Liouville theorems for the polyharmonic Henon-Lane-Emden system

2013/08/01 by Fazly, Mostafa
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.0073

Abstract

We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden system \(-Δ)^m u · amp;= · amp; |x|^av^p in ℝⁿ,
(-Δ)^m v · amp;= · amp; |x|^bu^q in ℝⁿ,. when m,p,q ≥ 1, pq≠1, a,b≥0. The main conjecture states that (u,v)=(0,0) is the unique nonnegative solution of this system whenever (p,q) is \it under the critical Sobolev hyperbola, i.e. (n+a)/(p+1)+(n+b)/(q+1)>n-2m. We show that this is indeed the case in dimension n=2m+1 for bounded solutions. In particular, when a=b and p=q, this means that u=0 is the only nonnegative bounded solution of the polyharmonic Hénon equation (-Δ)^m u= |x|^au^p in ℝⁿ in dimension n=2m+1 provided p is the subcritical Sobolev exponent, i.e., 1

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