On critical double phase Choquard problems with singular nonlinearity

Abstract

In this article, we consider the following double phase problem with singular term and convolution term −Δpu−Δqu=λu−γ+∫Ω|u|qμ∗|x−y|μdy|u|qμ∗−2uinΩ,u>0inΩ,u=0on∂Ω,where Ω is a bounded domain in RN with Lipschitz boundary ∂Ω, γ∈(0,1), 1<p<q<qμ∗, −Δ℘φ=div(|∇φ|℘−2∇φ), with ℘∈{p,q}, is the homogeneous ℘-Laplacian. λ>0 is a real parameter, 0<μ<N, N>p and qμ∗=(pN−pμ/2)/(N−p) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality.The existence of at least one weak solution is obtained for the above problem by using the Nehari manifold approach.