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On the existence of nontrivial solutions of inequalities in OrliczSobolev spaces
1.  Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, United States 
References:
[1] 
R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New YorkLondon, 1975. 
[2] 
J. Ball, J. Currie and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135174. doi: 10.1016/00221236(81)900859. 
[3] 
K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102129. doi: 10.1016/0022247X(81)900950. 
[4] 
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990. 
[5] 
L. Gasiński 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. 
[6] 
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, 69, Kluwer Academic Publishers, Boston, MA, 2003. 
[7] 
D. Goeleven and D. Motreanu, "Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. II. Unilateral Problems," Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston, MA, 2003. 
[8] 
J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163205. doi: 10.1090/S00029947197403428542. 
[9] 
M. A. Krasnosels'kiĭ and J. Rutickiĭ, "Convex Functions and Orlicz Spaces," Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. 
[10] 
A. Kufner, O. John and S. Fučic, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. 
[11] 
V. K. Le, Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts, J. Diff. Int. Eq., 15 (2002), 839862. 
[12] 
V. K. Le and D. Motreanu, On nontrivial solutions of variationalhemivariational inequalities with slowly growing principal parts, Z. Anal. Anwend., 28 (2009), 277293. 
[13] 
R. Livrea and S. A. Marano, Nonsmooth critical point theory, in "Handbook of Nonconvex Analysis and Applications" (eds. D. Y. Gao and D. Motreanu), 295351, International Press, 2010. 
[14] 
D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. 
[15] 
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77109. 
show all references
References:
[1] 
R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New YorkLondon, 1975. 
[2] 
J. Ball, J. Currie and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135174. doi: 10.1016/00221236(81)900859. 
[3] 
K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102129. doi: 10.1016/0022247X(81)900950. 
[4] 
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990. 
[5] 
L. Gasiński 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. 
[6] 
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, 69, Kluwer Academic Publishers, Boston, MA, 2003. 
[7] 
D. Goeleven and D. Motreanu, "Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. II. Unilateral Problems," Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston, MA, 2003. 
[8] 
J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163205. doi: 10.1090/S00029947197403428542. 
[9] 
M. A. Krasnosels'kiĭ and J. Rutickiĭ, "Convex Functions and Orlicz Spaces," Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. 
[10] 
A. Kufner, O. John and S. Fučic, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. 
[11] 
V. K. Le, Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts, J. Diff. Int. Eq., 15 (2002), 839862. 
[12] 
V. K. Le and D. Motreanu, On nontrivial solutions of variationalhemivariational inequalities with slowly growing principal parts, Z. Anal. Anwend., 28 (2009), 277293. 
[13] 
R. Livrea and S. A. Marano, Nonsmooth critical point theory, in "Handbook of Nonconvex Analysis and Applications" (eds. D. Y. Gao and D. Motreanu), 295351, International Press, 2010. 
[14] 
D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. 
[15] 
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77109. 
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