Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids

Amick, Charles J.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 11 (1984) no. 3, pp. 441-499.

[1] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), pp. 487-513. | MR | Zbl

[2] K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations, 45 (1982), pp. 113-127. | MR | Zbl

[3] J. Bona - D.K. Bose - R.E.L. Turner, Finite amplitude steady waves in stratified fluid, J. Math. Pures Appl., 62 (1983), pp. 389-439. | MR | Zbl

[4] R.E.L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci., VIII (4) (1981), pp. 513-573. | Numdam | MR | Zbl

[5] C.J. Amick - J.F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal., 76 (1981), pp. 9-95. | MR | Zbl

[6] C.J. Amick - J.F. Toland, On periodic water waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A, 303 (1981), pp. 633-673. | MR | Zbl

[7] C.J. Amick - L.E. Fraenkel - J.F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), pp. 193-214. | MR | Zbl

[8] C.J. Amick - R.E.L. Turner, A global theory for internal solitary waves in a two-fluid system (to appear). | Zbl

[9] A. Bongers - H.P. Heinz - T. Küpper, Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains, J. Differential Equations, 47 (1983), pp. 327-357. | MR | Zbl

[10] R. Chiapinelli - C.A. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. Differential Equations, 30 (1978), pp. 296-307. | MR | Zbl

[11] C.A. Stuart, Bifurcation from the continuous spectrum in the L2-theory of elliptic equations in RN, Atti 3° S.A.F.A. (1978), Confer. Sem. Mat. Univ. Bari (1979), pp. 157-180.

[12] C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc., 95 (1982), pp. 169-192. | MR | Zbl

[13] J.F. Toland, Global bifurcation for Neumann problems without eigenvalues, J. Differential Equations, 44 (1982), pp. 82-110. | MR | Zbl

[14] J.F. Toland, Positive solutions of nonlinear elliptic equations: existence and nonexistence of solutions with radial symmetry in Lp(R N) Trans. A.M.S., 282 (1983), pp. 335-365. | MR | Zbl

[15] K. Kirchgässner - J. Scheurle, Bifurcation from the continuous spectrum and singular solutions, in Trends in applications of pure mathematics to mechanics, Vol. III, pp. 138-154, R. Knops, ed., London, Pitman, 1980. | MR | Zbl

[16] K. Krichgässner - J. Scheurle, Bifurcation of nonperiodic solutions of some semilinear equations in unbounded domains, in : Applications of nonlinear analysis in the physical sciences, Vol. VI, pp. 41-59, H. Amann, N. Bazely, K. Kirchgässner, eds., London, Pitman, 1981. | MR | Zbl

[17] C.A. Stuart, Bifurcation from the essential spectrum, in Equadiff 82, Springer Lecture Notes no. 1017, pp. 575-596, H. Knoblock, K. Schmitt, eds., Berlin, Springer-Verlag, 1983. | MR | Zbl

[18] C.A. Stuart, A variational approach to bifurcation in L p on an unbounded symmetrical domain, Math. Ann., 263 (1983), pp. 51-59. | EuDML | MR | Zbl

[19] M. Esteban, Nonlinear elliptic problems in strip-like domains. Symmetry of positive vortex rings, Nonlinear Anal. T.M.A., 7 (4) (1983), pp. 365-380. | MR | Zbl

[20] C.J. Amick - J.F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip : global theory of bifurcation and asymptotic bifurcation, Math. Ann., 262 (1983), pp. 313-342. | EuDML | MR | Zbl

[21] A.M. Ter-Krikorov, On internal waves in an inhomogeneous fluid, J. Appl. Math. Mech., 26 (1962), pp. 1617-1631. | MR | Zbl

[22] A.M. Ter-Krikorov, Théorie exacte des ondes longues stationnaires dans un liquide hétérogène, J. Mécanique, 2 (1963), pp. 351-376. | MR

[23] T.B. Benjamin, An exact theory of finite steady waves in continuously stratified fluids, Fluid Mechanics Research Institute, University of Essex, report no. 48.

[24] C.S. Yih, Dynamics of nonhomogeneous fluids, New York, MacMillan, 1965. | MR | Zbl

[25] T.B. Benjamin, Internal waves of finite amplitude and permanent form, J. Fluid Mech., 25 (1966), pp. 241-270. | Zbl

[26] T.B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), pp. 559-592. | Zbl

[27] T.B. Benjamin, A unified theory of conjugate flows, Philos. Trans. Roy. Soc. London Ser. A, 269 (1971), pp. 587-643. | MR | Zbl

[28] R.E. Davis - A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), pp. 593-607. | Zbl

[29] M. Yanowitch, Gravity waves in a heterogeneous incompressible fluid, Comm. Pure Appl. Math., 15 (1962), pp. 45-61. | MR | Zbl

[30] R.R. Long, Some aspects of the flow of stratified fluids, Part I : A theoretical investigations, Tellus, 5 (1953), pp. 42-57. | MR

[31] R.R. Long, Solitary waves in the one- and two-fluid systems, Tellus, 8 (1956), pp. 460-471.

[32] G.H. Keulegan, Characteristics of internal solitary waves, J. Res. Nat. Bur. Standards, 51 (1953), pp. 133-140. | Zbl

[33] C.S. Yih, On the flow of a stratified fluid, Proc. U.S. Nat. Congr. Appl. Mech., 3rd (1958), pp. 857-861. | MR

[34] C.S. Yih, Exact solutions for steady two-dimensional flow of a stratified fluid, J. Fluid Mech., 9 (1960), pp. 161-174. | MR | Zbl

[35] A.S. Peter - J.J. Stokes, Solitary waves in liquids having nonconstant density, Comm. Pure Appl. Math., 8 (1960), pp. 115-164. | Zbl

[36] C.J. Amick - J.F. Toland, The limiting form of internal waves, Proc. Roy. Soc. London A 394 (1984), pp. 329-344. | MR | Zbl

[37] S. Agmon, The Lp approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), pp. 405-448. | EuDML | Numdam | MR | Zbl

[38] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), pp. 304-318. | MR | Zbl

[39] M.G. Krein - M.A. Rutman, Linear operators which leave a cone in Banach space invariant, Amer. Math. Soc. Transl. Ser. 1, 10 (1950). | MR | Zbl