Existence of solutions for a class of singular elliptic systems with convection term

Abstract

We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines sub-and supersolution method with the pseudomonotone operator theory and perturbation arguments involving singular terms. Systems (S) ± can be see as a version of the singular scalar equations± with α, β > 0 and g : R N → R be a continuous function verifying some technical conditions. Several works are devoted to classes of problems covering (P ) ± . For instance, see the papers of Aranda [5], Ghergu and Radulescu [19,20], Giarrusso and Porru [21], Lair and Wood [24], Zhang [29] and references therein. Problem (P ) ± without a convection term, that is g = 0 was also investigated. Relevant contributions regarding this situation can be found in [8-12,14]. The main tools used in the aforementioned works are Sub-and Supersolution, Fixed Point Theorems, Bifurcation Theory and Galerkin Method. On the other hand, using variational technique, more precisely mountains pass theorem, de Figueiredo, Girardi and Matzeu [13] studied a class of elliptic problems without singularity, where the nonlinearity depends of the gradient of the solution. Related to systems (S) ± , to date, the only case considered in the literature, known to authors for g i = 0 is the paper due to Alves, Carrião and Faria [1]. For the case where g i = 0, we refer the reader to the survey paper by Alves and Corrêa [2], Alves, Corrêa and Gonçalves [3], El Manouni, Perera and Shivaji [15], Ghergu [17,18], Hernández, Mancebo and Vega [22], Montenegro and Suarez [26] and Motreanu and Moussaoui [27]. From the above commentaries, we observe that in recent years singular elliptic problems with convection term has received few attention. Motivated by this fact, our aim is to show the existence of solutions for a class of elliptic systems where the nonlinearity besides a singular term has a convection term. The proof combines results involving pseudomonotone operators, sub-and supersolution method and perturbation arguments involving singular terms. We emphasize that our study complete those made in [2,3] and [27], in the sense that in those papers the authors did not consider the case where the nonlinearity has a convection term, and also [1], because a different type of singular term was considered. The method used in the present work is different from those applied in the aforementioned papers.Our main result is the following theorem.

Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs

“…Different approaches in the study of PDEs with gradient terms have been proposed: for example sub-and super-solutions, topological degree theory, mountain pass techniques. We mention, for instance, the pioneering works of Amann and Crandall [3], Brézis and Turner [4], Mawhin and Schmitt [22,23], Pokhozhaev [25] and the more recent contributions [1,7,9,10,13,26,27,29]. See also the very recent survey [8] and references therein.…”

Existence of Solutions for a Quasilinear System with Gradient Terms

“…At this point, we quote that, to the best of our knowledge, it is the first time that the system (1) is considered under the conditions (H 1 )-(H 2 ). This manuscript also completes the study done in [2] due to the fact that a system version of the problem in [2] is studied and the papers [1,[3][4][5][6][7][8] in the sense that different hypotheses can be considered to study systems with convection terms and involving the p-Laplacian operator.…”

“…where h is a sublinear function and f may exhibit a growth higher than p in the gradient variable. For other interesting papers which consider problems involving gradient terms see [4][5][6][7][8] and the references therein.…”

Singular Quasilinear Elliptic Systems With Gradient Dependence

“…Unlike quasilinear systems, meaningful contributions are already available for semilinear singular convective systems (i.e., p 1 = p 2 = 2). Here, we quote for instance [2,3] and the references therein, where the linearity of the principal part has been crucial. The particular case in singular system (P) when the convection terms ∇u and ∇v are removed has received a special attention.…”