On solutions to a class of degenerate equations with the Grushin operator

The Grushin Laplacian $-{\mathrm{\Delta }}_{\alpha }$ is a degenerate elliptic operator in ${\mathbb{R}}^{h+k}$ that degenerates on $\{0\}\times {\mathbb{R}}^k$. We consider weak solutions of $-{\mathrm{\Delta }}_{\alpha }u=Vu$ in an open bounded connected domain $\mathrm{\Omega }$ with $V\in W^{1,\sigma }(\mathrm{\Omega })$ and $\sigma >Q\mathrm{/}2$, where $Q=h+(1+\alpha )k$ is the so-called homogeneous dimension of ${\mathbb{R}}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $\mathrm{\Omega }$. As an application we derive strong unique continuation properties for solutions.