Singular quasilinear elliptic systems in $${\mathbb {R}}^{N}$$RN

Abstract

The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder's fixed point theorem.2010 Mathematics Subject Classification. 35J75; 35J48; 35J92.

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.…”

“…Proof of Theorem 1.1. Consider (λ, β) ∈ R, where the set R was defined in (7). Let M, γ > 0 be the constants given in Lemma 2.2 and in the equation ( 4) respectively.…”

“…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.…”

Strongly singular convective elliptic equations in RN driven by a non-homogeneous operator

“…Nevertheless, a sub-super-solution approach is available in several contexts; see, e.g., [26]. Existence of solutions in R N is treated in [10] for non-singular equations driven by the (p, q)-Laplacian, while in [22] an existence result for singular p-Laplacian systems in the whole space is obtained. The very recent work [17] treats singular convective p-Laplacian systems in the whole R N .…”

Existence of entire solutions for singular quasilinear convective elliptic systems

“…We also point out that, in the nonsingular case α 1 = β 2 = 0, the required properties on ζ 1 (and analogously for ζ 2 ) appearing in (3) guarantee that the right-hand side of (1) belongs to L r (R N ), r > N p , which is the minimum requirement (among Lebesgue spaces) on the right-hand side of the p-Poisson equation −∆ p w = h(x) to get w ∈ L ∞ (R N ), making condition (3), in a certain sense, natural. The prototype of (1), obtained by setting γ i = δ i = 0 and m i = M i , has a cooperative structure, i.e., f is increasing in v and g is increasing in u; however, we require no monotonicity assumptions on f, g. The Dirichlet version of (1) in bounded domains has been investigated in [2], while [12] deals with (1) for α 2 = β 1 = 0 and without convection terms (i.e., terms depending on the gradient of solutions). The present investigation follows the direction of the recent papers [11,9,8], regarding singular convective problems in bounded domains, with different boundary conditions.…”