Liouville theorems for the polyharmonic Henon-Lane-Emden system · arXivDesk
1308.0073Aug 1, 201316 pages. Submitted. This is an extension of the work of Nassif Ghoussoub and the author entitled "On the Hénon-Lane-Emden conjecture" given in arXiv:1107.5611 to polyharmonic equations and systems
Liouville theorems for the polyharmonic Henon-Lane-Emden system
We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden system eqnarray* {arraylcl (-Δ)^m u&=& |x|^av^p \ \ in\ \ Rⁿ, (-Δ)^m v&=& |x|^bu^q \ \ in\ \ Rⁿ, array. eqnarray* when m,p,q≥1,pq=1, a,b≥0
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. The main conjecture states that
(u,v)=(0,0)
is the unique nonnegative solution of this system whenever
(p,q)
is under the critical Sobolev hyperbola, i.e.
p+1n+a+q+1n+b>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 equation* (-Δ)^m u= |x|^au^p \ \ in\ \ Rⁿ equation* in dimension
n=2m+1
provided
p
is the subcritical Sobolev exponent, i.e.,
1<p<1+4m+2a
. Moreover, we show that the conjecture holds for radial solutions in any dimensions. It seems the power weight functions
∣x∣a
and
∣x∣b
make the problem dramatically more challenging when dealing with nonradial solutions.