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A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term. (English) Zbl 1128.35341

Summary: By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions to the semilinear elliptic problem \[ \Delta u -|\nabla u|^q=b(x)g(u),\quad u>0 \quad\text{in }\Omega, \quad u\big|_{\partial \Omega}=+\infty, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(q\in (1, 2]\), \(g \in C[0,\infty)\cap C^1(0, \infty)\), \(g(0)=0\), \(g\) is increasing on \([0, \infty)\), and \(b\) is non-negative non-trivial in \(\Omega\), which may be singular or vanishing on the boundary.

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B50 Maximum principles in context of PDEs
35R05 PDEs with low regular coefficients and/or low regular data
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