On the Hénon-Lane-Emden conjecture

We consider the problem of non-existence of solutions for the following Hénon-Lane-Emden system  −∆u = |x|v in R , −∆v = |x|u in R , when pq > 1, p, q, a, b ≥ 0, and (p, q) are under the critical hyperbola, i.e. N+a p+1 + N+b q+1 > N − 2. We show that there is no positive bounded solution in dimension N = 3, extending a result established recently by Phan-Souplet in the scalar case. This solves the Hénon-Lane-Emden conjecture in dimension N = 3 for bounded positive solutions. For the scalar cases, whether 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 (resp., N ≥ 5), there is no positive solution with a finite Morse index, whenever p is below the corresponding critical exponent, i.e 1 < p < N+2+2a N−2 (resp., 1 < p < N+4+2a N−4 ). 2010 Mathematics Subject Classification. 35J47; 35B33; 35B45; 35B08.