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Nonradial blow-up solutions of sublinear elliptic equations with gradient term. (English) Zbl 1136.35362

Summary: Let \(f\) be a continuous non-decreasing function such that \(f>0\) on \((0,\infty), f(0)=0, \sup_{s\geq1}f(s)/s<\infty\), and let \(p\) be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation \(\Delta u+|\nabla u|=p(x)f(u)\) in \(\Omega\), where \(\Omega\) is either a smooth bounded domain or \(\Omega={\mathbb R}^N\). If \(\Omega\) is bounded, we prove that the above problem never has a blow-up boundary solution. Since \(f\) does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case \(\Omega={\mathbb R}^N\) a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.

MSC:

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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