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Explosive solutions of elliptic equations with absorption and non-linear gradient term. (English) Zbl 1032.35070

Summary: Let \(f\) be a non-decreasing \(C^1\)-function such that \(f>0\) on \((0,\infty)\), \(f(0)=0\), \(\int^\infty_11/ \sqrt{F(t)}dt <\infty\) and \(F(t)/f^{2/a} (t)\to 0\) as \(t\to\infty\), where \(F(t)=\int^t_0 f(s)ds\) and \(a\in (0,2]\). We prove the existence of positive large solutions to the equation \(\Delta u+q(x) |\nabla u|^a= p(x)f(u)\) in a smooth bounded domain \(\Omega \subset\mathbb{R}^N\), provided that \(p,q\) are non-negative continuous functions so that any zero of \(p\) is surrounded by a surface strictly included in \(\Omega\) on which \(p\) is positive. Under additional hypotheses on \(p\) we deduce the existence of solutions if \(\Omega\) is unbounded.

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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