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Strongly nonlinear multivalued elliptic equations on a bounded domain

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Abstract

In this work we study the existence of nontrivial solution for the following class of multivalued quasilinear problems

$$\begin{aligned} \displaystyle -\text{ div } ( \phi (|\nabla u|) \nabla u) - b(u)u \in \lambda \partial F(x,u)\;\text{ in }\;\Omega , \quad u=0\; \text{ on }\;\partial \Omega \end{aligned}$$

where \(\Omega \subset \mathbb{R }^N\) is a bounded domain, \(N\ge 2\) and \(\partial F(x,u)\) is a generalized gradient of \(F(x,t)\) with respect to \(t\). The main tools utilized are Variational Methods for Locally Lipschitz Functional and a Concentration Compactness Theorem for Orlicz space.

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References

  1. Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Elsevier Science, Oxford (2003)

    Google Scholar 

  2. Ambrosetti, A., Turner, R.E.L.: Some discontinuous variational problems. Diff. Int. Equ. 1, 341–349 (1988)

    Google Scholar 

  3. Badiale, M., Tarantello, G.: Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlinear Anal. 29, 639–677 (1997)

    Article  Google Scholar 

  4. Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functinals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  Google Scholar 

  5. Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev expoents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  Google Scholar 

  6. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer Monographs in Mathematics. Springer, New York (2007)

    Google Scholar 

  7. Carl, S., Le, V.K., Motreanu, D.: Existence and comparison principles for general quasilinear variational-hemivariational inequalities. J. Math. Anal. Appl. 302, 65–83 (2005)

    Article  Google Scholar 

  8. Carl, S., Naniewicz, Z.: Vector quasi-hemivariational inequalities and discontinuous elliptic systems. J. Glob. Optim. 34, 609–634 (2006)

    Article  Google Scholar 

  9. Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)

    Article  Google Scholar 

  10. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  11. Dacorogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2009)

    Google Scholar 

  12. Dal Maso, G., Murat, F.: Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal. 31, 405–412 (1998)

    Article  Google Scholar 

  13. Donaldson, T.K., Trudinger, N.S.: Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8, 52–75 (1971)

    Article  Google Scholar 

  14. Fukagai, N., Ito, M., Narukawa, M.: Quasilinear elliptic equations with slowly growing principal part and critical Orlicz-Sobolev nonlinear term. Proc. R. Soc. Edinb. 139A, 73–100 (2009)

    Article  Google Scholar 

  15. Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear qlliptic equations with critical Orlicz-Sobolev nonlinearity on \({\mathbb{R}}^N\). Funckcialaj Ekvacioj 49, 235–267 (2006)

    Article  Google Scholar 

  16. Fukagai, N., Narukawa, M.: Nonlinear eigenvalue problem for a model equation of an elastic surface. Hiroshima Math. J. 25, 19–41 (1995)

    Google Scholar 

  17. Giuffre, S., Idone, G.: Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity \(q {\>} 1\) and with natural growth. J. Glob. Optim. 40, 99–117 (2008)

    Article  Google Scholar 

  18. Gossez, J.P.: Orlicz-Sobolev, spaces and nonlinear elliptic boundary value problems (English). In: Fucik, S., Kufner, A. (eds.) Nonlinear Analysis, Function Spaces and Applications, Proceedings of a Spring School held in Horn Bradlo, 1978, vol. 1. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, pp. 59–94 (1978)

  19. Hu, S., Kourogenis, N., Papageorgiou, N.S.: Nonlinear elliptic eigenvalue problems with discontinuities. J. Math. Anal. Appl. 233, 406–424 (1999)

    Article  Google Scholar 

  20. Kristaly, A., Marzantowicz, W., Varga, C.: A non-smooth three critical points theorem with applications in differential inclusions. J. Glob. Optim. 46, 49–62 (2010)

    Article  Google Scholar 

  21. Kourogenis, N., Papageorgiou, N.S.: Three nontrivial solutions for a quasilinear elliptic differrential equation at resonance with discontinuous right hand side. J. Math. Anal. Appl. 238, 477–490 (1999)

    Article  Google Scholar 

  22. Livrea, R., Bisci, G.M.: Some remarks on nonsmooth critical point theory. J. Glob. Optim. 37, 245–261 (2007)

    Article  Google Scholar 

  23. Motreanu, D., Naniewicz, Z.: A minimax approach to semicoercive hemivariational inequalities. Optimization 52, 541–554 (2003)

    Article  Google Scholar 

  24. Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Nonconvex Optim. Appl., vol. 29. Kluwer, Dordrecht (1998)

  25. Naniewicz, Z.: On variational aspects of some nonconvex nonsmooth global optimization problem. J. Glob. Optim. 6, 383–400 (1995)

    Article  Google Scholar 

  26. Pohozaev, S.L.: Eingenfunctions for the equation \(\Delta u+\lambda f(u)=0\). Soviet Math. Dokl. 6, 1408–1411 (1965)

    Google Scholar 

  27. Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)

    Google Scholar 

  28. Tang, G.J., Huang, N.J.: Existence theorems of the variational-hemivariational inequalities. J. Glob. Optim. doi:10.1007/s10898-012-9884-5

  29. Teng, K.: Multiple solutions for semilinear resonant elliptic problems with discontinuous nonlinearities via nonsmooth double linking theorem. J. Glob. Optim. 46, 89–110 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, PB, CEP:58429-900, Brazil

    Claudianor O. Alves & Jefferson A. Santos

  2. Instituto de Matemática e Estatística-IME, Universidade Federal de Goiás, Goiânia, GO, CEP:74001-970, Brazil

    José V. Gonçalves

Authors
  1. Claudianor O. Alves
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  2. José V. Gonçalves
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  3. Jefferson A. Santos
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Correspondence to Claudianor O. Alves.

Additional information

This work was Partially supported by PROCAD/CAPES-Brazil. Claudianor O. Alves was partially supported by INCT-MAT, CNPq/Brazil 303080/2009-4. José V. Gonçalves was partially supported by CNPq/Brazil.

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Alves, C.O., Gonçalves, J.V. & Santos, J.A. Strongly nonlinear multivalued elliptic equations on a bounded domain. J Glob Optim 58, 565–593 (2014). https://doi.org/10.1007/s10898-013-0052-3

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