A unified asymptotic behavior of boundary blow-up solutions to elliptic equations. (English) Zbl 1299.35142

This paper is concerned with the qualitative study of solutions to the semilinear elliptic problem \(\Delta u=b(x)f(u)\) in \(\Omega \) that blow up at the boundary, that is, \(u(x)=\infty \) on \(\partial \Omega \). Here, \(\Omega \subset \mathbb {R}^N\) \((N\geq 3)\) is a bounded and smooth domain, \(b,f\geq 0\) and \(b=0\) on \(\partial \Omega \).
The authors establish a unified characterization of the rate at which the blow-up occurs at the boundary under the hypothesis that \(f(u)\) is either a normalized varying function or a rapidly varying function at infinity. More precisely, the authors determine explicitly the second term of the expansion of the solution near the boundary. The approach relies on Karamata regular variation theory combined with various comparison results.


35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35C20 Asymptotic expansions of solutions to PDEs
35J75 Singular elliptic equations
35B40 Asymptotic behavior of solutions to PDEs