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Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation
Infinitely many solutions for some singular elliptic problems
1. | Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden, Sweden |
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, (). Google Scholar |
[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, (). Google Scholar |
[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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