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Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary. (English) Zbl 1390.35319

Summary: We consider the problem \[ -\Delta u+(V_{\infty }+V(x)) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \] where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\geq 3\), \(V_{\infty}>0\), \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N})\), \(\inf _{\mathbb {R}^{N}}V>-V_{\infty}\) 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 \(m+1\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J20 Variational methods for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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