# American Institute of Mathematical Sciences

August  2012, 5(4): 809-818. doi: 10.3934/dcdss.2012.5.809

## On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces

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

Received  February 2011 Revised  April 2011 Published  November 2011

This paper is about an alternate variational inequality formulation for the boundary value problem $$\begin{array}{l} -{\rm div} (a(|\nabla u|) \nabla u) + \partial_u G(x,u) \ni 0 \;\mbox{ in } \;\Omega , \\ u=0 \;\mbox{ on } \;\partial\Omega , \end{array}$$ where the principal part may have non-polynomial or very slow growth. As a consequence of this formulation, we can apply abstract nonsmooth linking theorems to study the existence and multiplicity of nontrivial solutions to the above problem.
Citation: Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] J. Ball, J. Currie and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135-174. doi: 10.1016/0022-1236(81)90085-9. [3] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0. [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), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2. [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), 839-862. [12] V. K. Le and D. Motreanu, On nontrivial solutions of variational-hemivariational inequalities with slowly growing principal parts, Z. Anal. Anwend., 28 (2009), 277-293. [13] R. Livrea and S. A. Marano, Non-smooth critical point theory, in "Handbook of Nonconvex Analysis and Applications" (eds. D. Y. Gao and D. Motreanu), 295-351, 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), 77-109.

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##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] J. Ball, J. Currie and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135-174. doi: 10.1016/0022-1236(81)90085-9. [3] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0. [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), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2. [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), 839-862. [12] V. K. Le and D. Motreanu, On nontrivial solutions of variational-hemivariational inequalities with slowly growing principal parts, Z. Anal. Anwend., 28 (2009), 277-293. [13] R. Livrea and S. A. Marano, Non-smooth critical point theory, in "Handbook of Nonconvex Analysis and Applications" (eds. D. Y. Gao and D. Motreanu), 295-351, 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), 77-109.
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