Asymptotic bifurcation and second order elliptic equations on N
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1259-1281.

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation G(u)=λu, where G acts between real Hilbert spaces and λ, and then for square-integrable solutions of a second order non-linear elliptic equation on N . The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the H-asymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the H-asymptotic derivative if it is too close to the essential spectrum.

DOI: 10.1016/j.anihpc.2014.09.003
Classification: 35J91,  47J15
Keywords: Asymptotic linearity, Asymptotic bifurcation, Nonlinear elliptic equation
@article{AIHPC_2015__32_6_1259_0,
     author = {Stuart, C.A.},
     title = {Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$
      },
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1259--1281},
     publisher = {Elsevier},
     volume = {32},
     number = {6},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.09.003},
     zbl = {1330.35187},
     mrnumber = {3425262},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.003/}
}
TY  - JOUR
AU  - Stuart, C.A.
TI  - Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$
      
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
DA  - 2015///
SP  - 1259
EP  - 1281
VL  - 32
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.003/
UR  - https://zbmath.org/?q=an%3A1330.35187
UR  - https://www.ams.org/mathscinet-getitem?mr=3425262
UR  - https://doi.org/10.1016/j.anihpc.2014.09.003
DO  - 10.1016/j.anihpc.2014.09.003
LA  - en
ID  - AIHPC_2015__32_6_1259_0
ER  - 
%0 Journal Article
%A Stuart, C.A.
%T Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$
      
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 1259-1281
%V 32
%N 6
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2014.09.003
%R 10.1016/j.anihpc.2014.09.003
%G en
%F AIHPC_2015__32_6_1259_0
Stuart, C.A. Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$
      . Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1259-1281. doi : 10.1016/j.anihpc.2014.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.003/

[1] A. Ambrosetti, D. Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser, Basel (2011) | MR | Zbl

[2] E.N. Dancer, On bifurcation from infinity, Q. J. Math. 25 no. 2 (1974), 81 -84 | MR | Zbl

[3] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985) | MR | Zbl

[4] J.-P. Dias, J. Hernandez, A remark on a paper by J.F. Toland and some applications to unilateral problems, Proc. R. Soc. Edinb. 75 (1976), 179 -182 | MR | Zbl

[5] A.L. Dontchev, R.T. Rockafellar, Implicit Functions and Solution Mappings, Springer, Heidelberg (2009) | MR | Zbl

[6] A. Edelson, C.A. Stuart, The principal branch of solutions of a nonlinear elliptic eigenvalue problem on N , J. Differ. Equ. 124 (1996), 279 -301 | MR | Zbl

[7] G. Evéquoz, C.A. Stuart, Hadamard differentiability and bifurcation, Proc. R. Soc. Edinb. A 137 (2007), 1249 -1285 | MR | Zbl

[8] G. Evéquoz, C.A. Stuart, On differentiability and bifurcation, Adv. Math. Econ. 8 (2006), 155 -184 | MR | Zbl

[9] T.M. Flett, Differential Analysis, Cambridge University Press, Cambridge (1980) | MR | Zbl

[10] F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line, Nonlinear Anal. 74 (2011), 4533 -4543 | MR | Zbl

[11] F. Genoud, Global bifurcation for asymptotically linear Schrödinger equations, Nonlinear Differ. Equ. Appl. 20 (2013), 23 -35 | MR | Zbl

[12] H. Koch, D. Tataru, Carleman estimates and absence of embedded eigenvalues, Commun. Math. Phys. 267 (2006), 419 -449 | MR | Zbl

[13] M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford (1964)

[14] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen (1964)

[15] P.H. Rabinowitz, On bifurcation from infinity, J. Differ. Equ. 14 (1973), 462 -475 | MR | Zbl

[16] C.A. Stuart, An introduction to elliptic equation on N , A. Ambrosetti, K.-C. Chang, I. Ekeland (ed.), Nonlinear Functional Analysis and Applications to Differential Equations, World Scientific, Singapore (1998)

[17] C.A. Stuart, Bifurcation for some non-Fréchet differentiable problems, Nonlinear Anal. 69 (2008), 1011 -1024 | MR | Zbl

[18] C.A. Stuart, Bifurcation and decay of solutions for a class of elliptic equations on N , Contemp. Math. vol. 540 (2011), 203 -230 | MR | Zbl

[19] C.A. Stuart, Asymptotic linearity and Hadamard differentiability, Nonlinear Anal. 75 (2012), 4699 -4710 | MR | Zbl

[20] C.A. Stuart, Bifurcation at isolated singular points of the Hadamard derivative, Proc. R. Soc. Edinb. (2014) | MR | Zbl

[21] C.A. Stuart, Bifurcation at isolated eigenvalues for some elliptic equations on N , preprint, 2012. | MR

[22] C.A. Stuart, H.-S. Zhou, Global branch of solutions for non-linear Schrödinger equations with deepening potential well, Proc. Lond. Math. Soc. (3) 92 (2006), 655 -681 | MR | Zbl

[23] J.F. Toland, Asymptotic nonlinearity and nonlinear eigenvalue problems, Quart. J. Math. Oxford 24 (1973), 241 -250 | MR | Zbl

[24] J.F. Toland, Asymptotic linearity and a class of nonlinear Strum–Liouville problems on the half-line, Lect. Notes Math. vol. 415 , Springer (1974), 429 -434 | MR | Zbl

[25] J.F. Toland, Asymptotic linearity and nonlinear eigenvalue problems, Proc. R. Ir. Acad. 77 (1977), 1 -12 | MR | Zbl

[26] J.F. Toland, Bifurcation and asymptotic bifurcation for non-compact non-symmetric gradient operators, Proc. R. Soc. Edinb. 73 (1975), 137 -147 | MR | Zbl

[27] E. Zeidler, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985) | Zbl

Cited by Sources: