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January  2013, 33(1): 321-333. doi: 10.3934/dcds.2013.33.321

Infinitely many solutions for some singular elliptic problems

Andrzej Szulkin 1, and  Shoyeb Waliullah 1,

1. 

Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden, Sweden

Received  June 2011 Revised  November 2011 Published  September 2012

We prove the existence of an unbounded sequence of critical points of the functional \begin{equation*} J_{\lambda}(u) =\frac{1} {p} ∫_{\mathbb{R} ^N}{||x|^{-α\nabla^k} u|} ^p - λ h(x){||x|^{-α+k}u|} ^p - \frac{1} {q} ∫_{\mathbb{R} ^N}Q(x){||x|^{-b}u|} ^q \end{equation*} related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume $Q\le 0$ at 0 and infinity and consider two essentially different cases: $h\equiv 1$ and $h$ in a certain weighted Lebesgue space.
Keywords: infinitely many solutions., Nehari manifold, sign-changing weight function, Caffarelli-Kohn-Nirenberg inequality.
Mathematics Subject Classification: Primary: 35J35, 35J62; Secondary: 35J30, 58E3.
Citation: Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321
References:
[1]

R. B. Assunção, P. C. Carrião and O. H. Miyagaki, Critical singular problems via concentration-compactness lemma, J. Math. Anal. Appl., 326 (2007), 137-154. doi: 10.1016/j.jmaa.2006.03.002.  Google Scholar

[2]

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlin. Anal., 20 (1993), 1205-1216. doi: 10.1016/0362-546X(93)90151-H.  Google Scholar

[3]

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. PDE, 30 (2007), 113-136. doi: 10.1007/s00526-006-0086-1.  Google Scholar

[4]

A. K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlin. Anal., 26 (1996), 823-833. doi: 10.1016/0362-546X(94)00324-B.  Google Scholar

[5]

G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlin. Anal., 25 (1995), 41-59. doi: 10.1016/0362-546X(94)E0070-W.  Google Scholar

[6]

A. Bonnet, A deformation lemma on a $C^1$-manifold, Manuscr. Math., 81 (1993), 339-359. doi: 10.1007/BF02567863.  Google Scholar

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  Google Scholar

[8]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Eq., 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[9]

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

[10]

F. Catrina and Z. Q. 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]

F. Catrina and Z. Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 157-178.  Google Scholar

[12]

J. Chabrowski and D. G. Costa, On existence of positive solutions for a class of Caffarelli-Kohn-Nirenberg type equations, Colloq. Math., 120 (2010), 43-62. doi: 10.4064/cm120-1-4.  Google Scholar

[13]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2), 48 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[14]

S. de Valeriola and M. Willem, On some quasilinear critical problems, Adv. Nonlin. Stud., 9 (2009), 825-836.  Google Scholar

[15]

G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portug. Math., 58 (2001), 339-378.  Google Scholar

[16]

J. Dolbeault, M. J. Esteban, M. Loss and G. Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlin. Stud., 9 (2009), 713-726.  Google Scholar

[17]

P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar

[18]

A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Diff. Int. Eq., 18 (2005), 1201-1220.  Google Scholar

[19]

C. S. Lin, Interpolation inequalities with weights, Comm. Partial Diff. Eq., 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.  Google Scholar

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'' Springer-Verlag, Berlin, 1977.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1:1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana, 1:2 (1985), 45-121. doi: 10.4171/RMI/12.  Google Scholar

[23]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Reg. Conf. Series Math., Amer. Math. Soc., Providence, R. I., 65 (1986).  Google Scholar

[24]

J. Simon, Régularité de la solution d'une équation non linéaire dans RN， Lecture Notes in Math., Springer-Verlag, 665 (1978), 205-227.  Google Scholar

[25]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.  Google Scholar

[26]

A. Szulkin, Ljusternik-Schnirelmann theory on c1-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.  Google Scholar

[27]

A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems,, Complex Variables and Elliptic Equations, ().   Google Scholar

[28]

J. Tan and J. Yang, On the singular variational problems, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), 672-690.  Google Scholar

[29]

S. Waliullah, Minimizers and symmetric minimizers for problems with critical Sobolev exponent, Topol. Meth. Nonlin. Anal., 34 (2009), 291-326.  Google Scholar

[30]

S. Waliullah, Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type, J. Math. Anal. Appl., 385 (2012), 721-736. doi: 10.1016/j.jmaa.2011.07.005.  Google Scholar

[31]

Z. Q. Wang and M. Willem, Singular minimization problems, J. Diff. Eq., 161 (2000), 307-320. doi: 10.1006/jdeq.1999.3699.  Google Scholar

[32]

M. Willem, "Minimax Theorems,'' Birkhäuser, Boston, 1996.  Google Scholar

show all references

References:
[1]

R. B. Assunção, P. C. Carrião and O. H. Miyagaki, Critical singular problems via concentration-compactness lemma, J. Math. Anal. Appl., 326 (2007), 137-154. doi: 10.1016/j.jmaa.2006.03.002.  Google Scholar

[2]

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlin. Anal., 20 (1993), 1205-1216. doi: 10.1016/0362-546X(93)90151-H.  Google Scholar

[3]

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. PDE, 30 (2007), 113-136. doi: 10.1007/s00526-006-0086-1.  Google Scholar

[4]

A. K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlin. Anal., 26 (1996), 823-833. doi: 10.1016/0362-546X(94)00324-B.  Google Scholar

[5]

G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlin. Anal., 25 (1995), 41-59. doi: 10.1016/0362-546X(94)E0070-W.  Google Scholar

[6]

A. Bonnet, A deformation lemma on a $C^1$-manifold, Manuscr. Math., 81 (1993), 339-359. doi: 10.1007/BF02567863.  Google Scholar

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  Google Scholar

[8]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Eq., 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[9]

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

[10]

F. Catrina and Z. Q. 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]

F. Catrina and Z. Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 157-178.  Google Scholar

[12]

J. Chabrowski and D. G. Costa, On existence of positive solutions for a class of Caffarelli-Kohn-Nirenberg type equations, Colloq. Math., 120 (2010), 43-62. doi: 10.4064/cm120-1-4.  Google Scholar

[13]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2), 48 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[14]

S. de Valeriola and M. Willem, On some quasilinear critical problems, Adv. Nonlin. Stud., 9 (2009), 825-836.  Google Scholar

[15]

G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portug. Math., 58 (2001), 339-378.  Google Scholar

[16]

J. Dolbeault, M. J. Esteban, M. Loss and G. Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlin. Stud., 9 (2009), 713-726.  Google Scholar

[17]

P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar

[18]

A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Diff. Int. Eq., 18 (2005), 1201-1220.  Google Scholar

[19]

C. S. Lin, Interpolation inequalities with weights, Comm. Partial Diff. Eq., 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.  Google Scholar

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'' Springer-Verlag, Berlin, 1977.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1:1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana, 1:2 (1985), 45-121. doi: 10.4171/RMI/12.  Google Scholar

[23]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Reg. Conf. Series Math., Amer. Math. Soc., Providence, R. I., 65 (1986).  Google Scholar

[24]

J. Simon, Régularité de la solution d'une équation non linéaire dans RN， Lecture Notes in Math., Springer-Verlag, 665 (1978), 205-227.  Google Scholar

[25]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.  Google Scholar

[26]

A. Szulkin, Ljusternik-Schnirelmann theory on c1-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.  Google Scholar

[27]

A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems,, Complex Variables and Elliptic Equations, ().   Google Scholar

[28]

J. Tan and J. Yang, On the singular variational problems, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), 672-690.  Google Scholar

[29]

S. Waliullah, Minimizers and symmetric minimizers for problems with critical Sobolev exponent, Topol. Meth. Nonlin. Anal., 34 (2009), 291-326.  Google Scholar

[30]

S. Waliullah, Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type, J. Math. Anal. Appl., 385 (2012), 721-736. doi: 10.1016/j.jmaa.2011.07.005.  Google Scholar

[31]

Z. Q. Wang and M. Willem, Singular minimization problems, J. Diff. Eq., 161 (2000), 307-320. doi: 10.1006/jdeq.1999.3699.  Google Scholar

[32]

M. Willem, "Minimax Theorems,'' Birkhäuser, Boston, 1996.  Google Scholar

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