A nonlinear Dirichlet problem involving terms with logarithmic growth

Abstract

We study existence and regularity of weak solutions for a class of boundary value problems, whose form is $$\begin{array}{l}\hfill \left\{\begin{array}{cc}-div\mathrm{(}\frac{log\mathrm{(}1+|\nabla \mathit{u}|\mathrm{)}}{|\nabla \mathit{u}|}\mathit{m}\mathrm{(}\mathit{x}\mathrm{)}\nabla \mathit{u}\mathrm{)}+\mathit{u}|\nabla \mathit{u}|log\mathrm{(}1+|\nabla \mathit{u}|\mathrm{)}=\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\hfill & \text{in }\mathrm{\Omega }\hfill \\ \mathit{u}=0\mathrm{,}\hfill & \text{on }\partial \mathrm{\Omega }\hfill \end{array}\right.\end{array}$$ where both the principal part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function $\mathit{s}log\mathrm{(}1+|\mathit{s}|\mathrm{)}$ even for ${\mathit{L}}^1\mathrm{(}\mathrm{\Omega }\mathrm{)}$ data.