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)
Existence and multiplicity results for some quasilinear elliptic equation with weights. (English) Zbl 1136.35049

Author’s abstract: Using variational methods, we show the existence and multiplicity of solutions of singular boundary value problems of the type

-div(|x| -α A(|u|)u)=|x| -β f(x,u)inΩ,u=0onΩ,

where the numbers α and β, as well as the functions A and f satisfy certain conditions.

MSC:
35J70Degenerate elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35D10Regularity of generalized solutions of PDE (MSC2000)
References:
[1]Abdellaoui, B.; Peral, I.: On quasilinear elliptic equations related to some caffarelli – Kohn – Nirenberg inequalities, Commun. pure appl. Anal. 2, No. 4, 539-566 (2003) · Zbl 1148.35324 · doi:10.3934/cpaa.2003.2.539
[2]Ambrosetti, A.; Rabinowitz, P.: Dual variational methods in critical point theory and applications, J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[3]Bidaut-Véron, M. -F.; García-Huidobro, M.: Regular and singular solutions of a quasilinear equation with weights, Asymptot. anal. 28, No. 2, 115-150 (2001) · Zbl 1027.35037
[4]Browder, F.: Existence theorems for nonlinear partial differentials equations, Proc. sympos. Pure math. 16, 1-60 (1970) · Zbl 0211.17204
[5]Browder, F.: Fixed point theory and nonlinear problems, Bull. amer. Math. soc. (N.S.) 9, No. 1, 1-39 (1983) · Zbl 0533.47053 · doi:10.1090/S0273-0979-1983-15153-4
[6]Caffarelli, L.; Kohn, R.; Nirenberg, L.: First order interpolation inequalities with weights, Compos. math. 53, 259-275 (1984) · Zbl 0563.46024 · doi:numdam:CM_1984__53_3_259_0
[7]Catrina, F.; Wang, Z. -Q.: Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN, Ann. inst. H. Poincaré anal. Non linéaire 18, No. 2, 157-178 (2001) · Zbl 1005.35045 · doi:10.1016/S0294-1449(00)00061-5 · doi:numdam:AIHPC_2001__18_2_157_0
[8]Clément, P.; De Figueiredo, D. G.; Mitidieri, E.: Quasilinear elliptic equations with critical exponents, Topol. methods nonlinear anal. 7, No. 1, 133-170 (1996) · Zbl 0939.35072
[9]Clément, P.; Manásevich, R.; Mitidieri, E.: Some existence and non-existence results for a homogeneous quasilinear problem, Asymptot. anal. 17, No. 1, 13-29 (1998) · Zbl 0945.34011
[10]De Figueiredo, D. G.; Girardi, M.; Matzeu, M.: Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential integral equations 17, No. 1 – 2, 119-126 (2004) · Zbl 1164.35341
[11]Evans, L. C.: Partial differential equations, Grad stud. Math. 19 (1998) · Zbl 0902.35002
[12]Do Ó, J.; Ubilla, P.: A multiplicity result for a class of superquadratic Hamiltonian systems, Electron. J. Differential equations 2003, No. 15, 1-14 (2003) · Zbl 1034.35046 · doi:emis:journals/EJDE/Volumes/2003/15/abstr.html
[13]Felli, V.; Schneider, M.: A note on regularity of solutions to degenerate elliptic equations of caffarelli – Kohn – Nirenberg type, Adv. nonlinear stud. 3, No. 4, 431-443 (2003) · Zbl 1106.35025
[14]García-Huidobro, M.: Behavior of positive radial solutions of a quasilinear equation with a weighted Laplacian, Electron. J. Differ. equ. Conf. 6, 173-187 (2000) · Zbl 0971.34009 · doi:emis:journals/EJDE/conf-proc/06/g1/abstr.html
[15]Pucci, P.; García-Huidobro, M.; Manásevich, R.; Serrin, J.: Qualitative properties of ground states for singular elliptic equations with weights, Ann. mat. Pura appl. (4) 185, S205-S243 (2006) · Zbl 1115.35050 · doi:10.1007/s10231-004-0143-3
[16]Gidas, B.; Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations, Commun. partial differential equations 6, No. 8, 883-901 (1981) · Zbl 0462.35041 · doi:10.1080/03605308108820196
[17]Gilbarg, D.; Trudinger, N.: Elliptic partial differential equations of second order, (2001)
[18]Guedda, M.; Veron, L.: Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear anal. 13, 879-902 (1989) · Zbl 0714.35032 · doi:10.1016/0362-546X(89)90020-5
[19]Heinonen, J.; Kilpeläinen, T.; Martio, O.: Nonlinear potential theory of degenerate elliptic equations, Oxford math. Monogr. (1993) · Zbl 0780.31001
[20]Lieberman, G.: The natural generalizations of the natural conditions of ladyshenskaya and ural’tseva for elliptic equations, Comm. partial differential equations 16, No. 2 – 3, 311-361 (1991) · Zbl 0742.35028 · doi:10.1080/03605309108820761
[21]Murthy, M. K. V.; Stampacchia, G.: Boundary value problems for some degenerate elliptic operators, Ann. mat. Pura. appl. 80, 1-122 (1968) · Zbl 0185.19201 · doi:10.1007/BF02413623
[22]Necas, J.: Introduction to the theory of nonlinear elliptic equations, (1986) · Zbl 0643.35001
[23]Prado, H.; Ubilla, P.: Existence of nonnegative solutions for generalized p-Laplacians, Lect. notes pure appl. Math. 194, 289-298 (1998) · Zbl 0913.35044
[24]Rabinowitz, P. H.: Variational methods for nonlinear elliptic eigenvalue problems, Indiana univ. Math. J. 23, 729-745 (1974) · Zbl 0278.35040 · doi:10.1512/iumj.1974.23.23061
[25]Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS reg. Conf. ser. Math. 65 (1986) · Zbl 0609.58002
[26]Simon, J.: Régularitéde la solution d’une équation non linéaire dans RN, Lecture notes in math. 665, 205-227 (1978) · Zbl 0402.35017
[27]Ubilla, P.: Homoclinic orbits for a quasilinear Hamiltonian system, J. math. Anal. appl. 193, 573-587 (1995) · Zbl 0836.34046 · doi:10.1006/jmaa.1995.1254
[28]Ubilla, P.: Multiplicity results for quasilinear elliptic equations, Comm. appl. Nonlinear anal. 3, No. 2, 35-49 (1996) · Zbl 0857.35049
[29]Xuan, B. J.: The eigenvalue problem of a singular quasilinear elliptic equation, Electron. J. Differential equations 2004, No. 16, 1-11 (2004) · Zbl 1217.35131 · doi:emis:journals/EJDE/Volumes/2004/16/abstr.html
[30]Xuan, B. J.: Existence results for a superlinear singular equation of caffarelli – Kohn – Nirenberg type, Bol. mat. (N.S.) 10, No. 2, 47-58 (2003) · Zbl 1203.35102
[31]Xuan, B. J.: Multiple solutions to a caffarelli – Kohn – Nirenberg type equation with asymptotically linear term, Rev. colombiana mat. 37, No. 2, 65-79 (2003) · Zbl 1112.35074