2011/07/28 by Fazly, Mostafa, Ghoussoub, Nassif · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1107.5611
We consider Liouville-type theorems for the following Hénon-Lane-Emden system \hfill -Δu&=& |x|avp in ℝN, \hfill -Δv&=& |x|buq in ℝN, when pq>1, p,q,a,b≥0. The main conjecture states that there is no non-trivial non-negative solution whenever (p,q) is under the critical Sobolev hyperbola, i.e. (N+a)/(p+1)+(N+b)/(q+1)>N-2. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order (a=b and p=q) or of fourth order (a≥ 0=b and p>1=q), we show that for all dimensions N≥ 3 in the first case (resp., N≥ 5 in the second case), there is no positive solution with a finite Morse index, whenever p is below the corresponding critical exponent, i.e 1