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# Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity

Ponente: Nils Ackermann
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 27/08/2015 de 11:00 a 12:00 Salón de seminarios 3 vCal iCal

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.