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

[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.

[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.

[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.

[5]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

Y. Li and C. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968.

[25]

Y. Li and W. Ni, On conformal scalar curvature equations in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7.

[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.

[27]

C. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations, 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

Y. Li and C. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968.

[25]

Y. Li and W. Ni, On conformal scalar curvature equations in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7.

[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.

[27]

C. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations, 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.

[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.

[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.

[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.

[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.

[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.

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