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

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


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