zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The Nehari manifold for a semilinear elliptic equation involving a sublinear term. (English) Zbl 1130.35049

The author discusses the problem of existence and multiplicity of non-negative solutions to the problem

-Δu(x)=λu(x)+b(x)|u(x)| γ-2 u(x)foruΩu(x)=0forxΩ,(1)

where Ω N is a smooth bounded domain, b:Ω a smooth function, λ and 1<γ<2. When 1<γ<2 the problem (1) is asymptotically linear and the author establishes results on bifurcation from infinity when λ=λ 1 , the principal eigenvalue of the linear problem

-Δu(x)=λu(x)xΩ;u(x)=0xΩ·

By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form tJ(tu) where J is the Euler functional associated to (1)), the author studies how the Nehari manifold changes as λ varies. The bifurcation is then described in terms of the sign of the quantity Ω bφ 1 γ dx where φ 1 is the positive eigenfunction of the above linear problem corresponding to λ 1 .

MSC:
35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J25Second order elliptic equations, boundary value problems
47J15Abstract bifurcation theory
47J30Variational methods (nonlinear operator equations)
References:
[1]Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. Jour. Diff. Equations 193, 481-499 (2003) · Zbl 1074.35032 · doi:10.1016/S0022-0396(03)00121-9
[2]Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. Nonlinear Differential Equations and Applications 2, 553-572 (1995) · Zbl 0840.35035 · doi:10.1007/BF01210623
[3]Binding, P.A., Drabek, P., Huang, Y.X.: On Neumann boundary value problems for some quasilinear elliptic equations. Electronic Journal of Differential Equations 5, 1-11 (1997)
[4]Drabek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibrering method. Proc. Royal Soc. Edinburgh 127, 703-726 (1997)
[5]Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101-123 (1960) · doi:10.1090/S0002-9947-1960-0111898-8
[6]Toland, J.: Asymptotic linearity and nonlinear eigenvalue problems. Quart. J. Math. Oxford Ser. 24(2), 241-250 (1973) · Zbl 0256.47049 · doi:10.1093/qmath/24.1.241