Existence and concentration of ground state solutions for Choquard equations involving critical growth and steep potential well

Abstract

In the present paper, we are interested in the following Choquard type equation −Δu+(λV(x)−μ)u=(Iα∗|u|2α∗)|u|2α∗−2u+|u|p−2uinR3,where p∈(4,6), λ∈R+, μ∈R is a constant such that the operator Lλ≔−Δ+λV(x)−μ is non-degenerate, Iα is a Riesz potential of order α∈(0,3), 2α∗=3+α is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality, the function V∈C(R3,R) is nonnegative and has a potential well Ω≔intV−1(0), additionally, the operator −Δ has a sequence of Dirichlet eigenvalues in H01(Ω) expressed as 0<μ1<μ2<⋯<μn⟶n+∞. If μ<μ1, via the Nehari manifold techniques and the Ekeland variational principle, we prove the existence and concentration of positive ground state solutions for sufficiently large λ. If μ>μ1 and μ≠μj for all j∈N+, employing the Nehari–Pankov manifold methods and the constrained minimization arguments, we obtain the existence of ground state solution for λ large enough and verify the asymptotic behaviour of ground state solutions as λ→+∞.