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November  2014, 13(6): 2141-2154. doi: 10.3934/cpaa.2014.13.2141

Nonlinear Biharmonic Problems with Singular Potentials

Paulo Cesar Carrião 1, , R. Demarque 2, and  Olímpio H. Miyagaki 3,

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

Received  November 2011 Revised  May 2014 Published  July 2014

We deal with the problem \begin{eqnarray} \Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N) \end{eqnarray} where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.
Keywords: critical potential, Biharmonic operator, radially symmetric solution..
Mathematics Subject Classification: Primary: 35J30, 35J35; Secondary: 35J70, 35J9.
Citation: Paulo Cesar Carrião, R. Demarque, Olímpio H. Miyagaki. Nonlinear Biharmonic Problems with Singular Potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2141-2154. doi: 10.3934/cpaa.2014.13.2141
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Chabrowski and J. M. do Ó, On some fourth order semilinear elliptic problems in $\R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[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.  Google Scholar

[8]

E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $\R^N$, Funkcialaj Ekvacioj, 35 (1992), 533-543.  Google Scholar

[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.  Google Scholar

[10]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-172.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Willem, Minimax theorem, PNLDE 24, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Chabrowski and J. M. do Ó, On some fourth order semilinear elliptic problems in $\R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[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.  Google Scholar

[8]

E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $\R^N$, Funkcialaj Ekvacioj, 35 (1992), 533-543.  Google Scholar

[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.  Google Scholar

[10]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-172.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Willem, Minimax theorem, PNLDE 24, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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