Multiple solutions for a class of semilinear elliptic variational inclusion problems

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October 2011, 7(4): 991-1002. doi: 10.3934/jimo.2011.7.991

Multiple solutions for a class of semilinear elliptic variational inclusion problems

Rong Xiao ^1, and Yuying Zhou ^1,

1.	Department of Mathematics, Soochow University, Suzhou, 215006, China, China

Received September 2010 Revised July 2011 Published August 2011

Abstract
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In this paper, by using a local linking theorem, we obtain the existence of multiple solutions for a class of semilinear elliptic variational inclusion problems at non-resonance.

Keywords: Local linking, projective operator., (PS)-condition, coercive, inclusive mapping.

Mathematics Subject Classification: Primary: 35R70, 35R45; Secondary: 35J5.

Citation: Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991

References:

[1]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.
[2]	Z. Denkowski, L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance, Nonlinear Anal., 66 (2007), 1329-1340. doi: 10.1016/j.na.2006.01.019.
[3]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.
[4]	M. Filippakis, L. Gasinski and N. S. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential, Nonlinear Anal., 61 (2005), 61-75. doi: 10.1016/j.na.2004.11.012.
[5]	Z. Denkowski, L. Gasinski and N. S. Papageorgiou, Nontrivial solutions for resonant hemivariational inequalities, J. Global Optim., 34 (2006), 317-337. doi: 10.1007/s10898-005-4388-1.
[6]	L. Gasinski 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, FL, 2005.
[7]	X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671-684.
[8]	G. Idone and A. Maugeri, Variational inequalities and a transport planning for an elastic and continuum model, J. Ind. Manag. Optim., 1 (2005), 81-86.
[9]	N. S. Papageorgiou, S. R. Andrade Santos and V. Staicu, Eigenvalue problems for hemivariational inequalities, Set-Valued Anal., 16 (2008), 1061-1087. doi: 10.1007/s11228-008-0100-1.
[10]	C.-L. Tang and Q.-J. Gao, Elliptic resonant problems at higher eigenvalues with an unbounded nonlinear term, J. Diff. Equat., 146 (1998), 56-66. doi: 10.1006/jdeq.1998.3411.
[11]	C.-L. Tang, Multiple solutions of Neumann problem for elliptic equations, Nonlinear Anal., 54 (2003), 637-650. doi: 10.1016/S0362-546X(03)00091-9.
[12]	L. Wang, Y. Li and L. W. Zhang, A differential equation method for solving box constrained variational inequality problems, J. Ind. Manag. Optim., 7 (2011), 183-198.

show all references

References:

[1]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.
[2]	Z. Denkowski, L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance, Nonlinear Anal., 66 (2007), 1329-1340. doi: 10.1016/j.na.2006.01.019.
[3]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.
[4]	M. Filippakis, L. Gasinski and N. S. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential, Nonlinear Anal., 61 (2005), 61-75. doi: 10.1016/j.na.2004.11.012.
[5]	Z. Denkowski, L. Gasinski and N. S. Papageorgiou, Nontrivial solutions for resonant hemivariational inequalities, J. Global Optim., 34 (2006), 317-337. doi: 10.1007/s10898-005-4388-1.
[6]	L. Gasinski 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, FL, 2005.
[7]	X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671-684.
[8]	G. Idone and A. Maugeri, Variational inequalities and a transport planning for an elastic and continuum model, J. Ind. Manag. Optim., 1 (2005), 81-86.
[9]	N. S. Papageorgiou, S. R. Andrade Santos and V. Staicu, Eigenvalue problems for hemivariational inequalities, Set-Valued Anal., 16 (2008), 1061-1087. doi: 10.1007/s11228-008-0100-1.
[10]	C.-L. Tang and Q.-J. Gao, Elliptic resonant problems at higher eigenvalues with an unbounded nonlinear term, J. Diff. Equat., 146 (1998), 56-66. doi: 10.1006/jdeq.1998.3411.
[11]	C.-L. Tang, Multiple solutions of Neumann problem for elliptic equations, Nonlinear Anal., 54 (2003), 637-650. doi: 10.1016/S0362-546X(03)00091-9.
[12]	L. Wang, Y. Li and L. W. Zhang, A differential equation method for solving box constrained variational inequality problems, J. Ind. Manag. Optim., 7 (2011), 183-198.

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