*(English)*Zbl 1135.34007

Summary: This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem

where $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous, $\zeta ,\eta $ and $\lambda \in \mathbb{R}$ are parameters. Using the variational structure of the above boundary value problem and critical point theory, it is shown that the different locations of the pair $(\eta ,\zeta )$ and $\lambda \in \mathbb{R}$ lead to different existence results for the above boundary value problem. More precisely, if the pair $(\eta ,\zeta )$ is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for $\lambda \in (-\lambda ,0)$ and has infinitely many solutions for $\lambda \in (0,\infty )$; if $(\eta ,\zeta )$ is on the right side of the first eigenvalue line and $\lambda \in (-\infty ,0)$, then the above boundary value problem has two nontrivial solutions or has at least ${n}_{*}$ $({n}_{*}\in \mathbb{N})$ distinct pairs of solutions, which depends on the fact that the pair $(\eta ,\zeta )$ is located in the second or fourth (first) quadrant.