Boundary blow-up solutions to semilinear elliptic equations with nonlinear gradient terms. (English) Zbl 1288.35253

Liu, Shufang; Xu, Yonglin

Electron. J. Differ. Equ. 2014, Paper No. 09, 20 p. (2014).

Summary: In this article we study the blow-up rate of solutions near the boundary for the semilinear elliptic problem \[ \displaylines{ \Delta u\pm |\nabla u|^q=b(x)f(u), \quad x\in\Omega,\cr u(x)=\infty, \quad x\in\partial\Omega, } \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), and \(b(x)\) is a nonnegative weight function which may be bounded or singular on the boundary, and f is a regularly varying function at infinity. The results in this article emphasize the central role played by the nonlinear gradient term \(|\nabla u|^q\) and the singular weight \(b(x)\). Our main tools are the Karamata regular variation theory and the method of explosive upper and lower solutions.

MSC:

35J61	Semilinear elliptic equations
35J25	Boundary value problems for second-order elliptic equations
35B50	Maximum principles in context of PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35B44	Blow-up in context of PDEs

Keywords:

boundary blow-up solutions; nonlinear gradient terms; Karamata regular variation