Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

García-Melián, J.; Letelier-Albornoz, R.; Sabina de Lis, J.

doi:10.1090/S0002-9939-01-06229-3

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up
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by J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis PDF

Proc. Amer. Math. Soc. 129 (2001), 3593-3602 Request permission

Abstract:

In this paper we prove uniqueness of positive solutions to logistic singular problems $-\Delta u=\lambda (x) u -a(x) u^{p}$, $u_{|\partial \Omega }=+\infty$, $p>1$, $a>0$ in $\Omega$, where the main feature is the fact that $a_{|\partial \Omega }=0$. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near $\partial \Omega$. This expansion involves both the distance function $d(x)=\text {dist}(x,\partial \Omega )$ and the mean curvature $H$ of $\partial \Omega$.

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