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In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents:~ \begin{equation} \begin{cases} -\Delta u=|u|^{2^{*}-2}u+\frac{\alpha}{2^{*}}|u|^{\alpha-2}|v|^{\beta}u+\lambda|u|^{q-2}u,~~&\mbox{in}~\Omega,\\ -\Delta v=|v|^{2^{*}-2}v+\frac{\beta}{2^{*}}|u|^{\alpha}|v|^{\beta-2}v+\mu|v|^{q-2}v,~~&\mbox{in}~\Omega,\\ u,~v>0,~~&\mbox{in}~\Omega,\\ u=v=0,~~&\mbox{on}~\partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega\subset\mathbb{R}^{N}(N\geq3)$ is a bounded domain with smooth boundary,~$\lambda,~\mu>0,~11,~\beta>1,~\alpha+\beta=2^{*}=\frac{2N}{N-2}$. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.

Hydrogen energy is considered as an ideal energy with the advantages of green, sustainability, and high energy density, and water splitting is one of the efficient strategies for green hydrogen without carbon emission. As for cathodic hydrogen evolution reaction (HER), besides the Pt-based electrocatalysts with excellent electrocatalytic activities on HER, transition metal nitrides (TMNs) as cheap and facile-prepared electrocatalysts have shown remarkable electrocatalytic activities. Incorporation of N atom in metal interstitial lattice results in the unique structure of TMN with high electronic conductivity, strong chemical stability, and d-band contraction. Although the intrinsic electrocatalytic activities of TMNs are mostly lower than those of Pt, it also attracted much attention to the development of TMN with higher intrinsic activity by electronic structure modulation. Here, we review the recent improvement strategies for the intrinsic electrocatalytic activities of TMN catalysts on HER by electronic structure modulation, such as facet, alloying, doping, vacancy, heterostructure, and hybridization. Some important breakthroughs of TMNs have been made; however, the scale application of TMNs with high activity in commercial water electrolyzer is urgent to explore. The future development of TMNs is proposed to focus on developing facile synthesis methods, elucidating regulation mechanism and catalytic mechanism, and enhancing activity and stability.