In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $\lambda$ and $\mu$ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $\lambda$, and analyze the relationship between $\lambda$ and $\mu$. In the end, we investigate the concentration phenomenon of solutions when $\mu \to 0$, and obtain a specific lower bound of the parameter $\lambda$ which is independent of $\mu$.

We discuss a poly-Laplacian system involving concave-convex nonlinearities and parameters subject to the Dirichlet boundary condition on locally finite graphs. It is obtained that the system admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy based on the mountain pass theorem and the Ekeland's variational principle. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result due to Brown et al., which is based on the fibering method and the Nehari manifold, we get the existence of the ground-state solution to the single equation corresponding to the poly-Laplacian system. Especially, we present some ranges of parameters for all of the results.