The solvability of an elliptic system under a singular boundary condition

Abstract

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

Nondegeneracy and uniqueness for boundary blow-up elliptic problems

“…After [7], an enormous amount of works have dealt with these problems, mainly concerned with the issues of existence, uniqueness and behavior near the boundary for positive solutions, both for single equations (see [1,[3][4][5][6]8,9,11,12,15,17,19,[25][26][27][28][29][30][32][33][34][35][36]) and lately for systems (cf. [10,13,14,[20][21][22]31]) . We also mention that there have been some recent applications of this kind of problems, for instance to Liouville theorems for logistic-like equations in R N in [16] or to the analysis of blow-up for a parabolic equation with a nonlinear boundary condition in [2].…”

Boundary blow-up solutions to elliptic systems of competitive type

“…We quote [8] for predator-prey LotkaVolterra systems [11,12,16,25], competitive type systems and [17] cooperative systems. However, we remark that boundary blow-up solutions (sometimes called "large") are explicitly treated only in [16,17]. Also, this seems to be the first work with regard to the boundary conditions (SF).…”

Existences and Boundary Behavior of Boundary Blow-Up Solutions to Quasilinear Elliptic Systems With Singular Weights

“…More results to system with boundary blow up, we refer reader to [6,7,8,16,17,18,22,24,28,29,31,32] and references therein.…”