American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Bifurcation results on positive solutions of an indefinite nonlinear elliptic system
  • DCDS Home
  • This Issue
  • Next Article
    Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation
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 - 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.  Google Scholar

[2]

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem,, Nonlin. Anal., 20 (1993), 1205.  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.  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.  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.  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.  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.   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.  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.   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.  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.   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.  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.  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.   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.   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.   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.  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.   Google Scholar

[19]

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

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'', Springer-Verlag, (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.  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.  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., 65 (1986).   Google Scholar

[24]

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

[25]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227.  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.   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.   Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[32]

M. Willem, "Minimax Theorems,'', Birkhäuser, (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.  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.  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.  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.  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.  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.  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.   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.  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.   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.  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.   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.  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.  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.   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.   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.   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.  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.   Google Scholar

[19]

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

[20]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces. I,'', Springer-Verlag, (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.  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.  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., 65 (1986).   Google Scholar

[24]

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

[25]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227.  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.   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.   Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[32]

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

[1]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
[2]

Jijiang Sun, Shiwang Ma. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2317-2330. doi: 10.3934/cpaa.2014.13.2317
[3]

Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
[4]

Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
[5]

B. Abdellaoui, I. Peral. On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2003, 2 (4) : 539-566. doi: 10.3934/cpaa.2003.2.539
[6]

Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629
[7]

Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024
[8]

Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[9]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002
[10]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
[11]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057
[12]

Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013
[13]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[14]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[15]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[16]

Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
[17]

J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427
[18]

Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268
[19]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[20]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences