Liouville theorems for the polyharmonic Hénon-Lane-Emden system

Abstract

Abstract. We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden systemwhen 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 under the critical Sobolev hyperbola, i.e. > 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 equationin 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.

Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$

“…Fazly and Ghoussoub [11] proved the nonexistence of positive solution with finite Morse index to (5) with n ≥ 5, β ≥ 0, 1 < p < n+4+2β n−4 . For α = 2m, m ∈ N, 1 < 2m < n, Fazly [10] showed the nonexistence of positive bounded solution to (5) with 1 < p < n+2m+2β n−2m . Furthermore, for α = 2m, m ∈ N, 1 < 2m < n, Fazly [10] proved that there exists no positive bounded solution to (7) with n+β p+1 + n+γ q+1 > n − 2m.…”

“…For α = 2m, m ∈ N, 1 < 2m < n, Fazly [10] showed the nonexistence of positive bounded solution to (5) with 1 < p < n+2m+2β n−2m . Furthermore, for α = 2m, m ∈ N, 1 < 2m < n, Fazly [10] proved that there exists no positive bounded solution to (7) with n+β p+1 + n+γ q+1 > n − 2m. For α = 2, β = γ = 0, (5) and (7) have been received a lot of attentions, and the reader can see [12,13,18,31,24,29,30,33] and references therein .…”

Liouville-type Theorems for Polyharmonic Hénon-Lane-Emden System

“…then there is no positive solution in R N of system (2). So far, this conjecture was proved for the class of radial solutions in any dimension (see [6,10]). For nonradial solutions, this conjecture is true for dimension N ≤ 2m.…”

“…The critical exponent in this case is p S (m) = N +2m N −2m . When a = 0, the Liouville type result was proved in dimensions N ≤ 2m + 1 -see [14] for m = 1, Cowan [5] for m = 2, and Fazly [6] for any m ≥ 1.…”

A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients

“…Recently, the study on the criteria governing the existence and non-existence of solutions for both differential and integral versions of the HLS type systems has received much attention, especially since Liouville type theorems are crucial in deriving a priori estimates and singularity and regularity properties of solutions for a class of nonlinear elliptic problems [9,32]. More precisely, it is conjectured that either system (1.2) or (1.3) admits no positive solution in the subcritical case n+σ 1 1+q + n+σ 2 1+p > n − α (see [2,8,25,28,30,37] for partial results). In the case where α = 2 and σ i = 0, this is often referred to as the Lane-Emden conjecture and it too has only partial results.…”