Nontrivial Solutions of Quasilinear Elliptic Equations with Natural Growth Term

  • Marco DegiovanniEmail author
  • Alessandra Pluda
Part of the Springer INdAM Series book series (SINDAMS, volume 22)


We prove the existence of multiple solutions for a quasilinear elliptic equation containing a term with natural growth, under assumptions that are invariant by diffeomorphism. To this purpose we develop an adaptation of degree theory.


Degree theory Divergence form Invariance by diffeomorphism Multiple solutions Natural growth conditions Quasilinear elliptic equations 

2010 Mathematics Subject Classification

35J66 47H11 



This paper is dedicated to Gianni Gilardi.


  1. 1.
    Almi, S., Degiovanni, M.: On degree theory for quasilinear elliptic equations with natural growth conditions. In: Serrin, J.B., Mitidieri, E.L., Rădulescu, V.D. (eds.) Recent Trends in Nonlinear Partial Differential Equations II: Stationary Problems (Perugia, 2012). Contemporary Mathematics, vol. 595, pp. 1–20. American Mathematical Society, Providence, RI (2013)CrossRefGoogle Scholar
  2. 2.
    Amann, H., Crandall, M.G.: On some existence theorems for semi-linear elliptic equations. Indiana Univ. Math. J. 27(5), 779–790 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ambrosetti, A., Lupo, D.: On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal. 8(10), 1145–1150 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal. 3(5), 635–645 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Aubin, J.-P., Cellina, A.: Differential inclusions. Grundlehren der Mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984)Google Scholar
  6. 6.
    Boccardo, L., Murat, F., Puel, J.-P.: Résultats d’existence pour certains problèmes elliptiques quasilinéaires. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 11(2), 213–235 (1984)Google Scholar
  7. 7.
    Boccardo, L., Murat, F., Puel, J.-P.: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. (4) 152, 183–196 (1988)Google Scholar
  8. 8.
    Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. (N.S.) 9(1), 1–39 (1983)Google Scholar
  9. 9.
    Dal Maso, G.: On the integral representation of certain local functionals. Ricerche Mat. 32(1), 85–113 (1983)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ (1983)Google Scholar
  11. 11.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Saccon, C.: Multiple positive solutions for a nonsymmetric elliptic problem with concave convex nonlinearity. In: de Figueiredo, D.G., do Ó, J.M., Tomei, C. (eds.) Analysis and Topology in Nonlinear Differential Equations (João Pessoa, 2012). Progress Nonlinear Differential Equations Applications, vol. 85, pp. 387–403. Birkhäuser/Springer, Cham (2014)Google Scholar
  13. 13.
    Skrypnik, I.V.: Methods for analysis of nonlinear elliptic boundary value problems. Translations of Mathematical Monographs, vol. 139. American Mathematical Society, Providence, RI (1994)Google Scholar
  14. 14.
    Solferino, V., Squassina, M.: Diffeomorphism-invariant properties for quasi-linear elliptic operators. J. Fixed Point Theory Appl. 11(1), 137–157 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Struwe, M.: A note on a result of Ambrosetti and Mancini. Ann. Mat. Pura Appl. (4) 131, 107–115 (1982)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità Cattolica del Sacro CuoreBresciaItaly
  2. 2.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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