In this talk, I will introduce two Liouville theorems for nonlinear boundary value problems on extended Grushin manifolds. These problems are closely related to nonlinear fractional Grushin equations. We will prove that these problems do not possess nontrivial positive solution under some assumptions on the nonlinear terms. We use the moving plane method in an integral form to prove our results. Finally, I will also introduce some problems we are working on and some open questions in related areas. This is a joint work with Jinggang Tan in Universidad Tecnica Federico Santa Maria.

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