# American Institute of Mathematical Sciences

January  2013, 33(1): 255-276. doi: 10.3934/dcds.2013.33.255

## Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths

 1 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, United States

Received  August 2011 Revised  October 2011 Published  September 2012

This paper is about systems of variational inequalities of the form: $$\left\{ \begin{array}{l} ‹A_k U_k+ F_k (u) , v_k -u_k› ≥ 0,\; ∀ v_k ∈ K_k \\ u_k ∈ K_k , \end{array} \right.$$ $(k=1,\dots , m)$, where $A_k$ and $F_k$ are multivalued mappings with possibly non-power growths and $K_k$ is a closed, convex set. We concentrate on the noncoercive case and follow a sub-supersolution approach to obtain the existence and enclosure of solutions to the above system between sub- and supersolutions.
Citation: Vy Khoi Le. Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 255-276. doi: 10.3934/dcds.2013.33.255
##### References:
 [1] R. Adams, "Sobolev Spaces," Academic Press, New York, 1975. [2] J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2 1990. [3] F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. doi: 10.1016/0022-1236(72)90070-5. [4] S. Carl and V. K. Le, Enclosure results for quasilinear systems of variational inequalities, J. Differential Equations, 199 (2004), 77-95. doi: 10.1016/j.jde.2003.10.009. [5] S. Carl, V. K. Le and D. Motreanu, Existence, comparison, and compactness results for quasilinear variational-hemivariational inequalities, Int. J. Math. Math. Sci., (2005), 401-417. doi: 10.1155/IJMMS.2005.401. [6] ——, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007. [7] S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems, J. Global Optim., 34 (2006), 609-634. doi: 10.1007/s10898-005-1651-4. [8] P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var., 11 (2000), 33-62. doi: 10.1007/s005260050002. [9] T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Diff. Equations, 10 (1971), 507-528. doi: 10.1016/0022-0396(71)90009-X. [10] T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75. doi: 10.1016/0022-1236(71)90018-8. [11] M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On pricipal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonl. Diff. Eq. Appl., 6 (1999), 207-225. doi: 10.1007/s000300050073. [12] J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2. [13] J. P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909. doi: 10.1017/S030821050000192X. [14] J. P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis, 11 (1987), 379-392. doi: 10.1016/0362-546X(87)90053-8. [15] S. C. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I," Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, Theory, 419 1997. [16] M. A. Krasnosels'kii and J. Rutic'kii, "Convex Functions and Orlicz Spaces," Noordhoff, Groningen, 1961. [17] A. Kufner, O. John and S. Fučic, "Function Spaces," Noordhoff, Leyden, 1977. [18] V. K. Le, Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts, J. Diff. Int. Eq., 15 (2002), 839-862. [19] ——, Generic existence result for an eigenvalue problem with rapidly growing principal operator, ESAIM Control Optim. Calc. Var., 10 (2004), 677-691. doi: 10.1051/cocv:2004027. [20] ——, Some existence and bifurcation results for quasilinear elliptic equations with slowly growing principal operators, Houston J. Math., 32 (2006), 921-943. [21] ——, Some existence results and properties of solutions in quasilinear variational inequalities with general growths, Differ. Equ. Dyn. Syst., 17 (2009), 343-364. doi: 10.1007/s12591-009-0025-7. [22] ——, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4. [23] ——, Variational inequalities with general multivalued lower order terms given by integrals, Adv. Nonlinear Studies, 11 (2011), 1-24. [24] ——, On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents, To appear in J. Math. Anal. Appl., (2012). [25] V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients, J. London Math. Soc., 62 (2000), 852-872. doi: 10.1112/S0024610700001423. [26] ——, Equations and inequalities in Orlicz-Sobolev spaces: Selected topics, International Press, Boston, (2010), 295-351. [27] M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432. doi: 10.1016/j.jmaa.2006.07.082.

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##### References:
 [1] R. Adams, "Sobolev Spaces," Academic Press, New York, 1975. [2] J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2 1990. [3] F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. doi: 10.1016/0022-1236(72)90070-5. [4] S. Carl and V. K. Le, Enclosure results for quasilinear systems of variational inequalities, J. Differential Equations, 199 (2004), 77-95. doi: 10.1016/j.jde.2003.10.009. [5] S. Carl, V. K. Le and D. Motreanu, Existence, comparison, and compactness results for quasilinear variational-hemivariational inequalities, Int. J. Math. Math. Sci., (2005), 401-417. doi: 10.1155/IJMMS.2005.401. [6] ——, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007. [7] S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems, J. Global Optim., 34 (2006), 609-634. doi: 10.1007/s10898-005-1651-4. [8] P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var., 11 (2000), 33-62. doi: 10.1007/s005260050002. [9] T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Diff. Equations, 10 (1971), 507-528. doi: 10.1016/0022-0396(71)90009-X. [10] T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75. doi: 10.1016/0022-1236(71)90018-8. [11] M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On pricipal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonl. Diff. Eq. Appl., 6 (1999), 207-225. doi: 10.1007/s000300050073. [12] J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2. [13] J. P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909. doi: 10.1017/S030821050000192X. [14] J. P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis, 11 (1987), 379-392. doi: 10.1016/0362-546X(87)90053-8. [15] S. C. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I," Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, Theory, 419 1997. [16] M. A. Krasnosels'kii and J. Rutic'kii, "Convex Functions and Orlicz Spaces," Noordhoff, Groningen, 1961. [17] A. Kufner, O. John and S. Fučic, "Function Spaces," Noordhoff, Leyden, 1977. [18] V. K. Le, Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts, J. Diff. Int. Eq., 15 (2002), 839-862. [19] ——, Generic existence result for an eigenvalue problem with rapidly growing principal operator, ESAIM Control Optim. Calc. Var., 10 (2004), 677-691. doi: 10.1051/cocv:2004027. [20] ——, Some existence and bifurcation results for quasilinear elliptic equations with slowly growing principal operators, Houston J. Math., 32 (2006), 921-943. [21] ——, Some existence results and properties of solutions in quasilinear variational inequalities with general growths, Differ. Equ. Dyn. Syst., 17 (2009), 343-364. doi: 10.1007/s12591-009-0025-7. [22] ——, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4. [23] ——, Variational inequalities with general multivalued lower order terms given by integrals, Adv. Nonlinear Studies, 11 (2011), 1-24. [24] ——, On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents, To appear in J. Math. Anal. Appl., (2012). [25] V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients, J. London Math. Soc., 62 (2000), 852-872. doi: 10.1112/S0024610700001423. [26] ——, Equations and inequalities in Orlicz-Sobolev spaces: Selected topics, International Press, Boston, (2010), 295-351. [27] M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432. doi: 10.1016/j.jmaa.2006.07.082.
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