Existence of solutions and regularizing effect for some elliptic nonlinear problems with nonhomogeneous Neumann boundary conditions

We are interested in the regularizing effect of the strongly increasing lower term and also the one due to the summability of the boundary data for nonlinear elliptic problem with nonhomogeneous Neumann boundary conditions of the form -diva(x,Du)+|u|s-1u=0inΩa(x,Du)·η=ψon∂Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle -\hbox {div }{\mathbf{a}}(x,Du)+\vert u \vert ^{s-1} u =0 &{} \quad \hbox { in }\Omega \\ \displaystyle {\mathbf{a}}(x,Du)\cdot \eta = \psi &{} \quad \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$\end{document}