Higher order singular problems of Caffarelli–Kohn–Nirenberg–Lin type

Abstract

We prove the existence of nontrivial critical points of the functional J λ ( u ) = ∫ R N 1 p ( | | x | − a ∇ k u | p − λ h ( x ) | | x | − ( a + k ) u | p ) − 1 q Q ( x ) | | x | − b u | q d x , related to the Caffarelli–Kohn–Nirenberg inequality and its higher order variant by Lin. As a consequence we obtain nontrivial solutions of the degenerate elliptic equation Δ ( | x | − a p | Δ u | p − 2 Δ u ) − λ h ( x ) | x | − ( a + k ) p | u | p − 2 u = Q ( x ) | x | − b q | u | q − 2 u . We also show that when p = 2 , J λ has infinitely many critical points.