Weak solutions for singular quasilinear elliptic systems

Abstract

We investigate the quasilinear elliptic systemwhere Ω ⊂ R N (N ≥ 1) is a bounded and smooth domain, 1 < m < ∞, p, q, r, s > 0. Under certain conditions imposed on the exponents we obtain the existence and uniqueness of a weak solution (u, v) with u, v ∈ W 1,m 0(Ω) ∩ C(Ω). We also investigate the W 1,τ 0 (Ω) regularity of solution and determine the optimal range of τ ≥ m for such regularity.

Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity

“…In Giacomoni et al, 23 considering the nonlinear case 1 < p < ∞ and combining subsolutions–supersolutions method with Schauder's fixed‐point theorem, the authors proved the existence, uniqueness, and regularity of the weak solution to the following system: where 1 < p , q < ∞ , and the numbers α 1 , α 2 , β 1 , β 2 > 0 satisfy suitable restrictions. The required compactness of involved operators is ensured by a Hölder regularity result of independent interest they proved for weak energy solutions to a scalar problem associated to () (see also Singh 24 for related issues). Recently, Candito et al 25 and Chu et al 26 used the same approach to get the existence of positive solutions to other kinds of quasilinear elliptic and singular systems (see also previous studies 27–29 for further extensions).…”

Some recent results on singular $ p $-Laplacian systems

“…and suitable upper bounds regarding α i , β i , they proved that (16) admits a solution (u, v) ∈ X p,q 0 (Ω) ∩ C 0,α (Ω) 2 . When p = q, see also [70]. Existence results for problem (11) where the competitive structure (a 1 ) is allowed can be found in [60,61].…”

Some recent results on singular p-laplacian systems

“…and suitable upper bounds regarding α i , β i , they proved that (3.12) admits a solution (u, v) ∈ X p,q 0 (Ω) ∩ C 0,α (Ω) 2 . When p = q, see also [69].…”