Nonlinear Bound States on Weakly Homogeneous Spaces |
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Authors: | Hans Christianson Jeremy Marzuola Jason Metcalfe |
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Affiliation: | Department of Mathematics , UNC-Chapel Hill , Chapel Hill , North Carolina , USA |
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Abstract: | We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails. |
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Keywords: | Bound state Nonlinear Schrödinger equation Standing wave Weakly homogeneous space |
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