Nonradial blow-up solutions of sublinear elliptic equations with gradient term

Marius Ghergu; Vicenţiu Rădulescu

doi:10.3934/cpaa.2004.3.465

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2004, Volume 3, Issue 3: 465-474. Doi: 10.3934/cpaa.2004.3.465

This issue Previous Article Upper and lower bounds on Mathieu characteristic numbers of integer orders Next Article The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources

Nonradial blow-up solutions of sublinear elliptic equations with gradient term

Marius Ghergu^1, and
Vicenţiu Rădulescu¹

1.
Department of Mathematics, University of Craiova, Street A. I. Cuza No. 13, 200 585 Craiova

Received: March 2003

Revised: February 2004

Published: September 2004

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Abstract

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.

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Mathematics Subject Classification: 35B50, 35J60, 58J05.

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