Self-focusing multibump standing waves in expanding waveguides. (English) Zbl 1229.35285

Summary: Let \(M\) be a smooth \(k\)-dimensional closed submanifold of \(\mathbb R^N\), \(N\geq 2\), and let \(\Omega_R\) be the open tubular neighborhood of radius 1 of the expanded manifold \(M_R:= \{R_x: x\in M\}\). For \(R\) sufficiently large we show the existence of positive multibump solutions to the problem
\[ -\Delta u+\lambda u=f(u)\quad\text{in }\Omega_R, \qquad u=0\quad\text{on }\partial\Omega_R. \]
The function \(f\) is superlinear and subcritical, and \(\lambda>-\lambda _1\), where \(\lambda_1\) is the first Dirichlet eigenvalue of \(-\Delta\) in the unit ball in \(\mathbb R^{N-k}\).


35Q60 PDEs in connection with optics and electromagnetic theory
78A50 Antennas, waveguides in optics and electromagnetic theory
78A60 Lasers, masers, optical bistability, nonlinear optics
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