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Nonlinear Biharmonic Problems with Singular Potentials
1. | Departamento de Matematica, Universidade Federal de Minas Gerais, 31270-010 Belo Horizonte-MG, Brazil |
2. | Departamento de Física e Matemática, Universidade Federal Fluminense, Campus de Rio das Ostras, Rio das Ostras, RJ 28890-000, Brazil |
3. | Departmento de Matemática, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG |
References:
[1] |
C. O. Alves and J. M. do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458. |
[2] |
C. O. Alves, J. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal., 46 (2001), 121-133.
doi: 10.1016/S0362-546X(99)00449-6. |
[3] |
C. O. Alves, J. M. do Ó and O. H. Miyagaki, On a class singular biharmonic problems involving critical exponent, J. Math. Anal. Appl., 277 (2003), 12-26.
doi: 10.1016/S0022-247X(02)00283-4. |
[4] |
M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Rend. Lincei Mat. Appl., 17 (2006), 1-13.
doi: 10.4171/RLM/450. |
[5] |
Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240. |
[6] |
J. Chabrowski and J. M. do Ó, On some fourth order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal., 49 (2002), 861-884.
doi: 10.1016/S0362-546X(01)00144-4. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer, Berlin-Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $\mathbb{R}^{N}$, Funkcialaj Ekvacioj, 35 (1992), 533-543. |
[9] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, AMS, Providence, 1986. |
[10] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-172. |
[11] |
J. Su and R. Tian, Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Comm. on Pure and Appl. Anal., 9 (2010), 885-904.
doi: 10.3934/cpaa.2010.9.885. |
[12] |
J. Su, Z-Q. Wang and M. Willem, Nonlinear Schödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583.
doi: 10.1142/S021919970700254X. |
[13] |
J. Su, Z-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[14] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008. |
[15] |
Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125.
doi: 10.1016/j.jde.2009.02.016. |
[16] |
Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl., 355 (2009), 649-660.
doi: 10.1016/j.jmaa.2009.01.076. |
[17] |
M. Willem, Minimax theorem, PNLDE 24, Birkhauser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[18] |
H. Xiong and Y. T. Shen, Nonlinear biharmonic equations with critical potential, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1285-1294.
doi: 10.1007/s10114-004-0502-4. |
show all references
References:
[1] |
C. O. Alves and J. M. do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458. |
[2] |
C. O. Alves, J. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal., 46 (2001), 121-133.
doi: 10.1016/S0362-546X(99)00449-6. |
[3] |
C. O. Alves, J. M. do Ó and O. H. Miyagaki, On a class singular biharmonic problems involving critical exponent, J. Math. Anal. Appl., 277 (2003), 12-26.
doi: 10.1016/S0022-247X(02)00283-4. |
[4] |
M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Rend. Lincei Mat. Appl., 17 (2006), 1-13.
doi: 10.4171/RLM/450. |
[5] |
Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240. |
[6] |
J. Chabrowski and J. M. do Ó, On some fourth order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal., 49 (2002), 861-884.
doi: 10.1016/S0362-546X(01)00144-4. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer, Berlin-Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $\mathbb{R}^{N}$, Funkcialaj Ekvacioj, 35 (1992), 533-543. |
[9] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, AMS, Providence, 1986. |
[10] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-172. |
[11] |
J. Su and R. Tian, Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Comm. on Pure and Appl. Anal., 9 (2010), 885-904.
doi: 10.3934/cpaa.2010.9.885. |
[12] |
J. Su, Z-Q. Wang and M. Willem, Nonlinear Schödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583.
doi: 10.1142/S021919970700254X. |
[13] |
J. Su, Z-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[14] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008. |
[15] |
Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125.
doi: 10.1016/j.jde.2009.02.016. |
[16] |
Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl., 355 (2009), 649-660.
doi: 10.1016/j.jmaa.2009.01.076. |
[17] |
M. Willem, Minimax theorem, PNLDE 24, Birkhauser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[18] |
H. Xiong and Y. T. Shen, Nonlinear biharmonic equations with critical potential, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1285-1294.
doi: 10.1007/s10114-004-0502-4. |
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