Nondegeneracy and uniqueness for boundary blow-up elliptic problems

Abstract

In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem u = a(x)u p + g(x, u) in , with u = +∞ on * , has a unique positive solution for large enough , and determine its asymptotic behavior as → +∞. Here p > 1, a(x) is a continuous function which can be singular near * and g(x, u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u p by e u or a "logistic type" function f (u).

Boundary asymptotical behavior of large solutions to Hessian equations

“…There are many papers resolving existence, uniqueness and asymptotic behavior issues for blow-up solutions of semilinear/quasilinear elliptic equations: for instance [Osserman 1957;Resnick 1987;Véron 1992;Bandle and Marcus 1992;1995;García-Melián et al 2001;Chuaqui et al 2004;Cîrstea and Rȃdulescu 2006;García-Melián 2006].…”

Uniqueness for boundary blow-up problems with continuous weights

“…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). We refer the interested reader to [13] for an extensive list of references with more general nonlinearities f (u) in (1.1). We only mention the pioneering works [3] and [17].…”

“…Our intention in the present work is to give a simple sufficient condition on a(x) to ensure uniqueness of the positive solution. Our proof does not make use of boundary estimates, but of a refinement of an iterative procedure due to Safonov, which has been previously used in [18], [5], [11] and [13]. This is why we do not need a definite behavior of a(x) near ∂Ω, nor do we require any further regularity than continuity.…”

“…(d) The proof of Theorem 1 can be extended to cover some more general nonlinearities f (u), as in [13]. It suffices that f (u) ∼ u p as u → ∞, with f (u)/u increasing.…”

Large solutions to an anisotropic quasilinear elliptic problem

“…In fact, they arise in completely different fields as Riemannian geometry or population dynamics. We refer the interested reader to [9] and [22] for a complete updated account of references and applications.…”