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March  2015, 14(2): 439-455. doi: 10.3934/cpaa.2015.14.439

## Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents

 1 Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China 2 Department of Mathematics, Huazhong Normal University,Wuhan, 430079, China

Received  December 2013 Revised  September 2014 Published  December 2014

We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.
Citation: Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
##### References:
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Soc.}, 132 (2004), 1685.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar [29] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2,, \emph{Rev. Mat. Iberoamericana, 1 (1985), 145.   Google Scholar [30] R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, \emph{Nonlinear Anal.}, 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar [31] S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, \emph{Ann. Inst. H. Poinvar\'e Anal. Non Lin\'eaire}, 12 (1995), 319.   Google Scholar [32] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511.  doi: 10.1007/BF01174186.  Google Scholar

show all references

##### References:
 [1] T. Bartsch, S. Peng and Z. Zhang, Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities,, \emph{Calc. Var. Partial Differential Equations}, 30 (2007), 113.  doi: 10.1007/s00526-006-0086-1.  Google Scholar [2] M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics,, \emph{Arch. Ration. Mech. Anal.}, 163 (2002), 259.  doi: 10.1007/s002050200201.  Google Scholar [3] J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics,, \emph{Arch. Rational Mech. Anal.}, 93 (1986), 159.  doi: 10.1007/BF00279958.  Google Scholar [4] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259.   Google Scholar [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential,, \emph{J. Differential Equations}, 205 (2004), 521.  doi: 10.1016/j.jde.2004.03.005.  Google Scholar [7] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, \emph{J. Differential Equations}, 224 (2006), 332.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar [8] D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms,, \emph{J. Differential Equations}, 193 (2003), 424.  doi: 10.1016/S0022-0396(03)00118-9.  Google Scholar [9] D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential,, \emph{Calc. Var. Partial Differential Equations}, 38 (2010), 471.  doi: 10.1007/s00526-009-0295-5.  Google Scholar [10] F. Catrina and Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar [11] Z. Cheng and W. Zou, On an elliptic problem with critical exponent and Hardy potential,, \emph{J. Differential Equations}, 252 (2012), 969.  doi: 10.1016/j.jde.2011.09.042.  Google Scholar [12] J. Chern and C. Lin, Minimizer of Caffarelli-Kohn-Nirenberg inequlities with the singularity on the boundary,, \emph{Arch. Ration. Mech. Anal.}, 197 (2010), 401.  doi: 10.1007/s00205-009-0269-y.  Google Scholar [13] K. Chou and C. Chu, On the best constant for a weighted Sobolev-Hardy inequality,, \emph{J. London Math. Soc.}, 48 (1993), 137.  doi: 10.1112/jlms/s2-48.1.137.  Google Scholar [14] H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities,, \emph{Indiana Univ. Math. J.}, 38 (1989), 235.   Google Scholar [15] I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large,, \emph{Bull. Amer. Math. Soc.}, 39 (2002), 207.  doi: 10.1090/S0273-0979-02-00929-1.  Google Scholar [16] A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations,, \emph{J. Differential Equations}, 177 (2001), 494.  doi: 10.1006/jdeq.2000.3999.  Google Scholar [17] D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains,, \emph{Proc. Roy. Soc. Edinburgh Sect.}, 105 (1987), 205.  doi: 10.1017/S0308210500022046.  Google Scholar [18] N. Ghoussoub and X. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 21 (2004), 767.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar [19] N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities,, \emph{Geom. Funct. Anal.}, 16 (2006), 1201.  doi: 10.1007/s00039-006-0579-2.  Google Scholar [20] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 5703.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar [21] C. H. Hsia, C. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg,, \emph{J. Funct. Anal.}, 259 (2010), 1816.  doi: 10.1016/j.jfa.2010.05.004.  Google Scholar [22] D. Kang and S. Peng, Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents,, \emph{J. Math. Anal. Appl.}, 291 (2004), 488.  doi: 10.1016/j.jmaa.2003.11.012.  Google Scholar [23] Y. Li, On the positive solutions of the Matukuma equation,, \emph{Duke Math. J.}, 70 (1993), 575.  doi: 10.1215/S0012-7094-93-07012-3.  Google Scholar [24] Y. Li and C. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents,, \emph{Arch. Ration. Mech. Anal.}, 203 (2012), 943.   Google Scholar [25] Y. Li and W. Ni, On conformal scalar curvature equations in $\R^N$,, \emph{Duke Math. J.}, 57 (1988), 895.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar [26] Y. Li and W. Ni, On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalization,, \emph{Arch. Rational Mech. Anal.}, 108 (1989), 175.  doi: 10.1007/BF01053462.  Google Scholar [27] C. Lin, Interpolation inequalities with weights,, \emph{Comm. Partial Differential Equations, 11 (1986), 1515.  doi: 10.1080/03605308608820473.  Google Scholar [28] C. Lin and Z. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities,, \emph{Proc. Amer. Math. Soc.}, 132 (2004), 1685.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar [29] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2,, \emph{Rev. Mat. Iberoamericana, 1 (1985), 145.   Google Scholar [30] R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, \emph{Nonlinear Anal.}, 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar [31] S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, \emph{Ann. Inst. H. Poinvar\'e Anal. Non Lin\'eaire}, 12 (1995), 319.   Google Scholar [32] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511.  doi: 10.1007/BF01174186.  Google Scholar
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