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

Infinitely many solutions for some singular elliptic problems

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.
Citation: Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1016/j.jmaa.2006.03.002.

[2]

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

[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. doi: 10.1007/s00526-006-0086-1.

[4]

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

[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. doi: 10.1016/0362-546X(94)E0070-W.

[6]

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

[7]

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

[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. doi: 10.1016/S0022-0396(03)00121-9.

[9]

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

[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. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[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.

[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. doi: 10.4064/cm120-1-4.

[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. doi: 10.1112/jlms/s2-48.1.137.

[14]

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

[15]

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

[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.

[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. doi: 10.1017/S0308210500023787.

[18]

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

[19]

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

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'', Springer-Verlag, (1977).

[21]

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

[22]

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

[23]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Reg. Conf. Series Math., 65 (1986).

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

M. Willem, "Minimax Theorems,'', Birkhäuser, (1996).

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. doi: 10.1016/j.jmaa.2006.03.002.

[2]

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

[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. doi: 10.1007/s00526-006-0086-1.

[4]

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

[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. doi: 10.1016/0362-546X(94)E0070-W.

[6]

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

[7]

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

[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. doi: 10.1016/S0022-0396(03)00121-9.

[9]

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

[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. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[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.

[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. doi: 10.4064/cm120-1-4.

[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. doi: 10.1112/jlms/s2-48.1.137.

[14]

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

[15]

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

[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.

[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. doi: 10.1017/S0308210500023787.

[18]

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

[19]

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

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'', Springer-Verlag, (1977).

[21]

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

[22]

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

[23]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Reg. Conf. Series Math., 65 (1986).

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

M. Willem, "Minimax Theorems,'', Birkhäuser, (1996).

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