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Nonradial blow-up solutions of sublinear elliptic equations with gradient term
Let $f$ be a continuous and non-decreasing function such that
$f>0$ on $(0,\infty)$, $f(0)=0$, su$p_{s\geq 1} f(s)/s< \infty$
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 has never
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.