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