Uniqueness for boundary blow-up problems with continuous weights

Abstract

Abstract. In this paper, we prove that for p > 1 the problem ∆u = a(x)u p in a bounded C 2 domain Ω of R N has a unique positive solution with u = ∞ on ∂Ω. The nonnegative weight a(x) is continuous in Ω, but is only assumed to verify a "bounded oscillations" condition of local nature near ∂Ω, in contrast with previous works, where a definite behavior of a near ∂Ω was imposed.

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

“…For studies of other boundary blow-up problems, we also refer the reader to [1,2,5,7,17,18,[20][21][22]25,29,33] and the references therein. [13,15].…”

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

“…Very recently, Zhang [33] and Yang [23] extended the above results to the problem (1.3) and gained some new results with nonlinear gradient terms. Problem (1.3) was discussed in a number of works; see, [2,3,4,5,9,10,11,12,13,19,23,25,34], Now let us return to problem (1.1). When m = n = 2, system (1.1) becomes 4) in the paper [14], when a(x) = 1, b(x) = 1, under Dirichlet boundary conditions of three different types: both components of (u, v) are bounded on ∂Ω (finite case); one of them is bounded while the other blows up(semilinear case); or both components blow up simultaneously(infinite case), under the assumption that(a − 1)(e − 1) > bc, necessary and suffcient conditions for existence of positive solutions were found, and uniqueness or multiplicity were also obtained, together with the exact boundary behavior of solutions.…”

Large solutions for equations involving the p-Laplacian and singular weights

“…For this aim some a priori "rough" estimates of solutions are needed. They are provided by the next lemma (see [3], [25], [26], [6], [17] or [19] for related results).…”

“…We refer the reader to [35] and [5]. We also quote the works [28], [24], [2], [3], [40], [11], [10], [18], [15], [16] and [17] where several interesting features related to (1.1) with p = 2 are analyzed (see also [19], where p = 2 and q is allowed to be a variable positive function).…”

“…The basic question is to find sufficient conditions on the weight a to guarantee uniqueness. Some conditions of this type have been found in [7], [8], [9], [29], [30], [17].…”