Some Quantitative Properties of Solutions to the Buckling Type Equation

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation ${\mathrm{△}}^{2}u+\lambda \mathrm{△}u-k^{2}u=0$ in a bounded analytic domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ with the homogeneous boundary conditions $u=0$ and $\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where $\lambda ,k$ are nonnegative real constants, and $\nu$ is the outer unit normal vector on $\mathrm{\partial }\mathrm{\Omega }$. We obtain that, the upper bounds for the maximal vanishing order of $u$ and the $n-1$ dimensional Hausdorff measure of the nodal set of $u$ are both $C(\sqrt{\lambda }+\sqrt{k}+1)$, where $C$ is a positive constant only depending on $n$ and $\mathrm{\Omega }$. Moreover, we also give a quantitative result of the propagation of smallness of $u$.