Ground state solution to N-Kirchhoff equation with critical exponential growth and without Ambrosetti–Rabinowitz condition

This article is focused on the existence of a ground state solution to the Kirchhoff problem: -k∫Ω|∇u|NdxΔNu=f(x,u)|x|a+λg(x)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -k\left( \int _{\Omega } |\nabla u|^{N}dx \right) \Delta _{N}u &{} =\dfrac{f(x,u)}{|x|^{a}}+\lambda g(x) \ \ \hbox {in} \ \ \Omega , \\ \quad u &{} = 0 \ \ \hbox {on} \ \ \partial \Omega , \end{array}\right. \end{aligned}$$\end{document}where Ω⊆RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subseteq {\mathbb {R}}^{N}(N\ge 2)$$\end{document} is a bounded domain with smooth boundary and a∈[0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in [0, N)$$\end{document}. We assume that f satisfies critical exponential growth at infinity but does not satisfy the well-known Ambrosetti–Rabinowitz condition. We prove the existence of a ground state weak solution via mountain pass theorem and Nehari manifold technique.