Existence results for nonlinear elliptic equations related to Gauss measure in a limit case

Giuseppina di Blasio; Filomena Feo; Maria Rosaria Posteraro

doi:10.3934/cpaa.2008.7.1497

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2008, Volume 7, Issue 6: 1497-1506. Doi: 10.3934/cpaa.2008.7.1497

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Existence results for nonlinear elliptic equations related to Gauss measure in a limit case

1.
Seconda Universitá degli Studi di Napoli, Dipartimento di Matematica, Via Vivaldi 43, Caserta
2.
Universitá degli Studi di Napoli Parthenope, Dipartimento per le Tecnologie, Is. C4, Centro Direzionale, Napoli
3.
Universitá degli Studi di Napoli “Federico II”, Dipartimento di Matematica “R. Caccioppoli”, Complesso Monte S. Angelo, Napoli

Received: September 30, 2007

Revised: March 31, 2008

Published: October 2008

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Abstract

The aim of this paper is to prove existence results for nonlinear elliptic equations whose the prototype is -div$(|\nabla u|^{p-2}\nabla u\varphi) =g\varphi $ in a open subset $ \Omega $ of $R^n,$ $u=0$ on $\partial \Omega $, where $p\geq 2$, the function $\varphi (x)=(2\pi)^{-\frac{n}{2}}$exp$( -|x|^2 /2) $ is the density of Gauss measure and $g\in L^1$ (log $L)^{\frac{1}{2}}( \varphi, \Omega)$. This condition on the function $g$ is sharp in the class of Zygmund spaces.

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Mathematics Subject Classification: Primary: 35J25, 35J70; Secondary: 35B45.

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