# 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)
Critical singular problems via concentration-compactness lemma. (English) Zbl 1159.35373
Summary: We consider existence and multiplicity results of nontrivial solutions for a class of quasilinear degenerate elliptic equations in $\Bbb R^N$ of the form $$-\text{div}\big[|x|^{-ap}\nabla u|^{p-2}\nabla u\big]+ \lambda|x|^{-(a+1)p}|u|^{p-2}u= |x|^{-bq}|u|^{q-2}u+f, \tag P$$ where $x\in\Bbb R^N$, $1<p<N$, $q=q(a,b)\equiv Np/[N-p(a+1-b)]$, $\lambda$ is a parameter, $0\le a<(N-p)/p$, $a\le b\le a+1$, and $f\in(L_b^q(\Bbb R^N))^*$. We look for solutions of problem (P) in the Sobolev space ${\cal D}_a^{1,p}(\Bbb R^N)$ and we prove a version of a concentration-compactness lemma due to Lions. Combining this result with the Ekeland’s variational principle and the mountain-pass theorem, we obtain existence and multiplicity results.

##### MSC:
 35J70 Degenerate elliptic equations 35J20 Second order elliptic equations, variational methods 35D05 Existence of generalized solutions of PDE (MSC2000)
Full Text:
##### References:
 [1] Alves, C. O.: Multiple positive solutions for equations involving critical Sobolev exponent in RN. Electron. J. Differential equations 13, 1-10 (1997) · Zbl 0886.35056 [2] Badiale, M.; Tarantello, G.: A Sobolev -- Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. ration. Mech. anal. 163, 259-293 (2002) · Zbl 1010.35041 [3] Ben-Naoum, A. K.; Troestler, C.; Willem, M.: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear anal. 26, 823-833 (1996) · Zbl 0851.49004 [4] Berestycki, H.; Lions, P. L.: Nonlinear scalar field equations I: Existence of a ground state. Arch. ration. Mech. anal. 82, 313-345 (1983) · Zbl 0533.35029 [5] Bianchi, G.; Chabrowski, J.; Szulkin, A.: On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear anal. 25, 41-59 (1995) · Zbl 0823.35051 [6] Boccardo, L.; Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear anal. 19, 581-597 (1992) · Zbl 0783.35020 [7] Brézis, H.; Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. pure appl. Math. 36, 437-477 (1983) · Zbl 0541.35029 [8] Brézis, H.; Nirenberg, L.: A minimization problem with critical exponent and nonzero data. Ann. scuola norm. Sup. Pisa cl. Sci. 16, 129-140 (1989) · Zbl 0763.46023 [9] Caffarelli, L.; Kohn, R.; Nirenberg, L.: First order interpolation inequalities with weights. Compos. math. 53, 259-275 (1984) · Zbl 0563.46024 [10] Caldiroli, P.; Musina, R.: On the existence of extremal functions for a weighted Sobolev embedding with critical exponent. Calc. var. Partial differential equations 8, 365-387 (1999) · Zbl 0929.35045 [11] Cao, D. M.; Li, G. B.; Zhou, H. S.: Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent. Proc. roy. Soc. Edinburgh sect. A 124, 1177-1191 (1994) · Zbl 0814.35026 [12] Catrina, F.; Wang, Z. -Q.: On the caffarelli -- Kohn -- Nirenberg inequalities: sharp constants, existence and nonexistence and symmetry of extremal functions. Comm. pure appl. Math. 54, 229-258 (2001) · Zbl 1072.35506 [13] Chou, K. S.; Chu, W. S.: On the best constant for a weighted Sobolev -- Hardy inequality. J. London math. Soc. (2) 48, 137-151 (1993) · Zbl 0739.26013 [14] Cîrstea, F.; Motreanu, D.; Rădulescu, V.: Weak solutions of quasilinear problems with nonlinear boundary condition. Nonlinear anal. 43, 623-636 (2001) · Zbl 0972.35038 [15] Clément, P.; De Figueiredo, D. G.; Mitidieri, E.: Quasilinear elliptic equations with critical exponents. Topol. methods nonlinear anal. 7, 133-170 (1996) · Zbl 0939.35072 [16] Clément, P.; Manasevich, R.; Mitidieri, E.: Some existence and non-existence results for a homogeneous quasilinear problem. Asymptot. analysis 17, 13-29 (1998) · Zbl 0945.34011 [17] Dibenedetto, E.: $C1+\alpha$ local regularity of weak solutions of degenerate elliptic equations. Nonlinear anal. 7, 827-850 (1983) · Zbl 0539.35027 [18] Ding, W. Y.; Ni, W. M.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. ration. Mech. anal. 91, 283-308 (1986) · Zbl 0616.35029 [19] Azorero, J. Garcia; Alonso, I. Peral: Multiplicity of solutions for elliptic problems with critical exponent or a nonsymmetric term. Trans. amer. Math. soc. 323, 877-895 (1991) · Zbl 0729.35051 [20] Ghoussoub, N.; Yuan, C.: Multiple solutions for quasi-linear pdes involving the critical Sobolev and Hardy exponents. Trans. amer. Math. soc. 352, 5703-5743 (2000) · Zbl 0956.35056 [21] Lieb, E. H.: Sharp constants in the Hardy -- Littlewood -- Sobolev and related inequalities. Ann. of math. 118, 349-374 (1983) · Zbl 0527.42011 [22] Lions, P. L.: The concentration-compactness principle in the calculus of variations. The limit case I. Rev. mat. Iberoamericana 1, 145-201 (1985) · Zbl 0704.49005 [23] Lions, P. L.: The concentration-compactness principle in the calculus of variations. The limit case II. Rev. mat. Iberoamericana 1, 45-121 (1985) · Zbl 0704.49006 [24] Noussair, E. S.; Swanson, C. A.; Yang, J. F.: Quasilinear elliptic problems with critical exponents. Nonlinear anal. 20, 285-301 (1993) · Zbl 0785.35042 [25] Piccirillo, A. M.; Toscano, R.: Multiple solutions of some nonlinear elliptic problems containing the p-Laplacian. Differential equations 37, 1121-1132 (2001) · Zbl 1165.35382 [26] Rădulescu, V.; Smets, D.: Critical singular problems on infinite cones. Nonlinear anal. 54, 1153-1164 (2003) · Zbl 1035.35044 [27] Smets, D.: A concentration compactness lemma with applications to singular eigenvalue problems. J. funct. Anal. 167, 463-480 (1999) · Zbl 0942.35127 [28] Talenti, G.: Best constants in Sobolev inequality. Ann. mat. Pura appl. (4) 110, 353-372 (1976) · Zbl 0353.46018 [29] Tan, J.; Yang, J.: On the singular variational problems. Acta math. Sci. ser. B 24, 672-690 (2004) · Zbl 1086.35030 [30] Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. inst. H. Poincaré anal. Non linéaire 9, 281-304 (1992) · Zbl 0785.35046 [31] Wang, Z. -Q.; Willem, M.: Singular minimization problems. J. differential equations 161, 307-320 (2000) · Zbl 0954.35074