| Abstract: |
| In this talk we will consider the well-known generalized Lane-Emden equation $-\Delta_p u = |u|^{q-1} u$, involving the usual p-Laplace operator in the Euclidean space. I will discuss some non-existence and classification results for positive solutions of the subcritical and the critical p-Laplace equation. In particular in the critical case, using the moving planes method, it has been recently shown that positive solutions to the critical p-Laplace equation with finite energy can be completely classified. I will then present some recent results concerning the classification of positive solutions to the critical p-Laplace equation, possibly having infinite energy. If n=2, or if n=3 and p lies in (3/2,2), rigidity is obtained without any further assumptions. In the remaining cases the classification follows under energy growth conditions or suitable control of the solutions at infinity. These assumptions are much weaker than those already appearing in the literature. I will also discuss extensions of these results to the Riemannian setting. This is based on a recent joint work with G. Catino and A. Roncoroni (Politecnico di Milano). |
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