Spectral structure and global bifurcation for biharmonic equations with clamped plate boundary conditions

We consider the following semilinear biharmonic problem\begin{equation}\label{abstract-equation}\left\{ \begin{array}{ll} \Delta^2 u=\lambda f(u) & \text { in } B,\\ u=\frac{\partial u}{\partial n}=0 & \text { on } \partial B , \end{array}\right.\end{equation}where $B$ denotes the unit ball in $\mathbb{R}^{N}$, $N \geq 2$, $\lambda\in\mathbb{R}$ is a parameter,$\partial/\partial n$ is the outward normal derivative, $f:[0, +\infty) \rightarrow [0, +\infty)$ is a continuous function. We obtain the complete spectral structure of according eigenvalue problems and the basis of the eigen-subspace corresponding to each eigenvalue. Moreover, by employing bifurcation and topological techniques, we establish the existence of positive solutions. Under certain appropriate conditions on the function $f$, we demonstrate that (\ref{abstract-equation}) possesses two positive solutions. Additionally, we present several results concerning the nonexistence of solutions.