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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 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), 1363-1392.

[11]

D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems, Topol. Methods Nonlinear Anal., 34 (2009), 41-48.

[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.

[13]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 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), 1363-1392.

[11]

D. Motreanu and K. Perera, Multiple nontrivial solutions of Neumann $p$-Laplacian systems, Topol. Methods Nonlinear Anal., 34 (2009), 41-48.

[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.

[13]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[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.

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