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

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  • 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.

    Mathematics Subject Classification: Primary: 35J20, 35J25; Secondary: 35J65.


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  • [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), 275-284.


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


    J. Chabrowski, On the Neumann problem with the Hardy-Sobolev potential, Ann. Mat. Pura Appl., 186 (2007), 703-719.doi: doi:10.1007/s10231-006-0027-9.


    J. Chabrowski, The Neumann problem for semilinear elliptic equations with critical Sobolev exponent, Milan Journal of Mathematics, 75 (2007), 197-224.doi: doi:10.1007/s00032-006-0065-1.


    J. Chabrowski, On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential, Rev. Mat. Complut., 17 (2004), 195-227.


    J. Chabrowski, On a critical Neumann problem with a perturbation of lower order, Acta Mathematicae Applicatae Sinica, 24 (2008), 441-452.doi: doi:10.1007/s10255-008-8038-5.


    D. Cao and J. Chabrowski, Critical Neumann problem with competing Hardy potentials, Rev. Mat. Complut., 20 (2007), 309-338.


    D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372.doi: doi:10.1016/j.jde.2005.07.010.


    D. Cao and P. Han, A note on the positive energy solutions for elliptic equations involving critical Sobolev exponent, Appl. Math. Lett., 16 (2003), 1105-1113.doi: doi:10.1016/S0893-9659(03)90102-9.


    J. Chabrowski and M. Willem, Least energy solutions of a critical Neumann problem with a weight, Calc. Var. Partial Differential Equations, 15 (2002), 421-431.doi: doi:10.1007/s00526-002-0101-0.


    Y. B. Deng and L. Y. Jin, Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents, Nonlinear Anal., 67 (2007), 3261-3275.doi: doi:10.1016/j.na.2006.07.051.


    Y. B. Deng, L. Y. Jin and S. J. Peng, A Robin boundary problem with Hardy potential and critical nonlinearities, Journal d'Analyse Mathématique, 104 (2008), 125-154.


    L. Ding and C. L. Tang, Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions, Nonlinear Anal., 71 (2009), 3668-3689.doi: doi:10.1016/j.na.2009.02.017.


    V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495.doi: doi:10.1080/03605300500394439.


    N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincar'e Anal. Non Linaire, 21 (2004), 767-793.doi: doi:10.1016/j.anihpc.2003.07.002.


    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), 167-186.doi: doi:10.1016/S0362-546X(03)00223-2.


    C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.doi: doi:10.1016/0022-0396(88)90147-7.


    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), 351-368.doi: doi:10.1090/S0002-9947-1986-0849484-2.


    M. Struwe, "Variational Methods," 2nd edition, Springer, New York, 1996.


    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), 1302-1320.doi: doi:10.1016/j.na.2008.02.013.


    X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.doi: doi:10.1016/0022-0396(91)90014-Z.

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