**Strongly nonlinear problem of infinite order with $L^1$ data**

**A. Benkirane**, Department of Mathematics, Faculty of Sciences of Fez, Fez, Morocco**M. Chrif**, Department of Mathematics, Faculty of Sciences of Fez, Fez, Morocco**S. El Manouni**, Al-Imam University, Faculty of Sciences, Riyadh, KSA

E. J. Qualitative Theory of Diff. Equ., No. 15. (2009), pp. 1-12.

Communicated by F. Zanolin. | Received on 2008-03-11 Appeared on 2009-03-21 |

**Abstract: **In this paper, we prove the existence of solutions for the strongly nonlinear equation of the type $$ Au+g(x,u)=f $$ where $A$ is an elliptic operator of infinite order from a functional space of Sobolev type to its dual. $g(x,s)$ is a lower order term satisfying essentially a sign condition on $s$ and the second term $f$ belongs to $L^1(\Omega).$

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