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    Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

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    The Cartan-Hadamard conjecture states that, on every nn-dimensional Cartan-Hadamard manifold Mn \mathbb{M}^n , the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for n≤4n \le 4. It was also shown that its validity in dimension nn ensures that every pp-Sobolev inequality (1<p<n 1 < p < n ) holds on Mn \mathbb{M}^n with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the nn-dimensional Cartan-Hadamard conjecture, the only such manifold is Rn \mathbb{R}^n , and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension n≤4n \le 4. Optimal functions for the Sobolev inequality are weak solutions to the critical pp-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equation −Δpu=uq ,u>0 ,on Mn , -\Delta_p u = u^q \, , \quad u>0 \, , \qquad \text{on } \mathbb{M}^n \, , where q≥p∗−1q \ge p^*-1. We show that if there exists a radial finite-energy solution, then Mn\mathbb{M}^n is necessarily isometric to Rn\mathbb{R}^n, q=p∗−1q=p^*-1 and uu is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space W˙1,p(Mn)\dot{W}^{1,p}(\mathbb{M}^n), studying separately the pp-stochastically complete and incomplete cases
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