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Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up. (English) Zbl 0989.35044

The paper deals with uniqueness of positive solutions to the problem \(-\Delta u=\lambda(x)u-a(x)u^p,\) \(u|_{\partial\Omega}=+\infty,\) \(p>1,\) \(a(x)>0\) in \(\Omega\) and \(a|_{\partial\Omega}=0.\) Exact asymptotic estimates are provided describing the blow-up rate near \(\partial\Omega\) in terms of the distance function \(\text{dist }(x,\partial\Omega)\) and the mean curvature H of \(\partial\Omega.\)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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