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)
Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains. (English) Zbl 1173.35498
Summary: Existence and multiplicity results to the following Dirichlet problem $$\cases -\Delta u+u= \lambda f(x)|u|^{q-1}+ h(x)|u|^{p-1} &\text{ in }\Omega,\\ u>0 &\text{ in }\Omega,\\ u=0 &\text{ on }\partial\Omega,\endcases$$ are established, where $\Omega=\Omega'\times\Bbb R$, $\Omega'\subset\Bbb R^{N-1}$ is bounded smooth domain and $N\ge2$. Here $1<q<2<p<2^*$ $(2^*= \frac{2N}{N-2}$ if $N\ge 3$, $2^*=\infty$ if $N=2)$ $\lambda$ is a positive real parameter, the function $f$, among other conditions, can possibly change sign in $\Omega$, and the function $h$ satisfies suitable conditions. The study is based on the comparison of energy levels on Nehari manifold.

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
35B20Perturbations (PDE)
35D05Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI
[1] Ambrosetti, A.; Brézis, H.; Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. funct. Anal. 122, 519-543 (1994) · Zbl 0805.35028
[2] Adimurthi; Pacella, F.; Yadava, S.: On the number of positive solutions of some semilinear Dirichlet problems in a ball. Differential integral equations 10, 1157-1170 (1997) · Zbl 0940.35069
[3] 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
[4] Tang, M.: Exact multiplicity for semilinear Dirichlet problem involving concave and convex nonlinearities. Proc. roy. Soc. Edinburgh sect. A 133, 705-717 (2003) · Zbl 1086.35053
[5] Ambrosetti, A.; Azorero, J. Garcia; Alonso, I. Peral: Multiplicity results for some nonlinear elliptic equations. J. funct. Anal. 137, 219-242 (1996) · Zbl 0852.35045
[6] Azorero, J. Garcia; Manfredi, J. J.; Alonso, I. Peral: Sobolev versus hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. contemp. Math. 2, 385-404 (2000) · Zbl 0965.35067
[7] Brown, K. J.; Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. differential equations 193, 481-499 (2003) · Zbl 1074.35032
[8] Brown, K. J.; Wu, T. -F.: A fibering map approach to a semilinear elliptic boundary value problem. Electron. J. Differential equations 69, 1-9 (2007) · Zbl 1133.35337
[9] Cao, D. M.; Zhong, X.: Multiplicity of positive solutions for semilinear elliptic equations involving the critical Sobolev exponents. Nonlinear anal. 29, 46-483 (1997) · Zbl 0879.35055
[10] De Figueiredo, D. G.; Gossez, J. P.; Ubilla, P.: Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. eur. Math. soc. 8, 269-288 (2006) · Zbl 1245.35048
[11] Cao, D. M.: Multiple solutions of a semilinear elliptic equation in RN. Ann. inst. H. Poincaré anal. Non linéaire 10, 593-604 (1993)
[12] Bahri, A.; Lions, P. L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. inst. H. Poincaré anal. Non linéaire 14, 365-413 (1997) · Zbl 0883.35045
[13] Cerami, G.; Molle, R.; Passaseo, D.: Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary. Ann. inst. H. Poincaré anal. Non linéaire 24, 41-60 (2007) · Zbl 1123.35017
[14] Tehrani, H.: Solutions for indefinite semilinear elliptic equations in exterior domains. J. math. Anal. appl. 255, 308-318 (2001) · Zbl 0977.35052
[15] Hsu, T. S.: Multiple solutions for semilinear elliptic equations in unbounded cylinder domains. Proc. roy. Soc. Edinburgh sect. A 134, 719-731 (2004) · Zbl 1152.35371
[16] T.-F. Wu, Multiple positive solutions for Dirichlet problems involving concave and convex nonlinearities, doi:10.1016/j.na.2007.10.056
[17] Chabrowski, J.; Do O, J. M. B.: On semilinear elliptic equations involving concave and convex nonlinearities. Math. nachr. 233--234, 55-76 (2002)
[18] Gonçalves, J. C.; Miyagaki, O. H.: Multiple positive solutions for semilinear elliptic equations in RN involving subcritical exponents. Nonlinear anal. 32, 41-51 (1998) · Zbl 0891.35031
[19] Ra\check{}dulescu, V.; Smets, D.: Critical singular problems on infinite cones. Nonlinear anal. 54, 1153-1164 (2003) · Zbl 1035.35044
[20] Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. inst. H. Poincaré anal. Non linéaire 9, 281-304 (1992) · Zbl 0785.35046
[21] 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
[22] Brézis, H.; Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. amer. Math. soc. 88, 486-490 (1983) · Zbl 0526.46037
[23] Lien, W. C.; Tzeng, S. Y.; Wang, H. C.: Existence of solutions of semilinear elliptic problems on unbounded domains. Differential and integral equations 6, 1281-1298 (1993) · Zbl 0837.35051
[24] Chen, K. J.; Chen, K. C.; Wang, H. C.: Symmetry of positive solutions of semilinear elliptic equations in infinite strip domains. J. differential equations 148, 1-8 (1998) · Zbl 0912.35014