Bifurcation of solutions of $U(1)$-equivariant semilinear boundary value problems
DOI:
https://doi.org/10.12775/TMNA.2022.056Keywords
Equivariant bifurcation, semilinear Fredholm maps, index bundle, elliptic BVPAbstract
Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.References
M.S. Agranovich, Elliptic Boundary Problems, Partial Differential Equations IX, Encyclopedia of Mathematical Sciences, Springer–Verlag, 1997.
J.C. Alexander, Bifurcation of zeroes of parameterized functions, J. Funct. Anal. 29 (1978), 37–53.
J.C. Alexander and J.Yorke, Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), 1–9.
M.F. Atiyah, Thom complexes, Proc. Lond. Math. Soc. 11 (1961), 291–310.
M.F. Atiyah, I.M. Singer, The index of elliptic operators IV, V, Ann. of Math. 93(1971), pp. 119-149.
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied equivariant degree, AIMS Series on Differential Equations and Dynamical Systems (AIMS), vol. 1, Springfield, MO, 2006.
J.C. Becker and R. Schultz, Equivariant function spaces and stable homotopy theory I, Comment. Math. Helv. 49 (1974), 1–34.
W. Browder, Surgery and the theory of differentiable transformation groups, Proceedings of the Conference of Transformation Groups (New Orláns, 1967), Springer–Verlag, New York, 1968, pp. 1-46.
E.N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc. (3) 26 (1973), 359–384.
B.V. Fedosov, A periodicity theorem in the algebra of symbols, Math. USSR Sb. 34 (1978), 382–410.
P. Fitzpatrick and J. Pejsachowicz, Orientation and the Leray–Schauder theory for fully nonlinear elliptic boundary value problems, Mem. Amer. Math. Soc. 101 (1993), no. 483.
W. Greub, S. Halperin and R.Vanstone, Connections, Curvature, and Cohomology, vol. I, Academic Press, New York, 1973.
D. Husemoller, Fibre Bundles, Springer–Verlag, 1975.
J. Ize, Bifurcation theory for Fredholm operators, Mem. Am. Math. Soc. 174 (1976).
J. Ize, Topological bifurcation, Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations 15 (1995), 341–463.
J. Ize and A. Vignoli, Equivariant Degree Theory, Walter de Gruyter & Co., Berlin, 2003.
G. López Garza and S. Rybicki, Equivariant bifurcation index, Nonlinear Anal. 73 (2010), 2779–2791.
T. Matsuoka, Equivariant function spaces and bifurcation, J. Math. Soc. Japan 35 (1983), 43–52.
J. Pejsachowicz, K-theoretic methods in bifurcation theory, Contemp. Math. 72 (1988), 193–205.
J. Pejsachowicz, The Leray–Schauder reduction and bifurcation for parametrized families of nonlinear elliptic boundary value problems, Topol. Methods Nonlinear Anal. 18 (2001), 243–268.
J. Pejsachowicz, Bifurcation of Fredholm maps I; Index bundle and bifurcation, Topol. Methods Nonlinear Anal. 38 (2011), no. 1, 115–168.
J. Pejsachowicz, Bifurcation of Fredholm maps II; The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal. 38 (2011), no. 2, 291–305.
J. Pejsachowicz, The family index theorem and bifurcation of solutions of nonlinear elliptic BVP, J. Differential Equations 252 (2011), 4942–4961.
J. Pejsachowicz, The index bundle and bifurcation from the infinity of solutions of nonlinear elliptic boundary value problems, J. Fixed Point Theory Appl. 17 (2015), 43–64.
J. Pejsachowicz and P.J. Rabier, Degree theory for C 1 -Fredholm mappings of index 0, J. Anal. Math. 76 (1998), 289–319.
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