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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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[1] N. Ackermann, M. Clapp, and F. Pacella, Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains, 40 pages, preprint, 2011. · Zbl 1273.35132
[2] T. Bartsch, M. Clapp, M. Grossi, and F. Pacella, Asymptotically radial solutions in expanding annular domains, Preprint. · Zbl 1276.35083
[3] H. Berestycki, L.A. Caffarelli, and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. I, Duke Math. J. 81 (1996), no. 2, 467-494, A celebration of John F. Nash, Jr. · Zbl 0860.35004
[4] Byeon J.: Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Differential Equations 136(1), 136–165 (1997) · Zbl 0878.35043
[5] Catrina F., Wang Z.Q.: Nonlinear elliptic equations on expanding symmetric domains. J. Differential Equations 156(1), 153–181 (1999) · Zbl 0944.35026
[6] Coffman C.V.: A nonlinear boundary value problem with many positive solutions. J. Differential Equations 54(3), 429–437 (1984) · Zbl 0569.35033
[7] Dancer E.N.: Real analyticity and non-degeneracy. Math. Ann. 325(2), 369–392 (2003) · Zbl 1040.35033
[8] Dancer E.N., Yan S.: Multibump solutions for an elliptic problem in expanding domains. Comm. Partial Differential Equations 27(1-2), 23–55 (2002) · Zbl 1011.35059
[9] Fromm S.J.: Potential space estimates for Green potentials in convex domains. Proc. Amer. Math. Soc. 119(1), 225–233 (1993) · Zbl 0789.35047
[10] Lee M.G., Lin S.S.: Multiplicity of positive solutions for nonlinear elliptic equations on annulus. Chinese J. Math. 19(3), 257–276 (1991) · Zbl 0757.35006
[11] Li Y.Y.: Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differential Equations 83(2), 348–367 (1990) · Zbl 0748.35013
[12] Lin S.S.: Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus. J. Differential Equations 103(2), 338–349 (1993) · Zbl 0803.35053
[13] C. Sulem and P.L. Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999, Self-focusing and wave collapse. · Zbl 0928.35157
[14] Suzuki T.: Positive solutions for semilinear elliptic equations on expanding annuli: mountain pass approach. Funkcial. Ekvac. 39(1), 143–164 (1996) · Zbl 0861.35029
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