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# Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary

Ponente: Yéferson Fernández
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 26/05/2016 de 10:00 a 11:00 Salón 2 vCal iCal

We consider the problem

\begin{equation*}
-\Delta u + (V_\infty + V (x)) u = |u|^{p-2} u,\qquad    u \in  H_0^1 (\Omega ),
\end{equation*}
where $$\Omega$$ is either $$\mathbb{R}^N$$ or a smooth domain in $$\mathbb{R}^N$$ with unbounded boundary, $$N\ge 3$$, $$V_\infty > 0$$, $$V \in \mathcal{C}^0 (\mathbb{R}^N )$$, $$\inf_{\mathbb{R}^N}V>-V_\infty$$  and $$2 < p < \frac{2N}{N-2}$$. We assume $$V$$ is periodic in the first $$m$$ variables, and decays exponentially to zero in the remaining ones. We also assume that $$\Omega$$ is periodic in the first $$m$$ variables and has bounded complement in the other ones. Then, assuming that $$\Omega$$ and $$V$$ are invariant under some suitable group of symmetries on the last $$N - m$$ coordinates of $$\mathbb{R}^N$$, we establish existence and multiplicity of sign-changing solutions to this problem.

This is joint work with Mónica Clapp (IM-UNAM).