Liouville type theorems for the generalized higher order Choquard-Pekar equation

Abstract

We study the generalized higher order Choquard-Pekar equation(⁎)(−Δ)mu=(∫RNΨ(|x−y|)up(y)dy)uqin RN, where N,m≥1 are integers, p,q>0. Under suitable assumption on Ψ, when N>2m the Liouville theorems in the ranges(i)(p≥1 or p+q≥2) and ∫|y|>1|y|−p(N−2m)Ψ(|y|)dy=∞; or(ii)p+q≥2 and limsupr→∞r2N−(N−2m)(p+q)Ψ(r)>0 for nonnegative classical solutions of (⁎) were got in previous studies. By using ‘‘Bootstrap’’ iteration procedure, we extend the above results by establishing the Liouville theorems for nonnegative classical solutions of (⁎) with N>2m in the ranges p≥1 and p+q<NN−2m. Moreover, we also consider the higher order Choquard-Pekar equation, i.e., the problem (⁎) with Ψ(r)=r−σ and 0<σ<N. By using the method of scaling sphere we also establish the Liouville theorems for nonnegative classical solutions with N>2m in the ranges 1≤p≤2N−σN−2m, 0<q≤N+2m−σN−2m, and p+q<3N+2m−2σN−2m, which also extend the recent studies on higher order Choquard-Pekar equation. Extensions to the integral representation of (⁎) with more general assumption p>0 are also included.