On the existence of a solution for a strongly nonlinear elliptic perturbed anisotropic problem of infinite order with variable exponents

In this work, we shall be interested in the existence of a solution to the following Dirichlet problem for a specific class of elliptical anisotropic equations of the type\begin{eqnarray}\label{P.۱}\left \{\begin{array}{rl}&A(u)+g(x,u)= f \ \ {\rm in}\\Omega%[۱.۵ex]\\%[۱ex]& u=۰\ \ {\rm on}\ {\partial \Omega},\end{array}\right.\end{eqnarray}where \Omega is a bounded open set of \mathbb{R}^{N}, A=\sum_{|\alpha|=۰}^{\infty}(-۱)^{|\alpha|}D^{\alpha}\big(a_{\alpha}|D^{\alpha}u|^{p_{\alpha}(x)-۲}D^{\alpha}u\big) is an operator of infinite order and g(x, s ) is a non-linear lower order term that verify some natural growth and sign conditions, where the data f is framed in L^۱(\Omega).

Keywords:

Strongly nonlinear elliptic equations of infinite order , monotonicity condition , variable exponents , sign condition