A class of hypersurfaces in ${\mathbb{E}}_s^{n+1}$ satisfying $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$

A nondegenerate hypersurface in a pseudo-Euclidean space ${\mathbb{E}}_s^{n+1}$ is called to have proper mean curvature vector if its mean curvature $\overrightarrow{H}$ satisfies $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ for a constant $\lambda$. In 2013, Arvanitoyeorgos and Kaimakamis conjectured ^[1]: any hypersurface satisfying $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ in pseudo-Euclidean space ${\mathbb{E}}_s^{n+1}$ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $M_r^n$ in ${\mathbb{E}}_s^{n+1}$ satisfying $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ has constant mean curvature if $M_r^n$ has diagonalizable shape operator with less than seven distinct principal curvatures.