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We consider the following critical \(p\)-Laplace equation: \begin{equation}(1)\quad \Delta_p u+u^{p^{\ast}-1}=0 \quad \text{ in \(\mathbb{R}^n\)} \, , \end{equation} with \(n \geq 2\) and \(1 < p < n\). Equation (1) has been largely studied in the PDE's and geometric analysis' communities, since extremals of Sobolev inequality solve (1) and, for \(p=2\), the equation is related to the Yamabe's problem. In particular, it has been recently shown, exploiting the moving planes method, that positive solutions to (1) such that \(u\in L^{p^\ast}(\mathbb{R}^n)\) and \(\nabla u\in L^p(\mathbb{R}^n)\) can be completely classified. In the talk we will consider the anisotropic critical \(p\)-Laplace equation in convex cones of \(\mathbb{R}^n\). Since the moving plane method strongly relies on the symmetries of the equation and of the domain, in the talk a different approach to this problem will be presented. In particular this approach gives a complete classification of the solutions in an anisotropic setting. More precisely, we characterize solutions to the critical \(p\)-Laplace equation induced by a smooth norm inside any convex cone of \(\mathbb{R}^n\).
This is a joint work with G. Ciraolo and A. Figalli.