Existence of Ground State Solution for <i>p</i>-Choquard Equation with Upper Critical Exponent

该文讨论下列具有上临界指标的p-Choquard方程 { − Δ p u+V( x ) | u | p−2 u=[ ( − Δ D ) −α 2 ∗ | u | p α ∗ ] | u | p α ∗ −2 u, in ℝ N ; u∈ W 1,p ( ℝ N ), 其中 N≥3 ， p α * = p 2 ( N+α N−p ) 是关于Hardy-Littlewood-Sobolev不等式的上临界指数， 2≤pN ， Δ p :=div( | ∇u | p−2 ∇u ) ， ( − Δ D ) −α 2 是关于Dirichlet边界条件下的Riesz变分位势，指数 α∈( 0,N ) 。该文通过Nehari流形和山路引理证明上述方程存在基态解。In this paper, we are concerned with the following p-Choquard equation with upper critical exponent: { − Δ p u+V( x ) | u | p−2 u=[ ( − Δ D ) −α 2 ∗ | u | p α ∗ ] | u | p α ∗ −2 u, in ℝ N ; u∈ W 1,p ( ℝ N ), where N≥3 , p α * = p 2 ( N+α N−p ) is the upper critical Hardy-Littlewood-Sobolev exponent, 2≤pN , Δ p :=div( | ∇u | p−2 ∇u ) , ( − Δ D ) −α 2 is the Riesz fractional potential with Dirichlet boundary condition. α∈( 0,N ) . We study the existence of the ground state solution to above equation by using the methods of Nehari manifold and the Mountain Pass Lemma.