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

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A variational approach to a class of quasilinear elliptic equations not in divergence form

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

Abstract
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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 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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