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| In this talk we present a symmetric version of the Pucci and Serrin three critical points theorem, which we apply to an abstract eigenvalue problem in order to show the existence of three different symmetric solutions. Furthermore we illustrate the existence of nontrivial nonnegative solutions, which are invariant by $k$-spherical cap symmetrization, of quasilinear elliptic Dirichlet problems in either a ball of $\mathbb R^N$ or an annulus of $\mathbb R^N$, both centered at $0$. |
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