## 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