Nonlinear Elliptic Systems with Coupled Gradient Terms

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type: { − Δ u = | ∇ v | q + λ f in  Ω , − Δ v = | ∇ u | p + μ g in  Ω , u = v = 0 on  ∂ Ω , u , v ≥ 0 in  Ω , where Ω is a bounded domain of R N and p , q ≥ 1 . f , g are no...

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Bibliographic Details
Published in:Acta applicandae mathematicae 2020-12, Vol.170 (1), p.163-183
Main Authors: Attar, Ahmed, Bentifour, Rachid, Laamri, El-Haj
Format: Article
Language:English
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control
Optimization
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Nonlinear systems
Partial Differential Equations
Probability Theory and Stochastic Processes
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Summary:In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type: { − Δ u = | ∇ v | q + λ f in  Ω , − Δ v = | ∇ u | p + μ g in  Ω , u = v = 0 on  ∂ Ω , u , v ≥ 0 in  Ω , where Ω is a bounded domain of R N and p , q ≥ 1 . f , g are nonnegative measurable functions with additional hypotheses and λ , μ ≥ 0 . This extends previous similar results obtained in the case where the right-hand sides are potential and gradient terms, see (Abdellaoui et al. in Appl. Anal. 98(7):1289–1306, [ 2019 ], Attar and Bentifour in Electron. J. Differ. Equ. 2017:1, [ 2017 ]).
ISSN:0167-8019
1572-9036
DOI:10.1007/s10440-020-00329-7