‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour

Bandle, Catherine; Marcus, Moshe

doi:10.1007/bf02790355

Cited by 315 publications

(218 citation statements)

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“…Its study seems to have been started in [21], where the case a(x) = 1, p = (N + 2)/ (N − 2) was analyzed, and later continued in [19], [1], [2], [25], [24] (see also [10] for the p-Laplacian analogue), where a(x) ≥ a 0 > 0 in Ω and a linear term −λu is added in the equation in some cases. In all of them, uniqueness was obtained by means of precise boundary estimates, which took the form u ∼ Ad −α as d → 0, where A is given in terms of p and α = 2/(p − 1).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Uniqueness for boundary blow-up problems with continuous weights

García-Melián

2007

Proc. Amer. Math. Soc.

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Section: Introduction and Resultsmentioning

confidence: 99%

Uniqueness for boundary blow-up problems with continuous weights

García-Melián

2007

Proc. Amer. Math. Soc.

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“…holds true for smallx 1 The proof is completed by observing that d(x, y) = x 1 on points on the normal direction close enough to (0, 0).…”

Section: Boundary Behavior In General Domainsmentioning

confidence: 98%

“…Since the pioneering works of Bieberbach, [2], and Rademacher, [21], and further continuations by Keller, [14], Osserman, [20], Loewner-Nirenberg, [18], and Bandle-Marcus [1], a great amount of research has been devoted to study such problems. In fact, they arise in completely different fields as Riemannian geometry or population dynamics.…”

Section: Introductionmentioning

confidence: 99%

Large solutions to an anisotropic quasilinear elliptic problem

García-Melián

Rossi

Lis

2010

Annali di Matematica

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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).

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“…Motivated by certain geometric problems, they established the uniqueness for the case f (u) = u N +2 N −2 (N > 2). Bandle and Marcus [3] give results on asymptotic behaviour and uniqueness of the large solution for more general nonlinearities including f (u) = u p for any p > 1. Theorem 2.3 in [3] proves that when (A) holds and (B)…”

Section: Such That U(x) − Log(d(x)mentioning

confidence: 99%

Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type

Cîrstea

Rădulescu

2007

Trans. Amer. Math. Soc.

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Abstract. We establish the uniqueness of the positive solution for equations of the form −∆u = au − b(x)f (u) in Ω, u| ∂Ω = ∞. The special feature is to consider nonlinearities f whose variation at infinity is not regular (e.g., exp(u) − 1, sinh(u), cosh(u) − 1, exp(u) log(u + 1), u β exp(u γ ), β ∈ R, γ > 0 or exp(exp(u)) − e) and functions b ≥ 0 in Ω vanishing on ∂Ω. The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the nonregular variation of f at infinity with the blow-up rate of the solution near ∂Ω.

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‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour

Cited by 315 publications

References 4 publications

Uniqueness for boundary blow-up problems with continuous weights

Uniqueness for boundary blow-up problems with continuous weights

Large solutions to an anisotropic quasilinear elliptic problem

Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type

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