Infinitely many solutions for some singular elliptic problems

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	A. Bonnet, A deformation lemma on a $C^1$-manifold, Manuscr. Math., 81 (1993), 339-359. 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), 486-490.
[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.
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
[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.
[11]	F. Catrina and Z. Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in R^N, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 157-178.
[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.
[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.
[14]	S. de Valeriola and M. Willem, On some quasilinear critical problems, Adv. Nonlin. Stud., 9 (2009), 825-836.
[15]	G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portug. Math., 58 (2001), 339-378.
[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.
[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.
[18]	A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Diff. Int. Eq., 18 (2005), 1201-1220.
[19]	C. S. Lin, Interpolation inequalities with weights, Comm. Partial Diff. Eq., 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.
[20]	J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'' Springer-Verlag, Berlin, 1977.
[21]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1:1 (1985), 145-201. 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), 45-121. 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., Amer. Math. Soc., Providence, R. I., 65 (1986).
[24]	J. Simon, Régularité de la solution d'une équation non linéaire dans R^N， Lecture Notes in Math., Springer-Verlag, 665 (1978), 205-227.
[25]	C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.
[26]	A. Szulkin, Ljusternik-Schnirelmann theory on c¹-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.
[27]	A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems, Complex Variables and Elliptic Equations, iFirst.
[28]	J. Tan and J. Yang, On the singular variational problems, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), 672-690.
[29]	S. Waliullah, Minimizers and symmetric minimizers for problems with critical Sobolev exponent, Topol. Meth. Nonlin. Anal., 34 (2009), 291-326.
[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.
[31]	Z. Q. Wang and M. Willem, Singular minimization problems, J. Diff. Eq., 161 (2000), 307-320. doi: 10.1006/jdeq.1999.3699.
[32]	M. Willem, "Minimax Theorems,'' Birkhäuser, Boston, 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), 137-154. doi: 10.1016/j.jmaa.2006.03.002.
[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.
[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.
[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.
[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.
[6]	A. Bonnet, A deformation lemma on a $C^1$-manifold, Manuscr. Math., 81 (1993), 339-359. 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), 486-490.
[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.
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
[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.
[11]	F. Catrina and Z. Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in R^N, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 157-178.
[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.
[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.
[14]	S. de Valeriola and M. Willem, On some quasilinear critical problems, Adv. Nonlin. Stud., 9 (2009), 825-836.
[15]	G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portug. Math., 58 (2001), 339-378.
[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.
[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.
[18]	A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Diff. Int. Eq., 18 (2005), 1201-1220.
[19]	C. S. Lin, Interpolation inequalities with weights, Comm. Partial Diff. Eq., 11 (1986), 1515-1538. doi: 10.1080/03605308608820473.
[20]	J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'' Springer-Verlag, Berlin, 1977.
[21]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1:1 (1985), 145-201. 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), 45-121. 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., Amer. Math. Soc., Providence, R. I., 65 (1986).
[24]	J. Simon, Régularité de la solution d'une équation non linéaire dans R^N， Lecture Notes in Math., Springer-Verlag, 665 (1978), 205-227.
[25]	C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.
[26]	A. Szulkin, Ljusternik-Schnirelmann theory on c¹-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.
[27]	A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems, Complex Variables and Elliptic Equations, iFirst.
[28]	J. Tan and J. Yang, On the singular variational problems, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), 672-690.
[29]	S. Waliullah, Minimizers and symmetric minimizers for problems with critical Sobolev exponent, Topol. Meth. Nonlin. Anal., 34 (2009), 291-326.
[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.
[31]	Z. Q. Wang and M. Willem, Singular minimization problems, J. Diff. Eq., 161 (2000), 307-320. doi: 10.1006/jdeq.1999.3699.
[32]	M. Willem, "Minimax Theorems,'' Birkhäuser, Boston, 1996.

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