On a Class of Nonhomogeneous Elliptic Equation on Compact Riemannian Manifold Without Boundary

Let (M, g) be a compact Riemannian manifold of dimension n(n≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n\ge 2)$$\end{document} without boundary. In this paper, a fixed point result and a version of the Trudinger–Moser inequality are employed to establish sufficient conditions for the existence of solutions of quasilinear nonhomogeneous elliptic equation of the type -divg(a(x,∇gu))+|u|n-2u=f(x,u)+λhinM.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - {{\mathrm{div}}}_g(a(x,\nabla _g u))+|u|^{n-2}u=f(x,u)+ \lambda h\quad \text {in }\quad M. \end{aligned}$$\end{document}We assume that there exists a function A:M×T(M)→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A : M \times T(M) \rightarrow \mathbb {R}$$\end{document} such that a(x,ξ)=∇gA(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(x,\xi ) = \nabla _{g} A(x,\xi )$$\end{document} satisfies the so-called Leray–Lions conditions. Here 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, the nonlinearity f:M×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: M \times \mathbb {R} \rightarrow \mathbb {R}$$\end{document} can be discontinuous and enjoy exponential critical growth and h belongs to the dual space of W1,n(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,n}(M)$$\end{document} with h≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\not \equiv 0$$\end{document}.