Liouville Theorems for a p-Laplace equation with Hartree type nonlinearity

Abstract

Let u∈ C1(ℝN) be a weak solution to the equation −Δpu=(1|x|N−α∗|u|q)|u|q−2uinℝN, where 2 ≤ p < N and max{ 0 , N− 2 p} < α< N. We prove that if p < q < qc and u is stable, then u ≡ 0. Here qc is a new critical exponent, which equals to infinity when N+αN−p≥p+12. We also show that if p<q<p(N+α)2(N−p) and u is stable outside a compact set or has a finite Morse index, then u ≡ 0. Our proofs rely on several integral estimates and a Pohozaev type identity.