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

  • Mónica Clapp
  • Yéferson Fernández
Article
  • 130 Downloads

Abstract

We consider the problem
$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$
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. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)

Keywords

Nonlinear Schrödinger equation partially periodic potential exterior domain with unbounded boundary sign-changing solutions Lusternik–Schnirelmann theory for noncompact groups 

Mathematics Subject Classification

35Q55 (35J20, 35B06) 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoCoyoacánMexico

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