Abstract
This paper concerns the bifurcation of bound states \( u \in L^{2} (\mathbb{R}^{N}) \) for a class of second-order nonlinear elliptic eigenvalue problems that includes cases which are already known to exhibit some surprising behaviour. By treating a larger class of nonlinearities we cover new cases such as a situation where there is no bifurcation at a simple isolated eigenvalue lying at the bottom of the spectrum of the linearization. As an application of recent work on bifurcation for problems that are only Hadamard differentiable, we also establish bifurcation at all isolated eigenvalues of odd multiplicity which are sufficiently far from the essential spectrum.
Mathematics Subject Classification (2010).35J61, 35P30, 47J15.
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References
H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer- Verlag, Berlin 1987.
D.E. Edmunds and W.D. Evans, Spectral Theory and differential operators, Oxford University Press, Oxford 1987.
G. Evéquoz and C.A. Stuart, Hadamard differentiability and bifurcation, Proc. Royal Soc. Edinburgh, 137A (2007), 1249–1285.
G. Evéquoz and C.A. Stuart, On differentiability and bifurcation, Adv. Math. Ecom., 8 (2006), 155–184.
G.Evéquoz and C.A. Stuart, Bifurcation points of a degenerate elliptic boundaryvalue problem, Rend. Lincei Mat. Appl., 17 (2006), 309–334.
G.Evéquoz and C.A.Stuart, Bifurcation and concentration of radial solutions of a nonlinear degenerate elliptic eigenvalue problem, Adv. Nonlinear Studies, 6 (2006), 215–232.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer, Berlin 1983.
D. Idczak and A. Rogowski, On a generalization of Krasnoselskii’s theorem, J.Austral.Math.Soc., 72 (2002), 389–394.
M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, New York, 1964.
P.J. Rabier, Bifurcation in weighted spaces, Nonlinearity, 21 (2008), 841–856.
P.J. Rabier, Decay transference and Fredholmness of differential operators in weighted Sobolev spaces, Differential Integral Equations, 21 (2008), 1001–1018.
P.J. Rabier and C.A. Stuart, Fredholm properties of Schrödinger operators in \( L^{p} (\mathbb{R}^{N}) \), Diff. Integral Eqns., 13 (2000), 1429–144.
P.J. Rabier and C.A. Stuart, Global bifurcation for quasilinear elliptic equations on \( \mathbb{R}^{N} \), Math. Z., 237 (2001), 85–124.
B. Simon, Schrödinger semigroups, Bull. AMS, 7 (1982), 447–526.
C.A. Stuart, Bifurcation for some non-Fréchet differentiable problems, Nonlinear Anal., TMA, 69 (2008), 1011–1024.
C.A. Stuart, An introduction to elliptic equations on \( \mathbb{R}^{N} \), in Nonlinear functional analysis and applications to differential equations, ed., A. Ambrosetti, K.C. Chang and I. Ekeland, World Scientific, Singapore 1998.
C.A. Stuart, Bifurcation and decay of solutions for a class of elliptic equations on \( \mathbb{R}^{N} \), Cont. Math., 540 (2011), 203–230.
C.A. Stuart, Asymptotic linearity and Hadamard differentiability, Nonlinear Analysis, 75 (2012), 4699–4710.
C.A. Stuart, Bifurcation at isolated singular points of the Hadamard derivative, preprint 2012.
C.A. Stuart, Asymptotic bifurcation and second order elliptic equations on \( \mathbb{R}^{N} \), preprint 2012.
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Stuart, C.A. (2014). Bifurcation at Isolated Eigenvalues for Some Elliptic Equations on ℝN . In: de Figueiredo, D., do Ó, J., Tomei, C. (eds) Analysis and Topology in Nonlinear Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-04214-5_25
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DOI: https://doi.org/10.1007/978-3-319-04214-5_25
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