Existence of renormalized solutions for a nonlinear elliptic equation in Musielak framework and L1 data

Taghi Admedatt, Mohamed Saad Bouh Elemine Vall, Abdelmoujib Benkirane, Abdelfattah Touzani


In this paper, we prove existence result of renormalized solutions in the setting of Musielak-Orlicz spaces $W^1_0L_\varphi(\Omega)$ for the following strongly nonlinear Dirichlet problem

$$A(u)+g(x,u,\nabla u)=f\quad \textrm{in }\Omega,$$

where $A$ is a Leray-Lions operator acting from its domain $D(A)\subset W^1_0L_\varphi(\Omega)$ into its dual, while $g(x, u,\nabla u)$ is a nonlinear term having a growth conditions with respect only to $\nabla u$, and does not satisfy any sign condition. The right-hand side $f$ belongs to $ L^1(\Omega)$. 
A modular-inequality of Poincar\'e type in this setting is also proved (Lemma 2.5). 

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