Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary
Institución: IM-UNAM
Tipo de Evento: Researcher
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).