# American Institute of Mathematical Sciences

August  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

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.
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], C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.  Google Scholar [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), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar [3] S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar [4] S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.  Google Scholar [5] S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems, J. Global Optim., 34 (2006), 609-634. doi: 10.1007/s10898-005-1651-4.  Google Scholar [6] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625. doi: 10.1155/S1085337502207010.  Google Scholar [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), 212-238. doi: 10.1006/jdeq.1999.3645.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Series in Mathematical Analysis and Applications, 8, Chapman & Hall/CRC, Boca Raton, 2005.  Google Scholar [9] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [10] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.  Google Scholar [11] D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems, Topol. Methods Nonlinear Anal., 34 (2009), 41-48.  Google Scholar [12] D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Var. Anal., 19 (2011), 255-269. Google Scholar [13] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar [14] J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems, Nonlinear Anal., 71 (2009), 2840-2846. doi: 10.1016/j.na.2009.01.158.  Google Scholar

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], C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.  Google Scholar [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), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar [3] S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar [4] S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.  Google Scholar [5] S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems, J. Global Optim., 34 (2006), 609-634. doi: 10.1007/s10898-005-1651-4.  Google Scholar [6] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625. doi: 10.1155/S1085337502207010.  Google Scholar [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), 212-238. doi: 10.1006/jdeq.1999.3645.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Series in Mathematical Analysis and Applications, 8, Chapman & Hall/CRC, Boca Raton, 2005.  Google Scholar [9] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [10] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.  Google Scholar [11] D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems, Topol. Methods Nonlinear Anal., 34 (2009), 41-48.  Google Scholar [12] D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Var. Anal., 19 (2011), 255-269. Google Scholar [13] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar [14] J. Zhang and Z. Zhang, Existence results for some nonlinear elliptic systems, Nonlinear Anal., 71 (2009), 2840-2846. doi: 10.1016/j.na.2009.01.158.  Google Scholar
 [1] Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems, 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, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095 [3] 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 [4] Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151 [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] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 [7] Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 [8] Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 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] 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 [11] 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 [12] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [13] Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256 [14] Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737 [15] 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 [16] 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 [17] 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 [18] 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 [19] Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013 [20] 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

2020 Impact Factor: 2.425