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

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.

Keywords

• Boundary blow-up solutions
• nonlinear gradient terms
• Karamata regular variation