# American Institute of Mathematical Sciences

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

## Multiple solutions for a class of semilinear elliptic variational inclusion problems

 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.
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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