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)
On semilinear elliptic equations involving concave--convex nonlinearities and sign-changing weight function. (English) Zbl 1153.35036
The author studies the multiplicity results of positive solutions of the following elliptic equation: $$-\Delta u=u^p+\lambda f(x)u^q\text{ in } \Omega \quad 0\le u\in H_0^1(\Omega),\tag 1$$ where $\Omega$ is a bounded domain in $\bbfR^N$, $0<q<1<p<2^*$; $2^*=\frac{N+2}{N-2}$ if $N\ge 3$, $2^*=\infty$ if $N=2$, and $f$ is a given function, $\lambda>0$. The author under suitable assumptions on the data (1) proves that (1) possesses at least two positive solutions for $\lambda$ is sufficiently small.

35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J25Second order elliptic equations, boundary value problems
47J30Variational methods (nonlinear operator equations)
Full Text: DOI
[1] Adimurthy; Pacella, F.; Yadava, L.: On the number of positive solutions of some semilinear Dirichlet problems in a ball. Differential integral equations 10, 1157-1170 (1997) · Zbl 0940.35069
[2] Ambrosetti, A.; Brezis, H.; Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. funct. Anal. 122, 519-543 (1994) · Zbl 0805.35028
[3] Bahri, A.: Topological results on a certain class of functionals and applications. J. funct. Anal. 41, 397-427 (1981) · Zbl 0499.35050
[4] Bahri, A.; Berestycki, H.: A perturbation method in critical point theory and application. Trans. amer. Math. soc. 267, 1-32 (1981) · Zbl 0476.35030
[5] Cîrstea, F. S.; Rădulescu, V.: Multiple solutions of degenerate perturbed elliptic problems involving a subcritical Sobolev exponent. Topol. methods nonlinear anal. 15, 285-300 (2000) · Zbl 0979.35055
[6] Cao, D. M.; Zhou, H. S.: Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN. Proc. roy. Soc. Edinburgh sect. A 126, 443-463 (1996) · Zbl 0846.35042
[7] Damascelli, L.; Grossi, M.; Pacella, F.: Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle. Ann. inst. H. Poincaré anal. Non linéaire 16, 631-652 (1999) · Zbl 0935.35049
[8] Drábek, P.; Kufner, A.; Nicolosi, F.: Quasilinear elliptic equations with degenerations and singularities. De gruyter ser. Nonlinear anal. Appl. 5 (1997) · Zbl 0894.35002
[9] Ekeland, I.: On the variational principle. J. math. Anal. appl. 17, 324-353 (1974) · Zbl 0286.49015
[10] M. Ghergu, V. Rădulescu, Singular elliptic problems with lack of compactness, Ann. Mat. Pura Appl., in press
[11] Ouyang, T.; Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problem II. J. differential equations 158, 94-151 (1999) · Zbl 0947.35067
[12] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations. Regional conf. Ser. in math. (1986)
[13] Struwe, M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscripta math. 32, 335-364 (1980) · Zbl 0456.35031
[14] Tarantello, G.: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. inst. H. Poincaré anal. Non linéaire 9, 281-304 (1992) · Zbl 0785.35046
[15] Trudinger, N. S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. pure appl. Math. 20, 721-747 (1967) · Zbl 0153.42703
[16] Tang, M.: Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities. Proc. roy. Soc. Edinburgh sect. A 133, 705-717 (2003) · Zbl 1086.35053