American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 400-408. doi: 10.3934/proc.2015.0400

Existence and uniqueness of positive solutions for singular biharmonic elliptic systems

Luiz F. O. Faria 1,

1. 

Departamento de Matemática - ICE, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330, Minas Gerais, Brazil

Received  September 2014 Revised  January 2015 Published  November 2015

In this paper we prove existence and uniqueness of positive solutions of nonlinear singular biharmonic elliptic system in smooth bounded domains, with coupling of the equations, under Navier boundary condition. The solution is constructed through an approximating process based on a priori estimates, regularity up to the boundary and Hardy-Sobolev inequality.
Keywords: Biharmonic system, negative exponents, Hardy-Sobolev inequality., Navier boundary condition, approximation method.
Mathematics Subject Classification: 35B09, 35B45, 35J58, 35J7.
Citation: Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
References:
[1]

S. Agmon, The $L_p$ approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 405-448.  Google Scholar

[2]

C. O. Alves and F. J. S. A. Corrêa., On the existence of positive solutions for a class of singular systems involving quasilinear operators, Appl. Math. and Computation, 185 (2007), 727-736.  Google Scholar

[3]

C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods, Function spaces, differential operators and nonlinear analysis (Teistungen, 2001), 47-57, Birkhaüser, Basel, (2003).  Google Scholar

[4]

A. Alvino, V. Ferone and G. Trombetti, On the best constant in a HardySobolev inequality, Appl. Anal., 85 (2006), 171-180.  Google Scholar

[5]

E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differential Equations (2005), 20 pp.  Google Scholar

[6]

F. 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

[7]

J. F. Bonder and J. D. Rossi, A fourth order elliptic equation with nonlinear boundary conditions, Nonlinear Anal., 49 (2002), 1037-1047.  Google Scholar

[8]

P. C. Carrião, L. F. O. Faria and O.H. Miyagaki, A biharmonic elliptic problem with dependence on the gradient and the Laplacian, Electron. J. Differential Equations (2009), 12 pp.  Google Scholar

[9]

S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York, 1957.  Google Scholar

[10]

Q- Choi and T. Jung, Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation, Acta Math. Sci., 19 (1999), 361-374.  Google Scholar

[11]

Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234.  Google Scholar

[12]

D. G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems, Resenhas 2 (1996), 373-391.  Google Scholar

[13]

Y. Deng and G. Wang, On inhomogeneous biharmonic equations involving critical exponents, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 925-946.  Google Scholar

[14]

W. Fulks and J.S. Maybee, A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19.  Google Scholar

[15]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  Google Scholar

[16]

Y. G. Gu, Y. B. Deng and X. J. Wang, Existence of nontrivial solutions for critical semilinear biharmonic equations, Systems Sci. Math. Sci., 7 (1994), 140-152.  Google Scholar

[17]

G. L. Hernandez and Y. Choi, Existence of solutions in a singular biharmonic nonlinear problem, Proc. Edinburgh Math. Soc. (2), 36 (1993), 537-546.  Google Scholar

[18]

T. Jung and Q- Choi, Existence of nontrivial solutions of the nonlinear biharmonic system, Korean J. Math., 16 (2008), 135-143. Google Scholar

[19]

O. Kavian, Inegalité de Hardy-Sobolev et applications, Theése de Doctorate de 3eme cycle, Université de Paris, VI (1978). Google Scholar

[20]

S. Kesavan, Topics in functional analysis and application, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[21]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  Google Scholar

[22]

T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15.  Google Scholar

[23]

A. M. Micheletti, A. Pistoia and C. Saccon, Three solutions of a fourth order elliptic problem via variational theorems of mixed type, Appl. Anal., 75 (2000), 43-59.  Google Scholar

[24]

R. C. A. M. Van der Vorst, Fourth-order elliptic equations with critical growth, C. R. Acad. Sci. Paris Sr. I Math., 320 (1995), 295-299.  Google Scholar

[25]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2(\Omega) \cap H_0^1(\Omega)$ into $L^{2N/N-4}(\Omega)$, Differential Integral Equations, 6 (1993), 259-276.  Google Scholar

[26]

P. Villaggio, Mathematical models for elastic structures, Cambridge University Press, Cambridge, 1997.  Google Scholar

[27]

W. Wang, A. Zang and P. Zhao, Multiplicity of solutions for a class of fourth elliptic equations, Nonlinear Anal., 70 (2009), 4377-4385.  Google Scholar

[28]

X. Z. Zeng and Y. B. Deng, Existence of multiple solutions for a semilinear biharmonic equation with critical exponent, Acta Math. Sci. Ser. A Chin. Ed., 20 (2000), 547-554.  Google Scholar

[29]

J. H. Zhang and S. J. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230.  Google Scholar

show all references

References:
[1]

S. Agmon, The $L_p$ approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 405-448.  Google Scholar

[2]

C. O. Alves and F. J. S. A. Corrêa., On the existence of positive solutions for a class of singular systems involving quasilinear operators, Appl. Math. and Computation, 185 (2007), 727-736.  Google Scholar

[3]

C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods, Function spaces, differential operators and nonlinear analysis (Teistungen, 2001), 47-57, Birkhaüser, Basel, (2003).  Google Scholar

[4]

A. Alvino, V. Ferone and G. Trombetti, On the best constant in a HardySobolev inequality, Appl. Anal., 85 (2006), 171-180.  Google Scholar

[5]

E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differential Equations (2005), 20 pp.  Google Scholar

[6]

F. 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

[7]

J. F. Bonder and J. D. Rossi, A fourth order elliptic equation with nonlinear boundary conditions, Nonlinear Anal., 49 (2002), 1037-1047.  Google Scholar

[8]

P. C. Carrião, L. F. O. Faria and O.H. Miyagaki, A biharmonic elliptic problem with dependence on the gradient and the Laplacian, Electron. J. Differential Equations (2009), 12 pp.  Google Scholar

[9]

S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York, 1957.  Google Scholar

[10]

Q- Choi and T. Jung, Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation, Acta Math. Sci., 19 (1999), 361-374.  Google Scholar

[11]

Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234.  Google Scholar

[12]

D. G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems, Resenhas 2 (1996), 373-391.  Google Scholar

[13]

Y. Deng and G. Wang, On inhomogeneous biharmonic equations involving critical exponents, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 925-946.  Google Scholar

[14]

W. Fulks and J.S. Maybee, A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19.  Google Scholar

[15]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  Google Scholar

[16]

Y. G. Gu, Y. B. Deng and X. J. Wang, Existence of nontrivial solutions for critical semilinear biharmonic equations, Systems Sci. Math. Sci., 7 (1994), 140-152.  Google Scholar

[17]

G. L. Hernandez and Y. Choi, Existence of solutions in a singular biharmonic nonlinear problem, Proc. Edinburgh Math. Soc. (2), 36 (1993), 537-546.  Google Scholar

[18]

T. Jung and Q- Choi, Existence of nontrivial solutions of the nonlinear biharmonic system, Korean J. Math., 16 (2008), 135-143. Google Scholar

[19]

O. Kavian, Inegalité de Hardy-Sobolev et applications, Theése de Doctorate de 3eme cycle, Université de Paris, VI (1978). Google Scholar

[20]

S. Kesavan, Topics in functional analysis and application, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[21]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  Google Scholar

[22]

T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15.  Google Scholar

[23]

A. M. Micheletti, A. Pistoia and C. Saccon, Three solutions of a fourth order elliptic problem via variational theorems of mixed type, Appl. Anal., 75 (2000), 43-59.  Google Scholar

[24]

R. C. A. M. Van der Vorst, Fourth-order elliptic equations with critical growth, C. R. Acad. Sci. Paris Sr. I Math., 320 (1995), 295-299.  Google Scholar

[25]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2(\Omega) \cap H_0^1(\Omega)$ into $L^{2N/N-4}(\Omega)$, Differential Integral Equations, 6 (1993), 259-276.  Google Scholar

[26]

P. Villaggio, Mathematical models for elastic structures, Cambridge University Press, Cambridge, 1997.  Google Scholar

[27]

W. Wang, A. Zang and P. Zhao, Multiplicity of solutions for a class of fourth elliptic equations, Nonlinear Anal., 70 (2009), 4377-4385.  Google Scholar

[28]

X. Z. Zeng and Y. B. Deng, Existence of multiple solutions for a semilinear biharmonic equation with critical exponent, Acta Math. Sci. Ser. A Chin. Ed., 20 (2000), 547-554.  Google Scholar

[29]

J. H. Zhang and S. J. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230.  Google Scholar

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