General uniqueness results and variation speed for blow-up solutions of elliptic equations

Abstract

Let Ω be a smooth bounded domain in RN. We prove general uniqueness results for equations of the form − Δ u = au − b(x) f(u) in Ω, subject to u = ∞ on ∂ Ω. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow‐up behavior of u(x) with f(u) and b(x), where b ≡ 0 on ∂ Ω and a certain ratio involving b is bounded near ∂ Ω. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov. 2000 Mathematics Subject Classification 35J25 (primary), 35B40, 35J60 (secondary).

Uniqueness for boundary blow-up problems with continuous weights

“…is not required to vanish identically on ∂Ω, as in most of the previous results in the literature (for instance [6] or [23]). However, condition (A) is only meaningful at a point x 0 ∈ ∂Ω when a(x 0 ) = 0; otherwise it is trivially satisfied.…”

“…(b) The uniqueness result contained in Theorem 1 is complementary to those in [6] or [23]. However, it has the advantage to require mere continuity to the weight a(x).…”

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

“…López-Gómez [27] established the blow-up rate where b(x) is more general without assuming the decay rate of b(x) to be approximated by a power of the distance function near the boundary. Some other related work can be found in [6,8,9,12,28,30], etc. when m > 1, the existence, uniqueness (for the case a(x) ≥ 0) and asymptotic behavior had been studied by Delgado et al [14,15], Peng [32] when f (u) ∼ Ku p/m with p > m, and by Li et al [23] when the variation of f is regular, and [24] for some kinds of functions with non-regular variation at infinity.…”

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.…”