Spectrum of Navier $p$-biharmonic problem with sign-changing weight

Abstract: In this paper, we consider the following eigenvalue problem {{l} (|u"|^{p-2}u")"=\lambda m(x)|u|^{p-2}u, x\in (0,1), u(0)=u(1)=u"(0)=u"(1)=0, where $1<p<+\infty$, $\lambda$ is a real parameter and $m$ is sign-changing weight. We prove there exists a unique sequence of eigenvalues for above problem. Each eigenvalue is simple and continuous with respect to $p$, the $k$-th eigenfunction, corresponding to the $k$-th positive or negative eigenvalue, has exactly $k-1$ generalized simple zeros in $(0,1)$.