1D logistic reaction and p-Laplacian diffusion as p goes to one

This work discusses the existence of the limit as p goes to 1 of the nontrivial solutions to the one-dimensional problem: -|ux|p-2uxx=λ|u|p-2u-|u|q-2u0<x<1u(0)=u(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( |u_x|^{p-2} u_x\right) _x = \lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u}&{} \quad 0< x < 1\\ u(0)=u(1)=0, &{} \end{array}\right. } \end{aligned}$$\end{document}where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a positive parameter and the exponents p, q satisfy 1<p<q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p < q$$\end{document}. We prove that solutions do converge to a limit function, which solves in a proper sense a Dirichlet problem involving the 1-Laplacian operator.