Multiple positive solutions for a class of quasilinear singular elliptic systems
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Multiple positive solutions for a class of quasilinear singular elliptic systems Hana Didi1 · Abdelkrim Moussaoui2 Received: 5 June 2019 / Accepted: 28 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray– Schauder Topological degree. Keywords Singular system · p-Laplacian · Leray–Schauder degree · Regularity Mathematics Subject Classification 35J75 · 35J48 · 35J92
1 Introduction We consider the following system of quasilinear elliptic equations: ⎧ − p u = f (u, v) in , ⎪ ⎪ ⎨ −q v = g(u, v) in , u, v > 0 in , ⎪ ⎪ ⎩ u, v = 0 on ∂,
(P)
where is a bounded domain in R N (N ≥ 2) with C 1,α boundary ∂, α ∈ (0, 1), p and q , 1 < p, q < N , are the p-Laplacian and q-Laplacian operators, respectively, that is, p u = div(|∇u| p−2 ∇u) and q v = div(|∇v|q−2 ∇v). The nonlinearities f , g : (0, +∞) × (0, +∞) → (0, +∞) are continuous functions satisfying the growth condition: (H.1) For every L¯ > 0, there are constants m i , Mi > 0 (i = 1, 2) such that m 1 s α1 t β1 ≤ f (s, t) ≤ M1 s α1 t β1 , α2 β2
m2s t
B
α2 β2
≤ g(s, t) ≤ M2 s t ,
¯ and all t > 0, for all 0 < s < L, ¯ for all 0 < t < L, and all s > 0,
Hana Didi [email protected] Abdelkrim Moussaoui [email protected]
1
Mathematic Department, Badji-Mokhtar Annaba University, 23000 Annaba, Algeria
2
Biology Department, A. Mira Bejaia University, Targa Ouzemour, 06000 Bejaïa, Algeria
123
H. Didi , A. Moussaoui
with
−1 < α1 < 0 < β1 < p − 1 −1 < β2 < 0 < α2 < q − 1.
(1.1)
(H.2) For every L¯ ∗ > 0 there exist constants J1 > λ1, p and J2 > λ1,q such that f (s, t) = J1 for all 0 < t < L¯ ∗ , s p−1 g(s, t) = J2 for all 0 < s < L¯ ∗ . lim t→∞ t q−1 lim
s→∞
We provide an example where (H.1) and (H.2) are fulfilled. Notice that under the above assumptions system (P) is cooperative, that is, for u (resp. v) fixed the right term in the first (resp. second) equation of (P) is increasing in v (resp. u). Example 1 Let θ ∈ Cc (R) with θ (s) = 1 on bounded sets. Consider the functions f , g : (0, +∞) × (0, +∞) → (0, +∞) defined by the following: f (s, t) = θ (s)s α1 t β1 + (1 − θ (s))J1 s p−1 , for s, t > 0 and g(s, t) = θ (t)s α2 t β2 + (1 − θ (t))J2 t q−1 , for s, t > 0, which clearly verify assumptions (H.1) and (H.2). The study of singular elliptic problems is greatly justified because they arise in several physical situations such as fluid mechanics pseudoplastics flow, chemical heterogeneous catalysts, non-Newtonian fluids, biological pattern formation and so on. In Fulks and Maybee [10], the reader can find a very nice physical illustration of a practical problem which leads to singular problem. With respect to singular system it is worth to cite, among others, the important Gierer– Meinhardt system which is the stationary counterpart of a parabolic system proposed by Gierer–Meinhardt (see [8,15]) which occurs in the study
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