A Generalized Brezis–Lieb Lemma on Graphs and Its Application to Kirchhoff Type Equations

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.