American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times
  • JIMO Home
  • This Issue
  • Next Article
    A DC programming approach for sensor network localization with uncertainties in anchor positions
July  2014, 10(3): 827-837. doi: 10.3934/jimo.2014.10.827

A class of quasilinear elliptic hemivariational inequality problems on unbounded domains

Lijing Xi 1, , Yuying Zhou 2, and  Yisheng Huang 1,

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.
Keywords: Local linking, coercive, inclusive mapping, (PS)-condition, projective operator..
Mathematics Subject Classification: Primary: 47J20; Secondary: 35J20, 49J5.
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. Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley-interscience, (1983). Google Scholar

[3]

Z. Dályay and C. Varga, An existence result for hemivariational inequalities,, Electronic J. Differential Equations, 2004 (2004), 1. Google Scholar

[4]

K. Fan, A generalization of Tychonoff's fixed point theorem,, Math. Ann., 142 (1961), 305. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Chapman and Hall/CRC, (2005). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

Y. S. Huang, Positive solutions of quasilinear elliptic equations,, Topol. Meth. Nonl. Anal., 12 (1998), 91. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

D. Motreanu and P. D. Panagiotopoulos, Minimax Theorem and Qualitative Properties of the Solutions of Hemivariational Inequalities,, Nonconvex Optimization and its Applications, (1999). Google Scholar

[22]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Kluwer Academic Publishers, (2003). Google Scholar

[23]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer, (1993). doi: 10.1007/978-3-642-51677-1. Google Scholar

[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. Google Scholar

[25]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia. Math., 135 (1999), 191. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley-interscience, (1983). Google Scholar

[3]

Z. Dályay and C. Varga, An existence result for hemivariational inequalities,, Electronic J. Differential Equations, 2004 (2004), 1. Google Scholar

[4]

K. Fan, A generalization of Tychonoff's fixed point theorem,, Math. Ann., 142 (1961), 305. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Chapman and Hall/CRC, (2005). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

Y. S. Huang, Positive solutions of quasilinear elliptic equations,, Topol. Meth. Nonl. Anal., 12 (1998), 91. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

D. Motreanu and P. D. Panagiotopoulos, Minimax Theorem and Qualitative Properties of the Solutions of Hemivariational Inequalities,, Nonconvex Optimization and its Applications, (1999). Google Scholar

[22]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Kluwer Academic Publishers, (2003). Google Scholar

[23]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer, (1993). doi: 10.1007/978-3-642-51677-1. Google Scholar

[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. Google Scholar

[25]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia. Math., 135 (1999), 191. Google Scholar

[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. Google Scholar

[1]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315
[2]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
[3]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[4]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[5]

Yanqun Liu. An exterior point linear programming method based on inclusive normal cones. Journal of Industrial & Management Optimization, 2010, 6 (4) : 825-846. doi: 10.3934/jimo.2010.6.825
[6]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[7]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Local study of a renormalization operator for 1D maps under quasiperiodic forcing. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1171-1188. doi: 10.3934/dcdss.2016047
[8]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[9]

Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009
[10]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear coercive Neumann problems. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1957-1974. doi: 10.3934/cpaa.2009.8.1957
[11]

Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010
[12]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
[13]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[14]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007
[15]

Dubi Kelmer. Quadratic irrationals and linking numbers of modular knots. Journal of Modern Dynamics, 2012, 6 (4) : 539-561. doi: 10.3934/jmd.2012.6.539
[16]

John Banks. Topological mapping properties defined by digraphs. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83
[17]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493
[18]

Jungkai A. Chen and Meng Chen. On projective threefolds of general type. Electronic Research Announcements, 2007, 14: 69-73. doi: 10.3934/era.2007.14.69
[19]

Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[20]

Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences