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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

Vy Khoi Le 1,

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.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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