March  2011, 10(2): 527-540. doi: 10.3934/cpaa.2011.10.527

Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  January 2010 Revised  July 2010 Published  December 2010

In this paper, we consider the following semilinear elliptic equations with critical Hardy-Sobolev exponent:

$ -\Delta u+\lambda\frac{u}{|x-a|^2}-\gamma\frac{u}{|x|^2} =\frac{Q(x)}{|x|^s}|u|^{2^*(s)-2}u+g(x,u), u>0$ in $\Omega,$

$ \frac{\partial u}{\partial\nu}+\alpha(x)u=0 $ on $\partial\Omega. $

By variational method, the existence of positive solution is obtained.

Citation: Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
References:
[1]

Adimurthi and S. L. Yadava, Critical Sobolev exponent problem in $\R^N$ $(N\geq 4)$ with Neumann boundary condition,, Proc. Indian Acad. Sci., 100 (1990), 275.   Google Scholar

[2]

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

[3]

J. Chabrowski, On the Neumann problem with the Hardy-Sobolev potential,, Ann. Mat. Pura Appl., 186 (2007), 703.  doi: doi:10.1007/s10231-006-0027-9.  Google Scholar

[4]

J. Chabrowski, The Neumann problem for semilinear elliptic equations with critical Sobolev exponent,, Milan Journal of Mathematics, 75 (2007), 197.  doi: doi:10.1007/s00032-006-0065-1.  Google Scholar

[5]

J. Chabrowski, On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential,, Rev. Mat. Complut., 17 (2004), 195.   Google Scholar

[6]

J. Chabrowski, On a critical Neumann problem with a perturbation of lower order,, Acta Mathematicae Applicatae Sinica, 24 (2008), 441.  doi: doi:10.1007/s10255-008-8038-5.  Google Scholar

[7]

D. Cao and J. Chabrowski, Critical Neumann problem with competing Hardy potentials,, Rev. Mat. Complut., 20 (2007), 309.   Google Scholar

[8]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: doi:10.1016/j.jde.2005.07.010.  Google Scholar

[9]

D. Cao and P. Han, A note on the positive energy solutions for elliptic equations involving critical Sobolev exponent,, Appl. Math. Lett., 16 (2003), 1105.  doi: doi:10.1016/S0893-9659(03)90102-9.  Google Scholar

[10]

J. Chabrowski and M. Willem, Least energy solutions of a critical Neumann problem with a weight,, Calc. Var. Partial Differential Equations, 15 (2002), 421.  doi: doi:10.1007/s00526-002-0101-0.  Google Scholar

[11]

Y. B. Deng and L. Y. Jin, Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents,, Nonlinear Anal., 67 (2007), 3261.  doi: doi:10.1016/j.na.2006.07.051.  Google Scholar

[12]

Y. B. Deng, L. Y. Jin and S. J. Peng, A Robin boundary problem with Hardy potential and critical nonlinearities,, Journal d'Analyse Math\'ematique, 104 (2008), 125.   Google Scholar

[13]

L. Ding and C. L. Tang, Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions,, Nonlinear Anal., 71 (2009), 3668.  doi: doi:10.1016/j.na.2009.02.017.  Google Scholar

[14]

V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity,, Comm. Partial Differential Equations, 31 (2006), 469.  doi: doi:10.1080/03605300500394439.  Google Scholar

[15]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar'e Anal. Non Linaire, 21 (2004), 767.  doi: doi:10.1016/j.anihpc.2003.07.002.  Google Scholar

[16]

P. G. Han and Z. X. Liu, Positive solutions for elliptic equations involving critical Sobolev exponents and Hardy terms with Neumann boundary conditions,, Nonlinear Anal., 55 (2003), 167.  doi: doi:10.1016/S0362-546X(03)00223-2.  Google Scholar

[17]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: doi:10.1016/0022-0396(88)90147-7.  Google Scholar

[18]

W. M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type,, Trans. Amer. Math. Soc., 297 (1986), 351.  doi: doi:10.1090/S0002-9947-1986-0849484-2.  Google Scholar

[19]

M. Struwe, "Variational Methods," 2nd, edition, (1996).   Google Scholar

[20]

Y. Y. Shang and C. L. Tang, Positive solutions for Neumann elliptic problems involving critical Hardy-Sobolev exponent with boundary singularities,, Nonlinear Anal., 70 (2009), 1302.  doi: doi:10.1016/j.na.2008.02.013.  Google Scholar

[21]

X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283.  doi: doi:10.1016/0022-0396(91)90014-Z.  Google Scholar

show all references

References:
[1]

Adimurthi and S. L. Yadava, Critical Sobolev exponent problem in $\R^N$ $(N\geq 4)$ with Neumann boundary condition,, Proc. Indian Acad. Sci., 100 (1990), 275.   Google Scholar

[2]

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

[3]

J. Chabrowski, On the Neumann problem with the Hardy-Sobolev potential,, Ann. Mat. Pura Appl., 186 (2007), 703.  doi: doi:10.1007/s10231-006-0027-9.  Google Scholar

[4]

J. Chabrowski, The Neumann problem for semilinear elliptic equations with critical Sobolev exponent,, Milan Journal of Mathematics, 75 (2007), 197.  doi: doi:10.1007/s00032-006-0065-1.  Google Scholar

[5]

J. Chabrowski, On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential,, Rev. Mat. Complut., 17 (2004), 195.   Google Scholar

[6]

J. Chabrowski, On a critical Neumann problem with a perturbation of lower order,, Acta Mathematicae Applicatae Sinica, 24 (2008), 441.  doi: doi:10.1007/s10255-008-8038-5.  Google Scholar

[7]

D. Cao and J. Chabrowski, Critical Neumann problem with competing Hardy potentials,, Rev. Mat. Complut., 20 (2007), 309.   Google Scholar

[8]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: doi:10.1016/j.jde.2005.07.010.  Google Scholar

[9]

D. Cao and P. Han, A note on the positive energy solutions for elliptic equations involving critical Sobolev exponent,, Appl. Math. Lett., 16 (2003), 1105.  doi: doi:10.1016/S0893-9659(03)90102-9.  Google Scholar

[10]

J. Chabrowski and M. Willem, Least energy solutions of a critical Neumann problem with a weight,, Calc. Var. Partial Differential Equations, 15 (2002), 421.  doi: doi:10.1007/s00526-002-0101-0.  Google Scholar

[11]

Y. B. Deng and L. Y. Jin, Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents,, Nonlinear Anal., 67 (2007), 3261.  doi: doi:10.1016/j.na.2006.07.051.  Google Scholar

[12]

Y. B. Deng, L. Y. Jin and S. J. Peng, A Robin boundary problem with Hardy potential and critical nonlinearities,, Journal d'Analyse Math\'ematique, 104 (2008), 125.   Google Scholar

[13]

L. Ding and C. L. Tang, Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions,, Nonlinear Anal., 71 (2009), 3668.  doi: doi:10.1016/j.na.2009.02.017.  Google Scholar

[14]

V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity,, Comm. Partial Differential Equations, 31 (2006), 469.  doi: doi:10.1080/03605300500394439.  Google Scholar

[15]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar'e Anal. Non Linaire, 21 (2004), 767.  doi: doi:10.1016/j.anihpc.2003.07.002.  Google Scholar

[16]

P. G. Han and Z. X. Liu, Positive solutions for elliptic equations involving critical Sobolev exponents and Hardy terms with Neumann boundary conditions,, Nonlinear Anal., 55 (2003), 167.  doi: doi:10.1016/S0362-546X(03)00223-2.  Google Scholar

[17]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: doi:10.1016/0022-0396(88)90147-7.  Google Scholar

[18]

W. M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type,, Trans. Amer. Math. Soc., 297 (1986), 351.  doi: doi:10.1090/S0002-9947-1986-0849484-2.  Google Scholar

[19]

M. Struwe, "Variational Methods," 2nd, edition, (1996).   Google Scholar

[20]

Y. Y. Shang and C. L. Tang, Positive solutions for Neumann elliptic problems involving critical Hardy-Sobolev exponent with boundary singularities,, Nonlinear Anal., 70 (2009), 1302.  doi: doi:10.1016/j.na.2008.02.013.  Google Scholar

[21]

X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283.  doi: doi:10.1016/0022-0396(91)90014-Z.  Google Scholar

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