References
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev.18, 620–709 (1976)
Amann, H., Zehnder, E.: Non-trivial solutions of a class of non-resonance problems and applications to nonlinear differential equations. Ann. Sc. Norm Super Pisa7, 539–603 (1980)
Berestycki, H.: Le nombre de solutions de certain problèmes semilinéaires elliptiques. J. Funct. Anal.40, 1–29 (1981)
Bott, R.: Lectures on Morse theory, old and new. Bull. Am. Math. Soc.7, 331–358 (1982)
Brown, K.J., Dunne, P.C., Gardner, R.A.: A semilinear parabolic system arising in the theory of semiconductivity. J. Differ. Equations40, 232–252 (1981)
Brown, P.N.: Decay to uniform states in ecological interactions. SIAM J Appl. Math.38, 22–37 (1980)
Cartan, H.: Differential calculus. Paris: Herman 1967
Clark, D.: A variant of the Lyusternik-Schnirelmann theory. Indiana U. Math. J.22, 65–74 (1973)
Conley, C.: Isolated invariant sets and the Morse index, CBMS regional conferences in mathematics 38. Providence, 1978
Conway, E.: Diffusion and the predator prey interaction: steady states with flux at the boundaries, pp. 215–234. In: Nonlinear partial differential equations, ed. J. Smoller. Am. Math. Soc. Providence, 1983
Crandall, M., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal.52, 161–180 (1973)
Dancer, E.N.: On the ranges of certain weakly nonlinear elliptic partial differential operators. J. Math. Pure Appl.57, 351–366 (1978)
Dancer, E.N.: Symmetries, degree, homotopy indices and asymptotically homogeneous problems. Nonlinear Anal.6, 667–686 (1982)
Dancer, E.N.: Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math.350, 1–22 (1984)
Dancer, E.N.: On non-radially-symmetric bifurcation. J. Lond. Math. Soc.20, 287–292 (1979)
Dancer, E.N.: TheG-invariant implicit function theorem in infinite dimensions. Proc. R. Soc. Edinb.92A, 13–30 (1982)
Dancer, E.N.: Bifurcation theory in real Banach space. Proc. Lond. Math. Soc.23, 699–734 (1971)
Dancer, E.N.: Bifurcation theory in Banach spaces. Ph.D. thesis, Cambridge University, 1972
Dancer, E.N.: On the indices of fixed points of mappings in cones and applications. J. Math. Anal. Appl.91, 131–151 (1983)
Dancer, E.N.: On positive solutions of some pairs of differential equations. Trans. Am. Math. Soc.284, 729–743 (1984)
Dancer, E.N.: On positive solutions of some pairs of differential equations II. To appear
Henry, D.: Geometric theory of semilinear parabolic equations. Berlin, Heidelberg, New York: Springer 1981
Hirsch, M.W.: Differential equations and convergence almost everywhere in strongly monotone semiflows, pp. 267–285. In: Nonlinear partial differential equations, ed. J. Smoller Am. Math. Soc. Providence, 1983
Hofer, H.: Variational and topological methods in partially ordered Hilbert spaces. Math. Ann.261, 493–514 (1982)
Lazer, A., McKenna, J.: On a conjecture related to the number of solutions of a nonlinear Dirichlet problem. Proc. R. Soc. Edinb.95A, 275–283 (1983)
Nussbaum, R.: Positive operators and elliptic eigenvalue problems. Math. Z.186, 247–264 (1984)
Pimbley, G.: Eigenfunction branches of nonlinear operators and their bifurcations. Lect. Notes Math. 104. Berlin, Heidelberg, New York: Springer 1969
Rybakowski, K.: On the homotopy index for infinite-dimensional semiflows. Trans. Am. Math. Soc.269, 351–381 (1982)
Rybakowski, K.: The Morse index, repellor attractor pairs and the connection index for flows on non-compact spaces. J. Differ. Equations47, 66–98 (1983)
Sattinger, D.: Group representation theory, bifurcation theory and pattern formulation. J. Funct. Anal.28, 58–101 (1978)
Saut, J.C., Teman, R.: Generic properties of nonlinear boundary value problems. Comm. Partial Differ. Equations4, 293–319 (1979)
Schaefer, H.H.: Topological vector spaces. New York: MacMillan 1966
Solimini, S.: Some remarks on the number of solutions of some nonlinear elliptic problems, preprint, SISSA, 1983
Vilenkin, N.: Special functions and the theory of group representation. Am. Math. Soc., Providence, 1968
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Dancer, E.N. Counterexamples to some conjectures on the number of solutions of nonlinear equations. Math. Ann. 272, 421–440 (1985). https://doi.org/10.1007/BF01455568
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DOI: https://doi.org/10.1007/BF01455568