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Asymptotic behavior of solutions of boundary blow-up problems. (English) Zbl 0811.35010

Summary: We consider the singular elliptic boundary value problem \(\Delta u(x) = f(u(x))\), \(x \in \Omega\), \(u(x) \to \infty\) as \(d(x) \to 0\), where \(d(x) = \text{dist} (x, \partial \Omega)\). Conditions are given which imply that for all such solutions there holds the asymptotic condition \(u(x) - Z(d(x)) \to 0\) as \(d(x) \to 0\), where \(Z\) is a blowup solution of \(Z''(r) = f(Z(r))\). These conditions also imply existence and uniqueness of \(u(x)\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
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