Electron. J. Diff. Equ., Vol. 2014 (2014), No. 09, pp. 1-20.

Boundary blow-up solutions to semilinear elliptic equations with nonlinear gradient terms

Shufang Liu, Yonglin Xu

Abstract:
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.

Submitted October 4, 2013. Published January 7, 2014.
Math Subject Classifications: 35J25, 35B50, 65J65.
Key Words: Boundary blow-up solutions; nonlinear gradient terms; Karamata regular variation.

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Shufang Liu
Department of Mathematics
Gansu Normal University for Nationalities
Hezuo, Gansu 747000, China
email: shuxueliushufang@163.com
Yonglin Xu
School of Mathematics and Computer Science Institute
Northwest University for Nationalities
Lanzhou, Gansu 730030, China
email: xuyonglin000@163.com

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