Signed power test for a Laplacian eigenfunction ($(\gamma -1)\int |u{|}^{\gamma -2}|Du{|}^2=\lambda \int |u{|}^{\gamma }$)

For a smooth Dirichlet Laplacian eigenfunction, testing $-\Delta u=\lambda u$ with $|u{|}^{\gamma -2}u$ for $\gamma \ge 2$ gives the displayed energy estimate. The Sobolev chain rule for $v=|u{|}^{\gamma /2}$ then yields $\int |Dv{|}^2=\lambda {\gamma }^2/[4(\gamma -1)]\int |u{|}^{\gamma }$. The absolute value and the sign in the test function permit nodal changes. At $\gamma =2$, use the Lipschitz absolute-value chain rule and the fact that a gradient of a Sobolev function vanishes on a level set.