zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bounded components of positive solutions of abstract fixed point equations: Mushrooms, loops and isolas. (English) Zbl 1066.47063
Let $U$ be an ordered Banach space whose positive cone is normal and has nonempty interior. This paper is devoted to the study of the nonlinear abstract equation ${\cal L}(\lambda)u+{\cal R}(\lambda,u)=0$ for $(\lambda ,u)\in \Bbb R\times U$, where ${\cal L}(\lambda)$ is a Fredholm operator of index $0$ and ${\cal R}\in C(\Bbb R\times U;U)$ is compact on bounded sets and $\lim_{u\rightarrow 0}{\cal R}(\lambda,u)/\Vert u\Vert =0$. The main result of the present paper concerns the bounded components of positive solutions emanating from $(\lambda,u)=(\lambda,0)$. The proofs are based on refined techniques from modern bifurcation theory.

47J05Equations involving nonlinear operators (general)
47J15Abstract bifurcation theory
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B32Bifurcation (PDE)
35B50Maximum principles (PDE)
Full Text: DOI
[1] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044
[2] Cano-Casanova, S.: Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type. Topol. meth. Nonlinear anal. 23, 45-72 (2004) · Zbl 1137.35374
[3] S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer, Isolas: compact solution components separated away from a given equilibrium curve, Hiroshima Math. J., in press. · Zbl 1152.35362
[4] Cingolani, S.; Gámez, J. L.: Positive solutions of a semilinear elliptic equation on RN with indefinite nonlinearity. Adv. differential equations 1, 773-791 (1996) · Zbl 0857.35036
[5] Crandall, M. G.; Rabinowitz, P. H.: Bifurcation from simple eigenvalues. J. funct. Anal. 8, 321-340 (1971) · Zbl 0219.46015
[6] Dancer, E. N.: Global solution branches for positive mappings. Arch. rational mech. Anal. 52, 181-192 (1973) · Zbl 0275.47043
[7] Dancer, E. N.: Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull. London math. Soc. 34, 533-538 (2002) · Zbl 1027.58009
[8] Esquinas, J.: Optimal multiplicity in local bifurcation theory, iigeneral case. J. differential equations 75, 206-215 (1988) · Zbl 0668.47043
[9] Esquinas, J.; López-Gómez, J.: Optimal multiplicity in local bifurcation theory, igeneralized generic eigenvalues. J. differential equations 71, 72-92 (1988) · Zbl 0648.34027
[10] Hess, P.; Kato, T.: On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. partial differential equations 5, 99-1030 (1980) · Zbl 0477.35075
[11] Kato, T.: Perturbation theory for linear operators, classics in mathematics. (1995) · Zbl 0836.47009
[12] Krein, M. G.; Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Amer. math. Soc. transl. 10, 199-325 (1962)
[13] López-Gómez, J.: The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems. J. differential equations 127, 263-294 (1996) · Zbl 0853.35078
[14] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001. · Zbl 0978.47048
[15] López-Gómez, J.; Molina-Meyer, M.: The maximum principle for cooperative weakly elliptic systems and some applications. Differential integral equations 7, 383-398 (1994) · Zbl 0827.35019
[16] J. López-Gómez, C. Mora-Corral, Finite Laurent developments and the logarithmic residue theorem in the real non-analytic case, Integral Equations Oper. Theory, in press. · Zbl 1085.47022
[17] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504