Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

Abstract

In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝN{\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝN{\mathbb{R}^{N}}.

Keywords

• singular elliptic equation
• hardy–sobolev–maz’ya type inequalities,refinement of hardy–littlewood–sobolev inequality
• critical nonlinearities
• variational methods
• 35a15
• 35b33
• 35d30