Three solutions with precise sign properties for systems of quasilinear elliptic equations

Previous Article
A variational approach to a class of quasilinear elliptic equations not in divergence form
DCDS-S Home
This Issue
Next Article
Multiplicity of solutions for variable exponent Dirichlet problem with concave term

2012, 5(4): 831-843. doi: 10.3934/dcdss.2012.5.831

Three solutions with precise sign properties for systems of quasilinear elliptic equations

1.	Université de Perpignan, Département de Mathématiques, 66860 Perpignan

Received January 2011 Revised February 2011 Published November 2011

Abstract
Related Papers

For a quasilinear elliptic system, the existence of two extremal solutions with components of opposite constant sign is established. If the system has a variational structure, the existence of a third nontrivial solution is shown.

Keywords: constant sign solutions, Quasilinear elliptic system, sub-supersolution, variational methods..

Mathematics Subject Classification: 35B30, 35J20, 49J4.

Citation: Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831

References:

[1]	A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids,, (French) [Simplicity and isolation of the first eigenvalue of the $p$-Laplacian with weight], 305 (1987), 725.
[2]	D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity,, Bull. Aust. Math. Soc., 77 (2008), 285. doi: 10.1017/S0004972708000282.
[3]	S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).
[4]	S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668. doi: 10.1016/j.na.2007.02.013.
[5]	S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems,, J. Global Optim., 34 (2006), 609. doi: 10.1007/s10898-005-1651-4.
[6]	S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2002), 613. doi: 10.1155/S1085337502207010.
[7]	M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian,, J. Differential Equations, 159 (1999), 212. doi: 10.1006/jdeq.1999.3645.
[8]	L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Series in Mathematical Analysis and Applications, 8 (2005).
[9]	L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Series in Mathematical Analysis and Applications, 9 (2006).
[10]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for constant sign and nodal solutions,, Adv. Differential Equations, 12 (2007), 1363.
[11]	D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems,, Topol. Methods Nonlinear Anal., 34 (2009), 41.
[12]	D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations,, Set-Valued Var. Anal., 19 (2011), 255.
[13]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optimization., 12 (1984), 191. doi: 10.1007/BF01449041.
[14]	J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems,, Nonlinear Anal., 71 (2009), 2840. doi: 10.1016/j.na.2009.01.158.

show all references

References:

[1]	A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids,, (French) [Simplicity and isolation of the first eigenvalue of the $p$-Laplacian with weight], 305 (1987), 725.
[2]	D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity,, Bull. Aust. Math. Soc., 77 (2008), 285. doi: 10.1017/S0004972708000282.
[3]	S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).
[4]	S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668. doi: 10.1016/j.na.2007.02.013.
[5]	S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems,, J. Global Optim., 34 (2006), 609. doi: 10.1007/s10898-005-1651-4.
[6]	S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2002), 613. doi: 10.1155/S1085337502207010.
[7]	M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian,, J. Differential Equations, 159 (1999), 212. doi: 10.1006/jdeq.1999.3645.
[8]	L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Series in Mathematical Analysis and Applications, 8 (2005).
[9]	L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Series in Mathematical Analysis and Applications, 9 (2006).
[10]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for constant sign and nodal solutions,, Adv. Differential Equations, 12 (2007), 1363.
[11]	D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems,, Topol. Methods Nonlinear Anal., 34 (2009), 41.
[12]	D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations,, Set-Valued Var. Anal., 19 (2011), 255.
[13]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optimization., 12 (1984), 191. doi: 10.1007/BF01449041.
[14]	J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems,, Nonlinear Anal., 71 (2009), 2840. doi: 10.1016/j.na.2009.01.158.

[1]	Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[2]	Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
[3]	Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[4]	Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
[5]	Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
[6]	Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[7]	Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[8]	Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[9]	M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819
[10]	Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
[11]	Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[12]	Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125
[13]	A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[14]	Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[15]	Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
[16]	Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure & Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175
[17]	Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[18]	Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[19]	Antonio Ambrosetti, Zhi-Qiang Wang. Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 55-68. doi: 10.3934/dcds.2003.9.55
[20]	Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences