The Grushin Laplacian $- \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 $- \Delta_\alpha u= Vu$ in an open bounded connected domain $\Omega$ with $V \in W^{ 1,\sigma}(\Omega)$ and $\sigma > Q/2$, 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 $\Omega$. As an application we derive strong unique continuation properties for solutions.
Categories: math.AP