Existence of radial solutions for an elliptic problem involving exponential nonlinearities

M. Grossi

doi:10.3934/dcds.2008.21.221

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2008, Volume 21, Issue 1: 221-232. Doi: 10.3934/dcds.2008.21.221

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Existence of radial solutions for an elliptic problem involving exponential nonlinearities

M. Grossi^1,

1.
Dip. di Matematica, Università di Roma "La Sapienza", P.le A.Moro 2 - 00185 - Roma

Received: January 31, 2007

Revised: August 31, 2007

Published: December 2007

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Abstract

Let us consider the problem

$-\Delta u+a(|x|)u=\lambda e^u$in$\ B_1,$ (0.1)
$u=0$ on$ \partial B_1.$

where $B_1$ is the unit ball in $R^N$, $N\ge2$, $\lambda>0$ and $a(|x|)\ge0$ is a smooth radial function.
Under some suitable assumptions on the regular part of the Green function of the operator $-u''- \frac{N-1}{r}u+a(r)u$, we prove the existence of a radial solution to (0.1) for $\lambda$ small enough.

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Mathematics Subject Classification: 35J60.

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