Quasilinear and singular systems: The cooperative case

Abstract

We investigate the following quasilinear and singular system,            −∆ p1 u = u α1 v β1 in Ω −∆ p2 v = u α2 v β2 in Ω u > 0, v > 0, u, v = 0 on ∂Ω, (P) where Ω is an open bounded domain with smooth boundary, 1 < p i < ∞ and α i + β i < 0 for any i = 1, 2. We employ monotone methods in order to show the existence of a unique (positive) solution of problem (P) in some cone. When α i + β i > −1 for i = 1, 2, we prove a regularity result for solutions to problem (P) in C 1,β (Ω) with some β ∈ (0, 1). Furthermore, we show that min i=1,2 α i + β i > −1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C 1 (Ω).

Quasilinear and singular elliptic systems

“…Remark 2.2. Instead of conditions (2.3) and (2.4), as in [12], we can rather suppose that there exist κ 1 , κ 2 > 0 and α 1 , α 2 > 0 such that for all (u, v) ∈ C,…”

“…Related problems for singular quasilinear systems have been also studied in [16] and [12]. Accordingly, we study in our paper a more general situation that handle more singular cases.…”

Singular Quasilinear Elliptic Systems with (super-) Homogeneous Condition

“…It also arises in the study of population dynamics [21], quasiconformal mappings [11] and other topics in geometry [22]. Recently, singular system (1.1) with cooperative structure was mainly studied in [6,7,16]. In [16] existence and boundedness theorems for (1.1) was established by using the sub-supersolution method for systems combined with perturbation techniques.…”

Positive solutions for infinite semipositone/positone quasilinear elliptic systems with singular and superlinear terms

“…Relevant contributions regarding the cooperative case of system (1.4), that is α 2 , β 1 > 0, can be found in [8,9,18]. With regard to the complementary situation α 2 , β 1 < 0 which is the so called competitive structure for system (1.4), we quote the papers [9,17,19].…”