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.

[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.

[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).

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

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.

[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.

[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).

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

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