Elliptic Problems with Nonhomogeneous Differential Operators and Multiple Solutions

  • Dumitru Motreanu
  • Patrick Winkert


The present survey aims to report on recent advances in the study of non-linear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δ p with \(1<p<+\infty\). More general operators can be considered, possibly having completely different properties, for instance not satisfying any homogeneity requirement. A major objective of our work is to provide existence theorems of multiple solutions for boundary value problems governed by such general operators. In this direction, a three nontrivial solutions theorem is presented. In the case of problems determined by the p-Laplacian, we give a theorem ensuring the existence of at least four nontrivial solutions. Moreover, a complete sign information is available: two positive solutions, a negative solution and a nodal (sign-changing) solution. Finally, we provide a theorem guaranteeing the existence of a positive solution for a problem involving the \((p,q)\)-Laplacian operator \(\Delta_p+\Delta_q\), with \(1<q<p\), and a nonlinearity depending on the solution and its gradient.


Nonlinear elliptic boundary value problem Nonhomogeneous differential operator p-Laplacian Eigenvalue problem Multiple solutions Critical point Minimizer Regularity Maximum principle 


  1. 1.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity. Discret. Contin. Dyn. Syst. 25(2), 431–456 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barletta, G., Papageorgiou, N.S.: A multiplicity theorem for the Neumann p-Laplacian with an asymmetric nonsmooth potential. J. Glob. Optim. 39(3), 365–392 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brezis, H., Nirenberg, L.: H 1versus C 1local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317(5), 465–472 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Carl, S., Perera, K.: Sign-changing and multiple solutions for the p -Laplacian. Abstr. Appl. Anal. 7(12), 613–625 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Chang, K.-C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 6. Birkhäuser, Boston, (1993)Google Scholar
  7. 7.
    Cuesta, M., de Figueiredo, D.G., Gossez, J.-P.: The beginning of the Fu\v cik spectrum for the p-Laplacian. J. Differ. Equ. 159(1), 212–238 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fan, X.: On the sub-supersolution method for p(x)-Laplacian equations. J. Math. Anal. Appl. 230(1), 665–682 (2007)CrossRefGoogle Scholar
  9. 9.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D.: Comparison and positive solutions for problems with \((p,q)\)-Laplacian and convection term. Proc. Edinburgh Math. Soc. (2014). doi:10.1017/S0013091513000576MathSciNetGoogle Scholar
  10. 10.
    Garc\'ıa Azorero, J.P., Peral Alonso, I., Manfredi, J.J.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2(3), 385–404 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman Hall/CRC Press, Boca Raton (2006)zbMATHGoogle Scholar
  12. 12.
    Gasiński, L., Papageorgiou, N.S.: Anisotropic nonlinear Neumann problems. Calc. Var. Partial Differ. Equ. 42(3–4), 323–354 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13(8), 879–902 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, Z., Zhang, Z.: \(W1,p\) versus C 1local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286(1), 32–50 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Iannizzotto, A., Papageorgiou, N.S.: Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities. Nonlinear Anal. 70(9), 3285–3297 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khan, A.A., Motreanu, D.: Local minimizers versus X-local minimizers. Optim. Lett. (2012). doi:10.1007/s11590-012-0474-8Google Scholar
  17. 17.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equat. 16(2–3), 311–361 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer-Verlag, New York (1989)Google Scholar
  20. 20.
    Miyajima, S., Motreanu, D., Tanaka, M.: Multiple existence results of solutions for the Neumann problems via super- and sub-solutions, J. Funct. Anal. 262(4), 1921–1953 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Motreanu, D., Papageorgiou, N.S.: Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential. Nonlinear Anal. 56(8), 1211–1234 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Motreanu, D., Papageorgiou, N.S.: Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator. Proc. Amer. Math. Soc. 139(10), 3527–3535 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Motreanu, D., Tanaka, M.: Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application. Pacific J. Math. 265(1), 151–184 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Motreanu, D., Winkert, P.: On the Fu\v cik spectrum for the p-Laplacian with Robin boundary condition. Nonlinear Anal. 74(14), 4671–4681 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Motreanu, D., Winkert, P.: The Fu\v c\'ı k spectrum for the negative p-Laplacian with different boundary conditions. Nonlinear Analysis. Springer Optimization and Its Applications, vol. 68, pp. 471–485. Springer, New York (2012)Google Scholar
  26. 26.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Nonlinear Neumann problems near resonance. Indiana Univ. Math. J. 58(3), 1257–1279 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5). 10(3), 729–755 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: On p-Laplace equations with concave terms and asymmetric perturbations. Proc. Roy. Soc. Edinb. A. 141A (1), 171–192 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  30. 30.
    Vazquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Winkert, P.: Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values. Adv. Differ. Equat. 15(5–6), 561–599 (2010)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Winkert, P.: Local \(C^1({\Omega})\)-minimizers versus local \(W1,p(\Omega)\)-minimizers of nonsmooth functionals. Nonlinear Anal. 72(11), 4298–4303 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Winkert, P., Zacher, R.: A priori bounds of solutions to elliptic equations with nonstandard growth. Discret. Contin. Dyn. Syst. Ser. S 5(4), 865–878 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de PerpignanPerpignanFrance
  2. 2.Technische Universität BerlinInstitut für MathematikBerlinGermany

Personalised recommendations