Electron. J. Differential Equations, Vol. 2017 (2017), No. 113, pp. 1-10.

Existence of positive solutions to nonlinear elliptic systems involving gradient term and reaction potential

Ahmed Attar, Rachid Bentifour

Abstract:
In this note we study the elliptic system
$$\displaylines{ -\Delta u = z^p+f(x) \quad \text{in }\Omega , \cr -\Delta z = |\nabla u|^{q}+g(x) \quad \text{in }\Omega , \cr z,u > 0 \quad \text{in }\Omega ,\cr z=u= 0 \quad \text{on }\partial \Omega, }$$
where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain, p>0, $0<q\le 2$ with pq<1 and f,g are two nonnegative measurable functions. The main result of this work is to analyze the interaction between the potential and the gradient terms in order to get the existence of a positive solution.

Submitted July 6, 2016. Published April 26, 2017.
Math Subject Classifications: 35K15, 35K55, 35K65, 35B05, 35B40.
Key Words: Elliptic system; Schauder fixed point theorem; gradient dependance.

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Ahmed Attar
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées
Département de
Mathématiques
Université Abou Bakr Belkaïd, Tlemcen
Tlemcen 13000, Algeria
email: ahm.attar@yahoo.fr
Rachid Bentifour
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées
Département de
Mathématiques
Université Abou Bakr Belkaïd, Tlemcen
Tlemcen 13000, Algeria
email: rachidbentifour@gmail.com

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