This paper concerns the existence of normalized solutions for the following N-Kirchhoff equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\left( a\epsilon ^N+b\epsilon ^{2N}\int _{{\mathbb {R}}^N}\left| \nabla u \right| ^{N}\mathrm {~d} x \right) \textrm{div}\left( {\left| {\nabla u} \right| }^{N - 2}\nabla u\right) = \lambda {\left| u \right| }^{N - 2}u \\ +\displaystyle \frac{\epsilon ^{-(N-1)}f(\epsilon u)}{| x|^{\beta }},\ \ x\in {\mathbb {R}}^N,\\ u \in {W^{1,N}}({{\mathbb {R}}^N}),\int _{{\mathbb {R}}^N}| u |^N\mathrm {~d} x = \rho , \end{array}\right. } \end{aligned}\) where \(N \ge 2\) , \(a,b > 0\) , \(0< \beta < N,\) \(\rho > 0\) , \(\lambda \in {\mathbb {R}}\) is a Lagrange multiplier, \(\epsilon >0\) and \(f\in C({\mathbb {R}})\) behaves like \(\exp (\alpha t^{\frac{N}{N-1}})\) with some \(\alpha >0\) as \(t\rightarrow \infty \) . Under some suitable assumptions, the existence of normalized ground state solutions is studied by restricting the discussion on Nehari–Pohozaev manifold and using the singular Trudinger–Moser inequality. As by-products, the regularity and exponential decay estimates are obtained. Moreover, another solution is discussed by the mountain pass theorem. In addition, the asymptotic behavior of solutions is also investigated as \(\epsilon \rightarrow 0\) . The main novelty of this paper is that we consider the Kirchhoff problem in the borderline case and the nonlinearity is singular.
