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EXISTENCE OF GUIDED CYLINDRICAL TM-MODES IN AN INHOMOGENEOUS SELF-FOCUSING DIELECTRIC

C. A. STUART

Station 8, Section de Mathémathiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

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and

HUAN-SONG ZHOU

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P. O. Box 71010, Wuhan 430071, P. R. China

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https://doi.org/10.1142/S0218202510004751Cited by:19 (Source: Crossref)

Abstract

We consider an eigenvalue problem for a certain type of quasi-linear second-order differential equation on the interval (0, ∞). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a self-focusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.

Keywords:

AMSC: 36Q60, 78A60, 34B16

References

G. Cerami, Ren. Acad. Sci. Let. Ist. Lombardo 112, 332 (1978). Google Scholar
G. Cerami, Ann. di Mat. 24, 161 (1980). Web of Science, Google Scholar
L. Cesari , Optimisation - Theory and Applications ( Springer-Verlag , 1993 ) . Google Scholar
D. E. Edmunds and W. D. Evans , Spectral Theory and Differential Operators ( Oxford Univ. Press , 1987 ) . Google Scholar
I. Ekeland , Convexity Methods in Hamiltonian Mechanics ( Springer-Verlag , 1990 ) . Crossref, Google Scholar
L. Jeanjean, Proc. Roy. Soc. Edinburgh A 129, 787 (1999). Crossref, Web of Science, Google Scholar
L. Jeanjean and K. Tanaka, ESAIM Control Optim. Calc. Var. 7, 597 (2002), DOI: 10.1051/cocv:2002068. Crossref, Web of Science, Google Scholar
R. E. Showalter , Monotone Operators in Banach Space and Nonlinear Partial Differential Equations , Mathematical Surveys and Monographs 49 ( Amer. Math. Soc. , 1997 ) . Google Scholar
C. A. Stuart, Math. Models Methods Appl. Sci. 6, 977 (1996), DOI: 10.1142/S0218202596000407. Link, Web of Science, Google Scholar
C. A. Stuart , Reaction Diffusion Systems , eds. G. Caristi and E. Mitidieri ( Marcel Dekker , 1997 ) . Google Scholar
C. A. Stuart, Localization of Palais–Smale and Cerami sequence in mountain pass geometries, to appear. . Google Scholar
C. A. Stuart and H. S. Zhou, Comm. PDE 24, 1731 (1999), DOI: 10.1080/03605309908821481. Crossref, Web of Science, Google Scholar
C. A. Stuart and H. S. Zhou, Ann. Inst. Henri Poincaré Anal. Nonlinéaire 18, 69 (2001), DOI: 10.1016/S0294-1449(00)00125-6. Crossref, Web of Science, Google Scholar
C. A. Stuart, Milan J. Math. 72, 107 (2004), DOI: 10.1007/s00032-004-0035-4. Crossref, Web of Science, Google Scholar
C. A. Stuart and H. S. Zhou, Calc. Var. 16, 335 (2003), DOI: 10.1007/s005260100153. Crossref, Web of Science, Google Scholar
C. A. Stuart and H. S. Zhou, SIAM J. Math. Anal. 37, 218 (2005), DOI: 10.1137/S0036141004441751. Crossref, Web of Science, Google Scholar
H. S. Zhou and H. B. Zhu, Quart. J. Math. 59, 523 (2008), DOI: 10.1093/qmath/ham047. Crossref, Web of Science, Google Scholar
M. Willem , Minimax Theorems ( Birkhäuser , 1996 ) . Crossref, Google Scholar