Abstract
In this paper, we study the elliptic equations
where \(G_{\alpha} =\Delta_{x}+ ( 1+\alpha )^{2}\lvert x\rvert^{2\alpha}\Delta_{y}\), \(\alpha > 0, \) is the Grushin operator. Here, the advection term \(c(\text {x})\) is a smooth, divergence free vector field satisfying certain decay condition and \(h(\text {x}) \) is a continuous function such that \(h({\rm x} )\geq C|{\rm x}|^l\), \(l\geq 0\), where \(|{\rm x}|\) is the Grushin norm of \({\rm x}\). We will prove that the equation has no stable solutions provided that
where \(N_\alpha:=N_1+(1+\alpha)N_2\) is the homogeneous dimension of \(\mathbb R^N\) associated to the Grushin operator.
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Quyet, D.T., Thang, D.M. On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms. Math Notes 112, 109–115 (2022). https://doi.org/10.1134/S0001434622070124
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DOI: https://doi.org/10.1134/S0001434622070124