Entropy solutions for some nonlinear and noncoercive unilateral elliptic problems
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Entropy solutions for some nonlinear and noncoercive unilateral elliptic problems Youssef Akdim1 · Mohammed Belayachi2 · Hassane Hjiaj3 · Mounir Mekkour1 Received: 26 April 2019 / Accepted: 5 December 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract This paper is concerned with the study of the existence results to the obstacle problem associated with the equation having degenerate coercivity, whose prototype is given by: − div(b(|u|)|∇u| p−2 ∇u) + d(|u|)|∇u| p = f (x, u) in , u=0
on ∂,
where is a bounded open set of R N (N ≥ 2), with 1 < p < N , and f (·, s) satisfying some growth condition. We show the existence of entropy solutions for this non-coercive unilateral elliptic equation, and we will conclude some regularity results. Keywords Nonlinear elliptic equations · Non-coercive problems · Entropy solutions · Obstacle problems Mathematics Subject Classification 35J60 · 46E30 · 46E35
1 Introduction Let be a bounded open set of R N (N ≥ 2), and let 1 < p < N .
B
Mohammed Belayachi [email protected] Youssef Akdim [email protected] Hassane Hjiaj [email protected] Mounir Mekkour [email protected]
1
LAMA Laboratory, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, P.O. Box 1796, Atlas Fez, Morocco
2
Laboratory LSI, Department of Mathematics, Physics and Informatics, Faculty Polydisciplinary of Taza, University Sidi Mohamed Ben Abdellah, P.O. Box 1223, Taza Gare, Morocco
3
Department of Mathematics, Faculty of Sciences Tétouan, University Abdelmalek Essaadi, BP 2121, Tétouan, Morocco
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Y. Akdim et al.
Boccardo, Murat and Puel have considered in [11] the strongly nonlinear Dirichlet problem Au + g(x, u, ∇u) = 0 in , (1.1) u=0 on ∂, where A is a Leray–Lions operator verifying the standard growth, monotony and coercivity conditions, the strongly nonlinear term g(x, s, ξ ) verify the sign and growth conditions. They have proved the existence of bounded solutions for the nonlinear elliptic unilateral associate to the problem (1.1). Bensoussan, Boccardo and Murat have studied in [5] the nonlinear elliptic problem Au + g(x, u, ∇u) = f (x) in , (1.2) u=0 on ∂,
where the data f (x) is assumed to be in the dual W −1, p (). They have proved the existence of weak solutions. Moreover some regularity results have been concluded. In the case of f (x) ∈ L 1 (), an existence result for the problem (1.2) was proved in [10] where the authors have assumed that the nonlinear lower order term g(x, s, ξ ) having natural growth (of order p) with respect to |∇u|, and satisfying the sign-condition g(x, s, ξ )s ≥ 0 (see the references therein). Also, the existence result was proved in [7,9] for the problem (1.2) in the case of g(x, s, ξ ) = 0. In [3], Benkirane and Elmahi have studied the problem (1.2), where g(x, s, ξ ) is a Carathéodory function that satisfying the following coercivity condition |g(x, s, ξ )| ≥ ν|ξ | p for |s| ≥ γ ,
(1.3)
where ν and γ are two positive constants. In [14], Elmahi
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