In this paper, with the help of potential function, we extend the classical Brezis–Lieb lemma on Euclidean space to graphs, which can be applied to the following Kirchhoff equation \(\begin{aligned} \left\{ \begin{array}{l} -\left( 1+b \int _{\mathbb { V}}|\nabla u|^2 d \mu \right) \Delta u+ \left( \lambda V(x) +1 \right) u=|u|^{p-2} u \ \text{ in } \mathbb { V}, \\ u \in W^{1,2}(\mathbb {V}), \end{array}\right. \end{aligned}\) on a connected locally finite graph \(G=(\mathbb {V}, \mathbb {E})\) , where \(b, \lambda >0\) , \(p>2\) and V(x) is a potential function defined on \(\mathbb {V}\) . The purpose of this paper is four-fold. First of all, using the idea of the filtration Nehari manifold technique and a compactness result based on generalized Brezis–Lieb lemma on graphs, we prove that there admits a positive solution \(u_{\lambda , b} \in E_\lambda \) with positive energy for \(b \in (0, b^*)\) when \(2<p<4\) . In the sequel, when \(p \geqslant 4\) , a positive ground state solution \(w_{\lambda , b} \in E_\lambda \) is also obtained by using standard variational methods. What’s more, we explore various asymptotic behaviors of \(u_{\lambda , b}, w_{\lambda , b} \in E_\lambda \) by separately controlling the parameters \(\lambda \rightarrow \infty \) and \(b \rightarrow 0^{+}\) , as well as jointly controlling both parameters. Finally, we utilize iteration to obtain the \(L^{\infty }\) -norm estimates of the solution.
