# American Institute of Mathematical Sciences

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

## Existence and uniqueness of positive solutions for singular biharmonic elliptic systems

 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.
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.   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.   Google Scholar [3] C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods,, Function spaces, (2003), 47.   Google Scholar [4] A. Alvino, V. Ferone and G. Trombetti, On the best constant in a HardySobolev inequality,, Appl. Anal., 85 (2006), 171.   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), (2005).   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.   Google Scholar [7] J. F. Bonder and J. D. Rossi, A fourth order elliptic equation with nonlinear boundary conditions,, Nonlinear Anal., 49 (2002), 1037.   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), (2009).   Google Scholar [9] S. Chandrasekhar, An introduction to the study of stellar structure,, Dover Publications, (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.   Google Scholar [11] Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents,, J. Differential Equations, 246 (2009), 216.   Google Scholar [12] D. G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems,, Resenhas 2 (1996), 2 (1996), 373.   Google Scholar [13] Y. Deng and G. Wang, On inhomogeneous biharmonic equations involving critical exponents,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 925.   Google Scholar [14] W. Fulks and J.S. Maybee, A singular non-linear equation,, Osaka Math. J., 12 (1960), 1.   Google Scholar [15] M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295.   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.   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.   Google Scholar [18] T. Jung and Q- Choi, Existence of nontrivial solutions of the nonlinear biharmonic system,, Korean J. Math., 16 (2008), 135.   Google Scholar [19] O. Kavian, Inegalité de Hardy-Sobolev et applications, Theése de Doctorate de 3eme cycle,, Université de Paris, (1978).   Google Scholar [20] S. Kesavan, Topics in functional analysis and application,, John Wiley & Sons, (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.   Google Scholar [22] T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings,, Appl. Math. Lett., 13 (2000), 11.   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.   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.   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.   Google Scholar [26] P. Villaggio, Mathematical models for elastic structures,, Cambridge University Press, (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.   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.   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.   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.   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.   Google Scholar [3] C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods,, Function spaces, (2003), 47.   Google Scholar [4] A. Alvino, V. Ferone and G. Trombetti, On the best constant in a HardySobolev inequality,, Appl. Anal., 85 (2006), 171.   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), (2005).   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.   Google Scholar [7] J. F. Bonder and J. D. Rossi, A fourth order elliptic equation with nonlinear boundary conditions,, Nonlinear Anal., 49 (2002), 1037.   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), (2009).   Google Scholar [9] S. Chandrasekhar, An introduction to the study of stellar structure,, Dover Publications, (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.   Google Scholar [11] Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents,, J. Differential Equations, 246 (2009), 216.   Google Scholar [12] D. G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems,, Resenhas 2 (1996), 2 (1996), 373.   Google Scholar [13] Y. Deng and G. Wang, On inhomogeneous biharmonic equations involving critical exponents,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 925.   Google Scholar [14] W. Fulks and J.S. Maybee, A singular non-linear equation,, Osaka Math. J., 12 (1960), 1.   Google Scholar [15] M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295.   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.   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.   Google Scholar [18] T. Jung and Q- Choi, Existence of nontrivial solutions of the nonlinear biharmonic system,, Korean J. Math., 16 (2008), 135.   Google Scholar [19] O. Kavian, Inegalité de Hardy-Sobolev et applications, Theése de Doctorate de 3eme cycle,, Université de Paris, (1978).   Google Scholar [20] S. Kesavan, Topics in functional analysis and application,, John Wiley & Sons, (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.   Google Scholar [22] T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings,, Appl. Math. Lett., 13 (2000), 11.   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.   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.   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.   Google Scholar [26] P. Villaggio, Mathematical models for elastic structures,, Cambridge University Press, (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.   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.   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.   Google Scholar
 [1] José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138 [2] Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191 [3] Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123 [4] Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061 [5] Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016 [6] Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 [7] Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018 [8] Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951 [9] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [10] Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561 [11] Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571 [12] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [13] Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211 [14] Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935 [15] Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439 [16] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [17] Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 [18] Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057 [19] Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171 [20] Yutian Lei. On the integral systems with negative exponents. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1039-1057. doi: 10.3934/dcds.2015.35.1039

Impact Factor: