The existence of positive solutions to the Choquard equation with critical exponent and logarithmic term

Abstract

We investigate the existence of solutions for the following nonlinear critical Choquard equation with the logarithmic term:{−Δu=(∫Ω|u(y)|2μ⁎|x−y|μdy)|u|2μ⁎−2u+βu+λulog⁡u2,x∈Ω,u=0,x∈∂Ω, where Ω is a bounded domain of RN with smooth boundary. λ, β are real parameters and 2μ⁎=2N−μN−2(N≥3) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We will show that there exists at last one positive solution to the above problem under some appropriate assumptions of λ and β.