Existence of positive solutions to a nonlinear elliptic system with nonlinearity involving gradient term

2017/09/10 by Abdellaoui, Boumediene, Attar, Ahmed, Laamri, El-Haj
#35D10 #35J55 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.03070

Abstract

In this work we analyze the existence of solutions to the nonlinear elliptic system: \ -Δu · amp; = · amp; v^q+\a g · amp; in Ω,
-Δv · amp; = · amp;|∇ u|^p+łf · amp;in Ω,
u=v · amp;= · amp; 0 · amp; on ∂ Ω,
u,v · amp; ≥ · amp; 0 · amp; in Ω,% . where Ω is a bounded domain of \ren and p≥ 1, q>0 with pq>1. f,g are nonnegative measurable functions with additional hypotheses and \a, ł≥ 0. As a consequence we show that the fourth order problem \ Δ² u · amp; = · amp;|∇ u|^p+ł f · amp;in Ω,
u=\D u · amp;= · amp; 0 · amp; on ∂ Ω,% . has a solution for all p>1, under suitable conditions on f and ł.

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