Upper bounds of nodal sets for solutions of bi-Laplace equations: II

We investigate the upper bounds of nodal sets for solutions of bi-Laplace equations without using frequency functions which play an essential role in the study of nodal sets in the celebrated work by Logunov \cite{Lo18}. We obtain some delicate monotonicity and propagation of smallness results by Carleman estimates. A polynomial upper bound for the nodal sets of solutions is obtained.