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

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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

Abstract
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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.

Keywords: Palais-Smale condition, Nonsmooth functional, Orlicz-Sobolev space., Mountain Pass Theorem.

Mathematics Subject Classification: Primary: 35B45, 35J65, 35J6.

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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