Existence of solutions to higher order Lane–Emden type systems

Abstract

We prove existence results for the Lane-Emden type systemwhere B1 is the unitary ball in R N , N > max{2α, 2β}, ν is the outward pointing normal, α, β ∈ N, α, β ≥ 1 and (−∆) α = −∆((−∆) α−1 ) is the polyharmonic operator. A continuation method together with a priori estimates will be exploited. Moreover, we prove uniqueness for the particular case α = 2, β = 1 and p, q > 1.

Uniqueness results for higher order Lane-Emden systems

“…Existence. Next we extend to system (8) the existence results obtained in [19] in the case of systems of two equations, see also [4] for p-Laplacian systems. In what follows we recall the main steps in the proof, and the necessary changes required to treat the case in which one has m > 2 equations.…”

“…where F (x, y, z) = (y, z, x p ). Since p > 1, then F is locally Lipschitz continuous, hence by the Gronwall Lemma, (19) For instance, assume that…”

“…with p, q > 1, see [11] and then extended in [8] to systems with more than two equations. More recently, the non-variational situation has been addressed in [19], where uniqueness of solutions is established for the following system…”

Existence and non-existence results for variational higher order elliptic systems

“…with α = β. In [28] we prove existence results for 10 on a ball by means of degree theory combined with moving planes methods. Here, we take into account the variational case α = β allowing for more general nonlinearities than power-like.…”

“…Here, we take into account the variational case α = β allowing for more general nonlinearities than power-like. Moreover, we deal with a larger class of bounded domains with respect to [28].…”