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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

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 & 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.  Google Scholar

[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.  Google Scholar

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L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Series in Mathematical Analysis and Applications, 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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