Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

Abstract

Hardy-Sobolev type integral equation Decay rates Integrable solution Bounded decaying solution In this paper, we are concerned with the Lane-Emden type 2m-order PDE with weightwhere n 3, p > 1, m ∈ [1, n/2), σ ∈ (−2m, 0], and the more general Hardy-Sobolev type integral equationthen there is not any positive solution to such an integral equation. Under the assumption of p > n+σ n−α , we obtain that the integrable solution u of the integral equation (i.e. u ∈ L n(p−1)α+σ (R n )) is bounded and decays fast with rate n − α. On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one α+σ p−1 . In addition, the classical solution u of the 2m-order PDE satisfies the integral equation with α = 2m. Therefore, for the 2m-order PDE, all the decay properties above are still true.

Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

“…Of course, these results hold for the scalar integral equation and its corresponding differential equation as well and we state them here for completeness sake. The following theorem, however, is essentially contained in [20]. The remaining two results are Liouville type theorems concerning radially symmetric, decreasing solutions for Hardy-Sobolev type systems with constant coefficients.…”

Liouville theorems for the polyharmonic Hénon-Lane-Emden system

“…al. in [14] in any dimensions and as far as we know only some partial results are given for the nonradial solutions in [12,14,28]. Note that Caristi, DAmbrosio and Mitidieri in [3] have proved Liouville theorems for supersolutions of system (1) and also they have explored the connection between (1) and the Hardy-Littlewood-Sobolev systems (HLS).…”

Mass and asymptotics associated to fractional Hardy–Schrödinger operators in critical regimes

“…Asymptotic properties of the positive extremals of µ 0,s,α (R n ) (i.e., when γ = 0 and 0 < s < α) were given by Y. Lei [23], Lu-Zhu [26], and Yang-Yu [34]. The latter proved that an extremalū(x) for µ 0,s,α (R n ) must have the following behaviour: There is C > 0 such that (4) C −1 1 + |x| 2 − (n−α) 2 ≤ū(x) ≤ C 1 + |x| 2 − (n−α) 2 for all x ∈ R n .…”