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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), 1685-1691. 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, Rev. Mat. Iberoamericana, 1 (1985), 145-201 and 2 (1985), 45-121.  Google Scholar [30] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986. 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, Ann. Inst. H. Poinvaré Anal. Non Linéaire, 12 (1995), 319-337.  Google Scholar [32] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517 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, Calc. Var. Partial Differential Equations, 30 (2007), 113-136. 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, Arch. Ration. Mech. Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201.  Google Scholar [3] J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183. doi: 10.1007/BF00279958.  Google Scholar [4] 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.  Google Scholar [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. 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, J. Differential Equations, 224 (2006), 332-372. 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, J. Differential Equations, 193 (2003), 424-434. 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, Calc. Var. Partial Differential Equations, 38 (2010), 471-501. 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, Comm. Pure Appl. Math., 54 (2001), 229-258. 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, J. Differential Equations, 252 (2012), 969-987. 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, Arch. Ration. Mech. Anal., 197 (2010), 401-432. 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, J. London Math. Soc., 48 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137.  Google Scholar [14] H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251. Google Scholar [15] I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc., 39 (2002), 207-265. 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, J. Differential Equations, 177 (2001), 494-522. 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, Proc. Roy. Soc. Edinburgh Sect., 105 (1987), 205-213. doi: 10.1017/S0308210500022046.  Google Scholar [18] 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.  Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] C. H. Hsia, C. 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.  Google Scholar [22] D. Kang and S. Peng, Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents, J. Math. Anal. Appl., 291 (2004), 488-499. doi: 10.1016/j.jmaa.2003.11.012.  Google Scholar [23] Y. Li, On the positive solutions of the Matukuma equation, Duke Math. J., 70 (1993), 575-589. 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, Arch. Ration. Mech. Anal., 203 (2012), 943-968.  Google Scholar [25] Y. Li and W. Ni, On conformal scalar curvature equations in $\R^N$, Duke Math. J., 57 (1988), 895-924. 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, Arch. Rational Mech. Anal., 108 (1989), 175-194. doi: 10.1007/BF01053462.  Google Scholar [27] C. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations, 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.  Google Scholar [28] C. Lin and Z. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132 (2004), 1685-1691. 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, Rev. Mat. Iberoamericana, 1 (1985), 145-201 and 2 (1985), 45-121.  Google Scholar [30] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986. 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, Ann. Inst. H. Poinvaré Anal. Non Linéaire, 12 (1995), 319-337.  Google Scholar [32] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517 doi: 10.1007/BF01174186.  Google Scholar
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