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July  2014, 10(3): 827-837. doi: 10.3934/jimo.2014.10.827

## A class of quasilinear elliptic hemivariational inequality problems on unbounded domains

 1 Department of Mathematics, Soochow University, Suzhou 215006, China, China 2 Department of Mathematics, Soochow University, Suzhou, 215006

Received  May 2013 Revised  June 2013 Published  November 2013

In this paper, we are concerned with the existence of solutions of a class of quasilinear elliptic hemivariational inequalities on unbounded domains. This kind of problems is more delicate due to the lack of compact embedding of the Sobolev spaces. By using the Clarke generalized directional derivatives for locally Lipschitz functions and some nonlinear function analysis techniques, such as the Ky Fan theorem for multivalued mappings, the theorem of finite intersection property, etc, we obtain the existence of solutions of the quasilinear elliptic hemivariational inequalities. Unlike those methods used in the references mentioned in this paper, we treat the case of unbounded domain by using the approximation of bounded domains.
Citation: Lijing Xi, Yuying Zhou, Yisheng Huang. A class of quasilinear elliptic hemivariational inequality problems on unbounded domains. Journal of Industrial & Management Optimization, 2014, 10 (3) : 827-837. doi: 10.3934/jimo.2014.10.827
##### References:
 [1] C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519. doi: 10.3934/jimo.2007.3.519. [2] F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley-interscience, (1983). [3] Z. Dályay and C. Varga, An existence result for hemivariational inequalities,, Electronic J. Differential Equations, 2004 (2004), 1. [4] K. Fan, A generalization of Tychonoff's fixed point theorem,, Math. Ann., 142 (1961), 305. [5] F. Faraci, A. Iannizzotto, P. Kupán and C. Varga, Existence and multiplicity results for hemivariational inequalities with two parameters,, Nonlinear Anal., 67 (2007), 2654. doi: 10.1016/j.na.2006.09.030. [6] M. Filippakis, L. Gasinski and N. S. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential,, Nonlinear Anal., 61 (2005), 61. doi: 10.1016/j.na.2004.11.012. [7] M. Filippakis, L. Gasinski and N. S. Papageorgiou, Nontrivial solutions for resonant hemivariational inequalities,, J. Global Optim., 34 (2006), 317. doi: 10.1007/s10898-005-4388-1. [8] L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Chapman and Hall/CRC, (2005). [9] F. Gazzola and V. Rădulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbbR^N$,, Differential Integral Equations, 13 (2000), 47. [10] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications,, Vol. I. Unilateral analysis and unilateral mechanics. Nonconvex Optimization and its Applications, (2003). [11] S. C. Hu and N. S. Papageorgiou, Positive solutions for nonlinear hemivariational inequalities,, J. Math. Anal. Appl., 310 (2005), 161. doi: 10.1016/j.jmaa.2005.01.051. [12] 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. doi: 10.3934/jimo.2007.3.671. [13] Y. S. Huang, Positive solutions of quasilinear elliptic equations,, Topol. Meth. Nonl. Anal., 12 (1998), 91. [14] Y. S. Huang and Y. Y. Zhou, Multiple solutions for a class of nonlinear elliptic problems with a p-Laplacian type operator,, Nonlinear Anal., 72 (2010), 3388. doi: 10.1016/j.na.2009.12.020. [15] A. Kristály, Multiplicity results for an eigenvalue problem for hemivariational inequalities,, Set-Valued Analysis, 13 (2005), 85. doi: 10.1007/s11228-004-6565-7. [16] A. Kristály, C. Varga and V. Varga, A nonsmooth principle of symmetric criticality and variational-hemivariational inequalities in strip-like domains,, J. Math. Anal. Appl., 325 (2007), 975. doi: 10.1016/j.jmaa.2006.02.062. [17] S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155. doi: 10.3934/jimo.2008.4.155. [18] H. Lisei, A. E. Molnr and C. Varga, On a class of inequality problems with lack of compactness,, J. Math. Anal. Appl., 378 (2011), 741. doi: 10.1016/j.jmaa.2010.12.041. [19] Z. H. Liu and D. Motreanu, A class of variational-hemivariational inequalities of elliptic type,, Nonlinearity, 23 (2010), 1741. doi: 10.1088/0951-7715/23/7/012. [20] I. I. Mezei and L. Săplăcan, Existence result and applications for general variational-hemivariational inequalities on unbounded domains,, Electronic J. Differential Equations, 2009 (2009), 1. [21] D. Motreanu and P. D. Panagiotopoulos, Minimax Theorem and Qualitative Properties of the Solutions of Hemivariational Inequalities,, Nonconvex Optimization and its Applications, (1999). [22] D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Kluwer Academic Publishers, (2003). [23] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer, (1993). doi: 10.1007/978-3-642-51677-1. [24] L. Scrimali, Mixed behavior network equilibria and quasi-variational inequalities,, J. Ind. Manag. Optim., 5 (2009), 363. doi: 10.3934/jimo.2009.5.363. [25] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia. Math., 135 (1999), 191. [26] R. Xiao and Y. Y. Zhou, Multiple solutions for a class of semilinear elliptic variational inclusion problems,, J. Ind. Manag. Optim., 7 (2011), 991. doi: 10.3934/jimo.2011.7.991.

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##### References:
 [1] C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519. doi: 10.3934/jimo.2007.3.519. [2] F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley-interscience, (1983). [3] Z. Dályay and C. Varga, An existence result for hemivariational inequalities,, Electronic J. Differential Equations, 2004 (2004), 1. [4] K. Fan, A generalization of Tychonoff's fixed point theorem,, Math. Ann., 142 (1961), 305. [5] F. Faraci, A. Iannizzotto, P. Kupán and C. Varga, Existence and multiplicity results for hemivariational inequalities with two parameters,, Nonlinear Anal., 67 (2007), 2654. doi: 10.1016/j.na.2006.09.030. [6] M. Filippakis, L. Gasinski and N. S. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential,, Nonlinear Anal., 61 (2005), 61. doi: 10.1016/j.na.2004.11.012. [7] M. Filippakis, L. Gasinski and N. S. Papageorgiou, Nontrivial solutions for resonant hemivariational inequalities,, J. Global Optim., 34 (2006), 317. doi: 10.1007/s10898-005-4388-1. [8] L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Chapman and Hall/CRC, (2005). [9] F. Gazzola and V. Rădulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbbR^N$,, Differential Integral Equations, 13 (2000), 47. [10] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications,, Vol. I. Unilateral analysis and unilateral mechanics. Nonconvex Optimization and its Applications, (2003). [11] S. C. Hu and N. S. Papageorgiou, Positive solutions for nonlinear hemivariational inequalities,, J. Math. Anal. Appl., 310 (2005), 161. doi: 10.1016/j.jmaa.2005.01.051. [12] 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. doi: 10.3934/jimo.2007.3.671. [13] Y. S. Huang, Positive solutions of quasilinear elliptic equations,, Topol. Meth. Nonl. Anal., 12 (1998), 91. [14] Y. S. Huang and Y. Y. Zhou, Multiple solutions for a class of nonlinear elliptic problems with a p-Laplacian type operator,, Nonlinear Anal., 72 (2010), 3388. doi: 10.1016/j.na.2009.12.020. [15] A. Kristály, Multiplicity results for an eigenvalue problem for hemivariational inequalities,, Set-Valued Analysis, 13 (2005), 85. doi: 10.1007/s11228-004-6565-7. [16] A. Kristály, C. Varga and V. Varga, A nonsmooth principle of symmetric criticality and variational-hemivariational inequalities in strip-like domains,, J. Math. Anal. Appl., 325 (2007), 975. doi: 10.1016/j.jmaa.2006.02.062. [17] S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155. doi: 10.3934/jimo.2008.4.155. [18] H. Lisei, A. E. Molnr and C. Varga, On a class of inequality problems with lack of compactness,, J. Math. Anal. Appl., 378 (2011), 741. doi: 10.1016/j.jmaa.2010.12.041. [19] Z. H. Liu and D. Motreanu, A class of variational-hemivariational inequalities of elliptic type,, Nonlinearity, 23 (2010), 1741. doi: 10.1088/0951-7715/23/7/012. [20] I. I. Mezei and L. Săplăcan, Existence result and applications for general variational-hemivariational inequalities on unbounded domains,, Electronic J. Differential Equations, 2009 (2009), 1. [21] D. Motreanu and P. D. Panagiotopoulos, Minimax Theorem and Qualitative Properties of the Solutions of Hemivariational Inequalities,, Nonconvex Optimization and its Applications, (1999). [22] D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Kluwer Academic Publishers, (2003). [23] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer, (1993). doi: 10.1007/978-3-642-51677-1. [24] L. Scrimali, Mixed behavior network equilibria and quasi-variational inequalities,, J. Ind. Manag. Optim., 5 (2009), 363. doi: 10.3934/jimo.2009.5.363. [25] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia. Math., 135 (1999), 191. [26] R. Xiao and Y. Y. Zhou, Multiple solutions for a class of semilinear elliptic variational inclusion problems,, J. Ind. Manag. Optim., 7 (2011), 991. doi: 10.3934/jimo.2011.7.991.
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