Some Remarks on a Class of Nonuniformly Elliptic Equations of p-Laplacian Type

This paper deals with the existence of weak solutions in W01(Ω) to a class of elliptic problems of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathop{\mathrm{div}}({a({x,\nabla u})})=\lambda_{1}\left|u\right |^{p-2}u+g\left(u\right)-h$$\end{document} in a bounded domain Ω of ℝN. Here a satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{a\left({x,\xi}\right)}\right|\leqq c_{0}\left({h_{0}\left(x\right)+h_{1}\left(x\right)\left|\xi\right|^{p-1}}\right)$$\end{document} for all ξ∈ℝN, a.e. x∈Ω, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{0}\in L^{\frac{p}{{p-1}}}(\Omega)$\end{document} , h1∈Lloc1(Ω), h1(x)≧1 for a.e. x in Ω; λ1 is the first eigenvalue for −Δp on Ω with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.