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September  2013, 12(5): 1943-1957. doi: 10.3934/cpaa.2013.12.1943

Elliptic equations with cylindrical potential and multiple critical exponents

Xiaomei Sun 1, and  Yimin Zhang 2,

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
2. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103

Received  November 2011 Revised  August 2012 Published  January 2013

In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.
Keywords: Variational method, cylindrical weight, multiple critical exponents..
Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 35J2.
Citation: Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943
References:
[1]

J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials,, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707.  doi: 10.1017/S0308210509001401.  Google Scholar

[2]

M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations,, J. Eur. Math. Soc., 9 (2007), 355.   Google Scholar

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M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities,, Adv. Diff. Equ., 12 (2007), 1321.  doi: 10.1007/s00009-005-0055-5.  Google Scholar

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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), 252.  doi: 10.1007/s002050200201.  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 E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.2307/2044999.  Google Scholar

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D. Castorina, I. 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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D. M. Cao and P. G. 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. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator,, Methods and Applications of Analysis, 15 (2008), 081.  doi: 10.1.1.140.417.  Google Scholar

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451.   Google Scholar

[12]

R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities,, J. Math. Pures Appl., 91 (2009), 156.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[13]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

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N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar

[15]

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

[16]

G. Mancini, I. Fabbri 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

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635.   Google Scholar

[18]

R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, Nonlinear Anal., 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality,, Commun. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar

[20]

J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials,, J. Differential Equa., 250 (2011), 223.  doi: 10.1016/j.jde.2010.08.025.  Google Scholar

[21]

J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, J. Differential Equa., 238 (2007), 201.  doi: 10.1016/j.jde.2007.03.018.  Google Scholar

[22]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[23]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar

[24]

A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality,, Ann. Mat. Pura e Appl., 186 (2007), 645.  doi: 10.1007/s10231-006-0024-z.  Google Scholar

[25]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. Henry Poincar\'e-Analyse Nonlin\'eaire, 12 (1995), 319.   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Third edition, (2000).   Google Scholar

show all references

References:
[1]

J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials,, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707.  doi: 10.1017/S0308210509001401.  Google Scholar

[2]

M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations,, J. Eur. Math. Soc., 9 (2007), 355.   Google Scholar

[3]

M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities,, Adv. Diff. Equ., 12 (2007), 1321.  doi: 10.1007/s00009-005-0055-5.  Google Scholar

[4]

M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials,, Nonlinear Analysis, 72 (2010), 602.  doi: 10.1016/j.na.2009.06.111.  Google Scholar

[5]

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), 252.  doi: 10.1007/s002050200201.  Google Scholar

[6]

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

[7]

H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.2307/2044999.  Google Scholar

[8]

D. Castorina, I. 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

[9]

D. M. Cao and P. G. 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

[10]

D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator,, Methods and Applications of Analysis, 15 (2008), 081.  doi: 10.1.1.140.417.  Google Scholar

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451.   Google Scholar

[12]

R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities,, J. Math. Pures Appl., 91 (2009), 156.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[13]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[14]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar

[15]

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

[16]

G. Mancini, I. Fabbri 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

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635.   Google Scholar

[18]

R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, Nonlinear Anal., 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality,, Commun. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar

[20]

J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials,, J. Differential Equa., 250 (2011), 223.  doi: 10.1016/j.jde.2010.08.025.  Google Scholar

[21]

J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, J. Differential Equa., 238 (2007), 201.  doi: 10.1016/j.jde.2007.03.018.  Google Scholar

[22]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[23]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar

[24]

A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality,, Ann. Mat. Pura e Appl., 186 (2007), 645.  doi: 10.1007/s10231-006-0024-z.  Google Scholar

[25]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. Henry Poincar\'e-Analyse Nonlin\'eaire, 12 (1995), 319.   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Third edition, (2000).   Google Scholar

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