• 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

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.  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]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[2]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010

[3]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[4]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[5]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

[6]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[7]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124

[8]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[9]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

[10]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[11]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[12]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[13]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[14]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021004

[15]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[16]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[17]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049

[18]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[19]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

[20]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]