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Nonlinear Elliptic Systems with Coupled Gradient Terms Ahmed Attar1 · Rachid Bentifour1 · El-Haj Laamri2

Received: 4 September 2019 / Accepted: 3 April 2020 © Springer Nature B.V. 2020

Abstract In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type: ⎧ −u = |∇v|q + λf in , ⎪ ⎪ ⎨ −v = |∇u|p + μg in , u=v = 0 on ∂, ⎪ ⎪ ⎩ u, v ≥ 0 in , where  is a bounded domain of RN 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). Keywords Nonlinear elliptic systems · Hamilton Jacobi equation · Fixed point theorem · Apriori estimates Mathematics Subject Classification 35J55 · 35D10 · 35J60

The first and the second authors are partially supported by DGRSDT, Algeria.

B E.-H. Laamri

[email protected] A. Attar [email protected] R. Bentifour [email protected]

1

Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria

2

Institut Elie Cartan, Université Lorraine, B.P. 239, 54506 Vandœuvre lés Nancy, France

A. Attar et al.

1 Introduction The main goal of this work is to analyze the issue of existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems with gradient terms. More precisely, we deal with the following model: ⎧ −u = |∇v|q + λf ⎪ ⎪ ⎨ −v = |∇u|p + μg u=v = 0 ⎪ ⎪ ⎩ u, v ≥ 0

in in on in

, , ∂, ,

(1.1)

where  is a bounded domain of RN , (p, q) ∈ [1, +∞[2 , (λ, μ) ∈ (0, +∞)2 and f, g are nonnegative measurable functions that satisfy additional assumptions that will be specified later. The purpose of this paper is to find the conditions between (f, g) and (λ, μ) which allow us to establish existence of solutions to System (1.1). Moreover, we will prove some non-existence results that show, in some sense, the optimality of our assumptions. In this article, we focus our attention on the proof of the existence of weak solutions; by weak solution, we mean solution in the sense of distributions (see Definition 2.5). Let us mention that a class of nonlinear elliptic systems with gradient terms has come to the fore in some electrochemical models of engineering, and in some other models in fluid dynamics. We refer to [18] and [19] for more details and more applications of this class of systems. More recently, another class was introduced in [26, 27] in the theory of mean-field games; see also [14] for an in-depth analysis. For more general results on problems including gradient term, we refer the interested reader to [11, 12, 17, 20–25, 34] and the references therein. Now, let us recall some previously known results related to the present study. First Case: One Equation. In the case where the gradient t