The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold

In this paper, we consider the following elliptic problem: −divg(|∇gu|N−2∇gu)+V(x)|u|N−2u=f(x,u)a(x)in M,(Pa)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\mathtt{div}_{g}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \bigr)+V(x) \vert u \vert ^{N-2}u = \frac{f(x, u)}{a(x)}\quad \mbox{in } M, \qquad (P_{a}) $$\end{document} where (M,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(M, g)$\end{document} be a complete noncompact N-dimensional Riemannian manifold with negative curvature, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq2$\end{document}, V is a continuous function satisfying V(x)≥V0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x) \geq V_{0 }> 0$\end{document}, a is a nonnegative function and f(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x, t)$\end{document} has exponential growth with t in view of the Trudinger–Moser inequality. By proving some estimates together with the variational techniques, we get a ground state solution of (Pa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{a}$\end{document}). Moreover, we also get a nontrivial weak solution to the perturbation problem.