^*=2N/N-2, λ, λ∈(0,N), I_α is the Riesz potential and {align} I_α*_Ω u(x):=∫_Ω {Γ(N-α/2)}{Γ(α/2)π^N/22^α|x-y|N-α} u(y)dy. {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. {equation} \{ {array}{lr} -{Δ}u=λ I_α*_Ω u in Ω , λ,\,u∈ H_0^1(Ω). {array} . {equation}