On critical double phase Choquard problems with singular nonlinearity.

In this article, we consider the following double phase problem with singular term and convolution term − Δ p u − Δ q u = λ u − γ + ∫ Ω | u | q μ ∗ | x − y | μ d y | u | q μ ∗ − 2 u in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N with Lipschitz boundary ∂ Ω , γ ∈ (0 , 1) , 1 < p < q < q μ ∗ , − Δ ℘ φ = d i v ( | ∇ φ | ℘ − 2 ∇ φ) , with ℘ ∈ { p , q } , is the homogeneous ℘ -Laplacian. λ > 0 is a real parameter, 0 < μ < N , N > p and q μ ∗ = (p N − p μ / 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. • The background of critical double phase Choquard problems is rather outstanding. Moreover, the study of critical problems is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions. • The main innovation and difficulty of this problem is that the critical term and singular nonlinearity appear simultaneously. In order to overcome these difficulties, we establish a truncation parameter, and combine with a subtle gradient analysis, to verify the solution sequence converges to this problem virtually. • This paper extends some existence results of problem concerning the existence of solutions to this problem in the subcritical case. Moreover, the emergence of p and q -Laplacian operator makes the study of this problem more complicated and interesting. [ABSTRACT FROM AUTHOR]