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November  2013, 12(6): 2697-2713. doi: 10.3934/cpaa.2013.12.2697

Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions

Mateus Balbino Guimarães 1, and  Rodrigo da Silva Rodrigues 1,

1. 

Universidade Federal de São Carlos, Departamento de Matemática, Rod. Washington Luís, Km 235, CEP. 13565-905, São Carlos, SP, Brazil, Brazil

Received  September 2012 Revised  October 2012 Published  May 2013

In this article we establish the existence and nonexistence of a weak solution to singular elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions.
Keywords: Variational methods, existence of solution, critical Caffarelli-Kohn-Nirenberg exponent, nonexistence of solution, singular and sign-changing weights..
Mathematics Subject Classification: Primary: 35A15, 35B33, 35B25; Secondary: 35J6.
Citation: Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
References:
[1]

M. Bouchekif and A. Matallah, Singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions, Electronic Journal of Differential Equations, 2010 (2010), 1-12.  Google Scholar

[2]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc.Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[3]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[4]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275.  Google Scholar

[5]

N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Royal Society of Edinburgh, 131A (2001), 1275-1295. doi: 10.1017/S0308210500001396.  Google Scholar

[6]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 2 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[7]

L. C. Evans, "Partial Differential Equations,'' Graduate studies in mathematics 19, Amer. Math. Soc. Providence, Rhode Island, 1998.  Google Scholar

[8]

A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522. doi: 10.1006/jdeq.2000.3999.  Google Scholar

[9]

P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal., 61 (2005), 735-758. doi: 10.1016/j.na.2005.01.030.  Google Scholar

[10]

X. J. Huang, X. P. Wu and C. L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal., 74 (2011), 2602-2611. doi: 10.1016/j.na.2010.12.015.  Google Scholar

[11]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. doi: 10.1006/jdeq.1998.3589.  Google Scholar

[12]

M. Lin, Some further results for a class of weighted nonlinear elliptic equations, J. Math. Anal. Appl., 337 (2008), 537-546. doi: 10.1016/j.jmaa.2007.04.034.  Google Scholar

[13]

O. H. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth, Nonlinear Anal., 29 (7) (1997), 773-781. doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar

[14]

R. S. Rodrigues, On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function, Nonlinear Anal., 73 (2010), 857-880. doi: 10.1016/j.na.2010.03.053.  Google Scholar

[15]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990.  Google Scholar

[16]

B. J. Xuan, S. Su and Y. Yan, Existence results for Brézis-Nirenberg problems with Hardy potential and singular coefficients, Nonlinear Anal., 67 (2007), 2091-2106. doi: 10.1016/j.na.2006.09.018.  Google Scholar

[17]

B. J. Xuan, The solvability of quasilinear Brézis-Nirenberg-type problems with singular weights, Nonlinear Anal., 62 (2005), 703-725. doi: 10.1016/j.na.2005.03.095.  Google Scholar

show all references

References:
[1]

M. Bouchekif and A. Matallah, Singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions, Electronic Journal of Differential Equations, 2010 (2010), 1-12.  Google Scholar

[2]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc.Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[3]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[4]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275.  Google Scholar

[5]

N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Royal Society of Edinburgh, 131A (2001), 1275-1295. doi: 10.1017/S0308210500001396.  Google Scholar

[6]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 2 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[7]

L. C. Evans, "Partial Differential Equations,'' Graduate studies in mathematics 19, Amer. Math. Soc. Providence, Rhode Island, 1998.  Google Scholar

[8]

A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522. doi: 10.1006/jdeq.2000.3999.  Google Scholar

[9]

P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal., 61 (2005), 735-758. doi: 10.1016/j.na.2005.01.030.  Google Scholar

[10]

X. J. Huang, X. P. Wu and C. L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal., 74 (2011), 2602-2611. doi: 10.1016/j.na.2010.12.015.  Google Scholar

[11]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. doi: 10.1006/jdeq.1998.3589.  Google Scholar

[12]

M. Lin, Some further results for a class of weighted nonlinear elliptic equations, J. Math. Anal. Appl., 337 (2008), 537-546. doi: 10.1016/j.jmaa.2007.04.034.  Google Scholar

[13]

O. H. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth, Nonlinear Anal., 29 (7) (1997), 773-781. doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar

[14]

R. S. Rodrigues, On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function, Nonlinear Anal., 73 (2010), 857-880. doi: 10.1016/j.na.2010.03.053.  Google Scholar

[15]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990.  Google Scholar

[16]

B. J. Xuan, S. Su and Y. Yan, Existence results for Brézis-Nirenberg problems with Hardy potential and singular coefficients, Nonlinear Anal., 67 (2007), 2091-2106. doi: 10.1016/j.na.2006.09.018.  Google Scholar

[17]

B. J. Xuan, The solvability of quasilinear Brézis-Nirenberg-type problems with singular weights, Nonlinear Anal., 62 (2005), 703-725. doi: 10.1016/j.na.2005.03.095.  Google Scholar

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