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 & Continuous Dynamical Systems - A, 2013, 33 (1) : 255-276. doi: 10.3934/dcds.2013.33.255
References:
[1]

R. Adams, "Sobolev Spaces,", Academic Press, (1975).   Google Scholar

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J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).   Google Scholar

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F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Functional Analysis, 11 (1972), 251.  doi: 10.1016/0022-1236(72)90070-5.  Google Scholar

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S. Carl and V. K. Le, Enclosure results for quasilinear systems of variational inequalities,, J. Differential Equations, 199 (2004), 77.  doi: 10.1016/j.jde.2003.10.009.  Google Scholar

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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.  doi: 10.1155/IJMMS.2005.401.  Google Scholar

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——, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).   Google Scholar

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S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems,, J. Global Optim., 34 (2006), 609.  doi: 10.1007/s10898-005-1651-4.  Google Scholar

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P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations,, Calc. Var., 11 (2000), 33.  doi: 10.1007/s005260050002.  Google Scholar

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T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Functional Analysis, 8 (1971), 52.  doi: 10.1016/0022-1236(71)90018-8.  Google Scholar

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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.  doi: 10.1007/s000300050073.  Google Scholar

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J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients,, Trans. Amer. Math. Soc., 190 (1974), 163.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

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J. P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891.  doi: 10.1017/S030821050000192X.  Google Scholar

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J. P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces,, Nonlinear Analysis, 11 (1987), 379.  doi: 10.1016/0362-546X(87)90053-8.  Google Scholar

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S. C. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I,", Mathematics and its Applications, 419 (1997).   Google Scholar

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M. A. Krasnosels'kii and J. Rutic'kii, "Convex Functions and Orlicz Spaces,", Noordhoff, (1961).   Google Scholar

[17]

A. Kufner, O. John and S. Fučic, "Function Spaces,", Noordhoff, (1977).   Google Scholar

[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.   Google Scholar

[19]

——, Generic existence result for an eigenvalue problem with rapidly growing principal operator,, ESAIM Control Optim. Calc. Var., 10 (2004), 677.  doi: 10.1051/cocv:2004027.  Google Scholar

[20]

——, Some existence and bifurcation results for quasilinear elliptic equations with slowly growing principal operators,, Houston J. Math., 32 (2006), 921.   Google Scholar

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——, Some existence results and properties of solutions in quasilinear variational inequalities with general growths,, Differ. Equ. Dyn. Syst., 17 (2009), 343.  doi: 10.1007/s12591-009-0025-7.  Google Scholar

[22]

——, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators,, Proc. Amer. Math. Soc., 139 (2011), 1645.  doi: 10.1090/S0002-9939-2010-10594-4.  Google Scholar

[23]

——, Variational inequalities with general multivalued lower order terms given by integrals,, Adv. Nonlinear Studies, 11 (2011), 1.   Google Scholar

[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).   Google Scholar

[25]

V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients,, J. London Math. Soc., 62 (2000), 852.  doi: 10.1112/S0024610700001423.  Google Scholar

[26]

——, Equations and inequalities in Orlicz-Sobolev spaces: Selected topics,, International Press, (2010), 295.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

show all references

References:
[1]

R. Adams, "Sobolev Spaces,", Academic Press, (1975).   Google Scholar

[2]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).   Google Scholar

[3]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Functional Analysis, 11 (1972), 251.  doi: 10.1016/0022-1236(72)90070-5.  Google Scholar

[4]

S. Carl and V. K. Le, Enclosure results for quasilinear systems of variational inequalities,, J. Differential Equations, 199 (2004), 77.  doi: 10.1016/j.jde.2003.10.009.  Google Scholar

[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.  doi: 10.1155/IJMMS.2005.401.  Google Scholar

[6]

——, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).   Google Scholar

[7]

S. Carl and Z. Naniewicz, Vector quasi-hemivariational inequalities and discontinuous elliptic systems,, J. Global Optim., 34 (2006), 609.  doi: 10.1007/s10898-005-1651-4.  Google Scholar

[8]

P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations,, Calc. Var., 11 (2000), 33.  doi: 10.1007/s005260050002.  Google Scholar

[9]

T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces,, J. Diff. Equations, 10 (1971), 507.  doi: 10.1016/0022-0396(71)90009-X.  Google Scholar

[10]

T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Functional Analysis, 8 (1971), 52.  doi: 10.1016/0022-1236(71)90018-8.  Google Scholar

[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.  doi: 10.1007/s000300050073.  Google Scholar

[12]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients,, Trans. Amer. Math. Soc., 190 (1974), 163.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[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.  doi: 10.1017/S030821050000192X.  Google Scholar

[14]

J. P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces,, Nonlinear Analysis, 11 (1987), 379.  doi: 10.1016/0362-546X(87)90053-8.  Google Scholar

[15]

S. C. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I,", Mathematics and its Applications, 419 (1997).   Google Scholar

[16]

M. A. Krasnosels'kii and J. Rutic'kii, "Convex Functions and Orlicz Spaces,", Noordhoff, (1961).   Google Scholar

[17]

A. Kufner, O. John and S. Fučic, "Function Spaces,", Noordhoff, (1977).   Google Scholar

[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.   Google Scholar

[19]

——, Generic existence result for an eigenvalue problem with rapidly growing principal operator,, ESAIM Control Optim. Calc. Var., 10 (2004), 677.  doi: 10.1051/cocv:2004027.  Google Scholar

[20]

——, Some existence and bifurcation results for quasilinear elliptic equations with slowly growing principal operators,, Houston J. Math., 32 (2006), 921.   Google Scholar

[21]

——, Some existence results and properties of solutions in quasilinear variational inequalities with general growths,, Differ. Equ. Dyn. Syst., 17 (2009), 343.  doi: 10.1007/s12591-009-0025-7.  Google Scholar

[22]

——, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators,, Proc. Amer. Math. Soc., 139 (2011), 1645.  doi: 10.1090/S0002-9939-2010-10594-4.  Google Scholar

[23]

——, Variational inequalities with general multivalued lower order terms given by integrals,, Adv. Nonlinear Studies, 11 (2011), 1.   Google Scholar

[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).   Google Scholar

[25]

V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients,, J. London Math. Soc., 62 (2000), 852.  doi: 10.1112/S0024610700001423.  Google Scholar

[26]

——, Equations and inequalities in Orlicz-Sobolev spaces: Selected topics,, International Press, (2010), 295.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

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