Advanced Search
Article Contents
Article Contents

Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

Abstract Related Papers Cited by
  • In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.
    Mathematics Subject Classification: 35J60, 35B33.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Al-aati, C. Wang and J. Zhao, Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term, Nonlinear Anal., 74 (2011), 4847-4861.doi: 10.1016/j.na.2011.04.057.


    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.doi: 10.1016/0022-1236(73)90051-7.


    A. Bahri and J. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.doi: 10.1002/cpa.3160410302.


    M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139.doi: 10.1016/j.jde.2008.12.011.


    H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Bonston, MA, 1987, 17-36.


    H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.


    H. Brezis 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.


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


    D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866.doi: 10.1090/S0002-9939-02-06729-1.


    D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424-434.doi: 10.1016/S0022-0396(03)00118-9.


    D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.doi: 10.1016/j.jfa.2012.01.006.


    D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), 471-501.doi: 10.1007/s00526-009-0295-5.


    A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poinceré Anal. Non Linéaire , 2 (1985), 463-470.


    D. Castorina, D. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206.doi: 10.1016/j.jde.2008.09.006.


    G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.doi: 10.1016/0022-1236(86)90094-7.


    J. L. Chern and C. S. Lin, Minimizer of Caffarelli-Kohn-Nirenberg inequlities with the singularity on the boundary, Arch. Ration. Mech. Anal., 197 (2010), 401-432.doi: 10.1007/s00205-009-0269-y.


    G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.


    D. Fabbri, G. Mancini and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations , 224 (2006), 258-276.doi: 10.1016/j.jde.2005.07.001.


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


    N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal., 16 (2006), 1201-1245.doi: 10.1007/s00039-006-0579-2.


    N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.doi: 10.1090/S0002-9947-00-02560-5.


    C. H. Hsia, C. S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849.doi: 10.1016/j.jfa.2010.05.004.


    Y. Li and C. S. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968.doi: 10.1007/s00205-011-0467-2.


    P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2, Rev. Mat. Iberoamericana, 1 (1985), 145-201; 2 (1985), 45-121.


    G. Mancini and K. Sandeep, Cylindrical symmetry of extremals of a Hardy-Sobolev inequality, Ann. Mat. Pura Appl., 183 (2004), 165-172.doi: 10.1007/s10231-003-0084-2.


    G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbbH^n$, Ann. Scuola. Norm. Sup. Pisa Cl. Sci., 7 (2008), 635-671.


    R. Musina, Ground state soluions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3927-3986.doi: 10.1016/j.na.2007.04.034.


    S. Peng and C. Wang, Infinitely many solutions for Hardy-Sobolev equation involving critical growth, Math. Meth. Appl. Sci., 38 (2015), 197-220.doi: 10.1002/mma.3060.


    P. H. Rabinowtz, Minimax Methods in Critical Points Theory with Applications to Differnetial Equations, CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.


    M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.doi: 10.1007/BF01174186.


    C. Wang and J. Wang, Infinitely many solutions for Hardy-Sobolev-Maz'ya equation involving critical growth, Commun. Contemp. Math., 14 (2012), 1250044, 38pp.doi: 10.1142/S0219199712500447.


    C. Wang and C. Xiang, Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry, arXiv:1407.7982v2, 2015.


    S. Yan and J. Yang, Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, Calc. Var. Partial Differential Equations, 48 (2013), 587-610.doi: 10.1007/s00526-012-0563-7.

  • 加载中

Article Metrics

HTML views() PDF downloads(103) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint