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

Abstract

In this paper we consider the boundary blow-up problem ∆ p u = a(x)u q in a smooth bounded domain Ω of R N , with u = +∞ on ∂Ω. Here ∆ p u = div(|∇u| p−2 ∇u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.

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.…”

“…This problem has been recently considered in [11], where all issues concerning existence, uniqueness and asymptotic behavior near the boundary of positive solutions were obtained. The following Lemma 2.1 contains the basic feature of problem (2.1), refer the reader to [11] for a proof.…”

Large solutions to an anisotropic quasilinear elliptic problem

“…Of particular importance are the special cases f (u) = u r and f (u) = e u . On the other hand, only few works have dealt with the p-Laplacian operator as left hand side in (1.1) (see [5], [19], [10], [11] and Remark 2).…”

Soluções blow-up para equações elípticas com peso singular ou expoente variável

“…( ver [5]). Alguns trabalhos, como [2], [13], [20] já provaram existência de solução para problemas parecidos com (2).…”

“…Outros trabalhos também estudam condições para a(x), para garantir existên-cia e unicidade de solução para (5). Vamos estudar, também, o problema (1) quando p = 2, a(x) = 1, Ω = R n e q : R n →…”