Large solutions to an anisotropic quasilinear elliptic problem

Abstract

Abstract. In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem divx(|∇xu| p−2 ∇xu)(x, y) + divy(|∇yu| q−2 ∇yu)(x, y) = u r (x, y) in a bounded domain Ω ⊂ R N × R M , together with the boundary condition u(x, y) = ∞ on ∂Ω. We prove that the necessary and sufficient condition for the existence of a solution u ∈ W 1,p,q loc (Ω) to this problem is r > max{p − 1, q − 1}. Assuming that r > q − 1 ≥ p − 1 > 0 we will show that the exponent q controls the blow-up rates near the boundary in the sense that all points of ∂Ω share the same profile, that depends on q and r but not on p, with the sole exception of the vertical points (where the exponent p plays a role).

Strong Maximum Principles for Anisotropic Elliptic and Parabolic Equations

“…Elliptic equations like (1.2) also received much attention in recent years. Possible references on elliptic equations like (1.2) are Alves-El Hamidi [1], Antontsev-Shmarev [4], Cianchi [13], D'Ambrosio [14], Di Castro [16], Di Castro-Montefusco [17],El Hamidi-Rakotoson [19,20], El Hamidi-Vétois [21], Fragalà-Gazzola-Kawohl [23], Fragalà-Gazzola-Lieberman [24], García-Melián-Rossi-Sabina de Lis [25], Li [28], Lieberman [29,30] [36], Skrypnik [42], Tersenov-Tersenov [43], and Vétois [45][46][47][48]. We refer to MercaldoRossi-Segura de León-Trombetti [33] for a description of the asymptotic behavior of solutions of equations like (1.2) as p − → 1, and we refer to Di Castro-Pérez-Llanos-Urbano [18] and Pérez-Llanos-Rossi [40] for the case p − → ∞, where p − = min (p 1 , .…”

Limit cases in an elliptic problem with a parameter-dependent boundary condition

“…The existence of v B is well documented (see, for instance, [13] and [7], Theorem 3). In conclusion,…”

Bifurcation for positive solutions of nonlinear diffusive logistic equations in R^N with indefinite weight

“…where c 7 > 0 is a constant depending on r 0 and ρ, but not on r. Now, we return to (15) and use (6), (9), (10), (14) and (15), obtaining…”

“…Problem (P λ ) settled in a bounded region with Neumann boundary condition was studied by Cantrell-Cosner-Hutson [4] and Umezu [25], and extensions to the p-Laplacian Dirichlet case can be found in the works of Dong [7], Garcia Melian-Sabina de Lis [15], Guedda-Veron [17], Kamin-Veron [19] and Papageorgiou-Papalini [21]. A related Neumann problem can be found also in Cardinali-Papageorgiou-Rubbioni [5].…”