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)
Three solutions of a quasilinear elliptic problem near resonance. (English) Zbl 0958.35052
Using critical point theory, the authors prove a multiplicity result near the first eigenvalue $\lambda_1$ for quasilinear elliptic problems of the form $$ -\Delta_pu-\lambda_1|u|^{p-2}u+\varepsilon |u|^{p-2}u=f(x,u)+h(x) $$ with Dirichlet conditions. It is proved that, for sufficiently small $\varepsilon >0$, the problem above has at least three solutions when $f$ and $h$ satisfy a Landesman-Lazer condition.

35J65Nonlinear boundary value problems for linear elliptic equations
58E30Variational principles on infinite-dimensional spaces
35A15Variational methods (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: EuDML
[1] AMBROSETTI A.-RABINOWITZ P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] ANANE A.: Simplicite et isolation de la premi√©re valeur propre du p-Laplacien avec poids. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), 725-728. · Zbl 0633.35061
[3] BADIALE M.-LUPO D.: Some remarks on a multiplicity result by Mawhin and Schmitt. Bull. Acad. R. Belgique Cl. Sci. 65 (1989), 210-224. · Zbl 0706.34020
[4] CHIAPPINELLI R.-MAWHIN J.-NUGARI R.: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities. Nonlinear Anal. T. M. A. 18 (1992), 1099-1112. · Zbl 0780.35038 · doi:10.1016/0362-546X(92)90155-8
[5] HIRANO N.: Multiple solutions for quasilinear elliptic equations. Nonlinear Anal. T. M. A. 15 (1990), 625-638. · Zbl 0723.35034 · doi:10.1016/0362-546X(90)90003-Y
[6] LUPO D.-RAMOS M.: Some multiplicity results for two-point boundary value problems near resonance. Rend. Sem. Mat. Univ. Politec. Torino 48 (1990), 125-135. · Zbl 0764.34017
[7] MAWHIN J.-SCHMITT K.: Nonlinear eigenvalue problems with the parameter near resonance. Ann. Polon. Math. LI (1990), 241-248. · Zbl 0724.34025
[8] RAMOS M.-SANCHEZ L.: A variational approach to multiplicity in elliptic problems near resonance. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 385-394. · Zbl 0869.35041 · doi:10.1017/S0308210500023696
[9] SANCHEZ L.: Boundary value problems for some fourth order ordinary differential equations. Applicable Anal. 38 (1990), 161-177. · Zbl 0682.34020 · doi:10.1080/00036819008839960
[10] ZONGMING, GUO: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Anal. T. M. A. 18 (1992), 957-971. · Zbl 0782.35053 · doi:10.1016/0362-546X(92)90132-X