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

Lazer, A. C.; McKenna, P. J.

Differ. Integral Equ. 7, No. 3-4, 1001-1019 (1994).

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

Keywords:

boundary blow-up problems; singular elliptic boundary value problem