Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo |
27/08/2015 de 11:00 a 12:00 |
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Dónde | Salón de seminarios 3 |
Agregar evento al calendario |
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We are concerned with the properties of weak solutions of the
stationary Schrödinger equation \(-\Delta u + Vu = f(u)\), \(u\in
H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\), where \(V\) is Hölder
continuous and \(\inf V>0\). Assuming \(f\) to be continuous and
bounded near \(0\) by a power function with exponent larger than \(1\)
we provide precise decay estimates at infinity for solutions in
terms of Green's function of the Schrödinger operator. In some
cases this improves known theorems on the decay of solutions. If
\(f\) is also real analytic on \((0,\infty)\) we obtain that the set of
positive solutions is locally path connected. For a periodic
potential \(V\) this implies that the standard variational functional
has discrete critical values in the low energy range and that a
compact isolated set of positive solutions exists, under additional
assumptions.