On the critical Choquard equation with potential well

Abstract

In this paper we are interested in the following nonlinear Choquard equationis the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function V ∈ C(R N , R) such that Ω := intV −1 (0) is a nonempty bounded set with smooth boundary. If β > 0 is a constant such that the operator −∆ + λV (x) − β is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int V −1 (0) for λ large enough and also characterize the asymptotic behavior of the solutions as the parameter λ goes to infinity. Furthermore, for any 0 < β < β 1 , we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where β 1 is the first eigenvalue of −∆ on Ω with Dirichlet boundary condition. Contents 1. Introduction and main results 1 2. Existence of solutions for the case 0 < β < β1 6 3. Proof of Theorem 1.3 9 4. Multiplicity of solutions for the case 0 < β < β1 12 5. Existence of solutions for the case β > β1 19 6. The proof of Theorem 1.5 22 References 25