Existence of solution for a partial differential inclusion in \(\pmb {{\mathbb {R}}}^{\varvec{N}}\) with steep potential well

  • Claudianor O. Alves
  • José V. A. Gonçalves
  • Jefferson A. SantosEmail author


In this work, we study the existence of nontrivial solutions for the following class of partial differential inclusion problem:
$$\begin{aligned} -\Delta u+\left( 1+\lambda V(x)\right) u\in \partial _t F(x,u) \quad \text {in} \quad \mathbb {R}^N, \qquad \quad (P_\lambda ) \end{aligned}$$
where \(N\ge 1\), \(\lambda >0\), V is a continuous function verifying some conditions and \(\partial _t F(x,u)\) is the generalized gradient of F(xt) with respect to t. Assuming that F(xt) is a mensurable function for each \(t\in \mathbb {R}\) and locally Lipschitzian for each \(x\in \mathbb {R}^N\), we have applied variational methods for locally Lipschitz functionals to get a solution for \((P_\lambda )\) when \(\lambda \) is large enough.


Orlicz–Sobolev spaces Discontinuous nonlinearity 

Mathematics Subject Classification

35A15 35J25 34A36 



  1. 1.
    Adams, A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)zbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Figueiredo, G.M., Santos, J.A.: Strauss–Lions Type Results for a Class of Orlicz–Sobolev Spaces and Applications, To appear in Topological Methods in Nonlinear Analysis (2014)Google Scholar
  3. 3.
    Alves, C.O., Bertone, A.M.: A discontinuous problem involving the \(p\)-Laplacian operator and critical exponent in \(\mathbb{R}^N\). Electron. J. Differ. Equ. 2003, 1–10 (2003)Google Scholar
  4. 4.
    Alves, C.O., Santos, J.A.: Multivalued elliptic equation with exponential critical growth in \(\mathbb{R}^{2}\). J. Differ. Equ. 261, 4758–4788 (2016)CrossRefGoogle Scholar
  5. 5.
    Alves, C.O., Bertone, A.M., Goncalves, J.V.: A variational approach to discontinuous problems with critical Sobolev exponents. J. Math. Anal. Appl. 265, 103–127 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ambrosetti, A., Turner, R.E.L.: Some discontinuous variational problems. Differ. Int. Equ. 1, 341–349 (1988)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ambrosetti, A., Calahorrano, M., Dobarro, F.: Global branching for discontinuous problems. Comment. Math. Univ. Carolinae 31, 213–222 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Badiale, M., Tarantello, G.: Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlinear Anal. 29, 639–677 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bartsch, T., Tang, Z.: Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete Contin. Dyn. Syst. 33, 7–26 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bartsch, T., Wang, Z.Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366–384 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bartsch, T., Pankov, A., Wang, Z.Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Carl, S., Dietrich, H.: The weak upper and lower solution method for elliptic equations with generalized subdifferentiable perturbations. Appl. Anal. 56, 263–278 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Carl, S., Heikkila, S.: Elliptic equations with discontinuous nonlinearities in \(\mathbb{R}^N\). Nonlinear Anal. 31, 217–227 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cerami, G.: Metodi variazionalli nello studio di problemi al contorno con parte nonlineare discontinua. Rend. Circ. Mat. Palermo 32, 336–357 (1983)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chang, K.C.: The obstacle problem and partial differential equations with discontinuous nonlinearities. Commun. Pure Appl. Math. 139–158 (1978)Google Scholar
  18. 18.
    Chang, K.C.: On the multiple solutions of the elliptic differential equations with discontinuous nonlinear terms. Sci. Sinica 21, 139–158 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, NY (1983)zbMATHGoogle Scholar
  21. 21.
    Donaldson, T.K., Trudinger, N.S.: Orlicz–Sobolev spaces and embedding theorems. J. Funct. Anal. 8, 52–75 (1971)CrossRefGoogle Scholar
  22. 22.
    Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 4(3), 539–564 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on \(\mathbb{R}^{N}\). Funkcial. Ekvac. 49, 235–267 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hu, S., Kourogenis, N., Papageorgiou, N.S.: Nonlinear elliptic eigenvalue problems with discontinuities. J. Math. Anal. Appl. 233, 406–424 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Motreanu, D., Varga, C.: Some critical point results for locally Lipschitz functionals. Commun. Appl. Nonlinear Anal. 4, 17–33 (1997)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Radulescu, V.: Mountain pass theorems for non-differentiable functions and applications. Proc. Jpn. Acad. 69(Ser.A), 193–198 (1993)CrossRefGoogle Scholar
  27. 27.
    Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Claudianor O. Alves
    • 1
  • José V. A. Gonçalves
    • 2
  • Jefferson A. Santos
    • 1
    Email author
  1. 1.Unidade Acadêmica de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

Personalised recommendations