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On solutions to a class of degenerate equations with the Grushin operator
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abstract
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.
Forward citations
Cited by 2 Pith papers
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Schauder estimates for elliptic equations degenerating on lower dimensional manifolds
Weak solutions to weighted elliptic equations with a distance-to-a-manifold weight are shown to be C^{0,α} or C^{1,α} up to the characteristic manifold under a homogeneous conormal condition.
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Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction
A positive weak solution exists for a Grushin problem in the whole space mixing singular, convective, and critical reactions, with decay at infinity in the non-convective case.
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