## 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 $$\mathbb R^{N-k}$$.

### MSC:

 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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### References:

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