Nonlinear Biharmonic Problems with Singular Potentials

Previous Article
Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators
CPAA Home
This Issue
Next Article

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

Abstract
Related Papers

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 and 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.
[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.

[1]	Lili Yan. Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022034
[2]	Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677
[3]	Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108
[4]	Thomas I. Vogel. Comments on radially symmetric liquid bridges with inflected profiles. Conference Publications, 2005, 2005 (Special) : 862-867. doi: 10.3934/proc.2005.2005.862
[5]	Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064
[6]	Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems and Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008
[7]	Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[8]	Huijiang Zhao, Qingsong Zhao. Radially symmetric stationary wave for two-dimensional Burgers equation. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2167-2185. doi: 10.3934/dcds.2020357
[9]	Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297
[10]	Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989
[11]	István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003
[12]	Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure and Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[13]	Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709
[14]	Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations and Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008
[15]	Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583
[16]	Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[17]	Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[18]	Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026
[19]	Jaakko Kultima, Valery Serov. Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022011
[20]	Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Paulo Cesar Carrião R. Demarque Olímpio H. Miyagaki
on Google Scholar
Paulo Cesar Carrião R. Demarque Olímpio H. Miyagaki

[Back to Top]