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November  2008, 7(6): 1497-1506. doi: 10.3934/cpaa.2008.7.1497

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

Giuseppina di Blasio 1, , Filomena Feo 2, and  Maria Rosaria Posteraro 3,

1. 

Seconda Universitá degli Studi di Napoli, Dipartimento di Matematica, Via Vivaldi 43, Caserta, Italy
2. 

Universitá degli Studi di Napoli Parthenope, Dipartimento per le Tecnologie, Is. C4, Centro Direzionale, Napoli, Italy
3. 

Universitá degli Studi di Napoli “Federico II”, Dipartimento di Matematica “R. Caccioppoli”, Complesso Monte S. Angelo, Napoli, Italy

Received  October 2007 Revised  April 2008 Published  August 2008

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.
Keywords: elliptic PDE's, rearrangements., Gauss measure.
Mathematics Subject Classification: Primary: 35J25, 35J70; Secondary: 35B4.
Citation: Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497
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