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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

Dumitru Motreanu 1,

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.
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], 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

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