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Liouville-type theorems for polyharmonic Hénon-Lane-Emden system (1405.4584v2)
Published 19 May 2014 in math.AP
Abstract: We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)mu=|x|avp,\; (-\Delta)mv=|x|buq$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2m$. It is shown by Fazly [6] that the conjecture holds for radial solutions in all dimensions and for classical solutions in dimension $N\leq 2m+1$. We here give some partial results in dimension $N\geq 2m+2$.