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

The Grushin Laplacian −Δ_α is a degenerate elliptic operator in R^{h+k} that degenerates on {0}×R^k. We consider weak solutions of −Δ_α u = Vu in an open bounded connected domain Ω with V ∈W^{1,σ}(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of 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 Ω. As an application we derive strong unique continuation properties for solutions.