March  2016, 36(3): 1603-1628. doi: 10.3934/dcds.2016.36.1603

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

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received  November 2014 Revised  April 2015 Published  August 2015

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.
Citation: Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
References:
[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. doi: 10.1016/j.na.2011.04.057. Google Scholar

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

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

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H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

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

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

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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. doi: 10.1016/S0022-0396(03)00118-9. Google Scholar

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

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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. doi: 10.1007/s00526-009-0295-5. Google Scholar

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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. Google Scholar

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

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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. doi: 10.1016/0022-1236(86)90094-7. Google Scholar

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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. doi: 10.1007/s00205-009-0269-y. Google Scholar

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

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

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

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

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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. doi: 10.1090/S0002-9947-00-02560-5. Google Scholar

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

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

[24]

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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

P. H. Rabinowtz, Minimax Methods in Critical Points Theory with Applications to Differnetial Equations,, CBMS Regional Conference Series in Mathematics, (1986). Google Scholar

[30]

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

[31]

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

[32]

C. Wang and C. Xiang, Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry,, , (2015). Google Scholar

[33]

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. doi: 10.1007/s00526-012-0563-7. Google Scholar

show all references

References:
[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. doi: 10.1016/j.na.2011.04.057. Google Scholar

[2]

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

[3]

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. doi: 10.1002/cpa.3160410302. Google Scholar

[4]

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

[5]

H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives,, in Directions in Partial Differential Equations (Madison, (1985), 17. Google Scholar

[6]

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

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[8]

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

[9]

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

[10]

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. doi: 10.1016/S0022-0396(03)00118-9. Google Scholar

[11]

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

[12]

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. doi: 10.1007/s00526-009-0295-5. Google Scholar

[13]

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. Google Scholar

[14]

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

[15]

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. doi: 10.1016/0022-1236(86)90094-7. Google Scholar

[16]

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. doi: 10.1007/s00205-009-0269-y. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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. doi: 10.1090/S0002-9947-00-02560-5. Google Scholar

[22]

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

[23]

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

[24]

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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

P. H. Rabinowtz, Minimax Methods in Critical Points Theory with Applications to Differnetial Equations,, CBMS Regional Conference Series in Mathematics, (1986). Google Scholar

[30]

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

[31]

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

[32]

C. Wang and C. Xiang, Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry,, , (2015). Google Scholar

[33]

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. doi: 10.1007/s00526-012-0563-7. Google Scholar

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