On the existence of positive solutions to semilinear elliptic systems involving gradient term

Boumediene Abdellaoui (1) , Ahmed Attar (1) , El-Haj Laamri (2)

1 Département de Mathematiques [Tlemcen]
2 IECL - Institut Élie Cartan de Lorraine

In this work we analyze the existence of solutions to the nonlinear elliptic system:        −∆u = v q + αg in Ω, −∆v = ||u| p + λf in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω, where Ω is a bounded domain of IR N and p ≥ 1, q > 0 with pq > 1. f, g are nonnegative measurable functions with additional hypotheses and α, λ ≥ 0. As a consequence we show that the fourth order problem ∆ 2 u = ||u| p + ˜ λ ˜ f in Ω, u = ∆u = 0 on ∂Ω, has a solution for all p > 1, under suitable conditions oñ f and˜λand˜ and˜λ.

Domaines: Equations aux dérivées partielles [math.AP]

Boumediene Abdellaoui, Ahmed Attar, El-Haj Laamri. On the existence of positive solutions to semilinear elliptic systems involving gradient term. Applicable Analysis, 2018, ⟨10.1080/00036811.2017.1419204⟩. ⟨hal-01748590⟩