Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional

We say that a hypersurface $\mathrm{\Sigma }\subset {\mathbb{R}}^{n+1}$ is $\alpha$-stationary if it is a critical point of the Euler-Dierkes-Huisken functional ${\mathcal{E}}_{\alpha }(\mathrm{\Sigma })={\int }_{\mathrm{\Sigma }}\mathrm{\mid }X{\mathrm{\mid }}^{\alpha }d{\mathcal{H}}^n$, introduced by Dierkes and Huisken in \cite{[DH-24]}. In this paper, we prove that every smooth, complete, connected, embedded $\alpha$-stationary hypersurface in ${\mathbb{R}}^{n+1}$ passing through the origin with $\alpha >0$ is a linear hyperplane.