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Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance

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Abstract

We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian. The hypotheses on the reaction term incorporate problems resonant at both ±∞ and zero. We consider both cases p>2 and 1<p<2 (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case p>2 we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method.

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References

  1. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 70 pp. (2008)

    Google Scholar 

  2. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: On p-superlinear equations with a nonhomogeneous differential operator. Nonlinear Differ. Equ. Appl. 20, 151–175 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amann, H.: Saddle points and multiple solutions of differential equations. Math. Z. 169, 127–166 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arcoya, D., Ruiz, D.: The Ambrosetti-Prodi problem for the p-Laplace operator. Commun. Partial Differ. Equ. 31, 849–865 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bartsch, T.: Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117–152 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bartsch, T., Li, S.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419–441 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Benci, V., D’Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Benguria, R., Brezis, H., Lieb, E.H.: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezis, H.: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brezis, H., Nirenberg, L.: H 1 versus C 1 local minimizers. C. R. Acad. Sci. Paris, Sér. I. 317, 465–472 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Castro, A., Lazer, A.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120, 113–137 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chang, K.C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  13. Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)

    Book  MATH  Google Scholar 

  14. Cingolani, S., Degiovanni, M.: Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cingolani, S., Vannella, G.: Critical groups computations on a class of Sobolev Banach spaces via Morse index. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, 271–292 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cingolani, S., Vannella, G.: Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces. Ann. Mat. Pura Appl. 186, 155–183 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Diaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris, Sér. I. 305, 521–524 (1987)

    MATH  MathSciNet  Google Scholar 

  18. Dugundji, J.: Topology. Alynn & Bacon, Boston (1966)

    MATH  Google Scholar 

  19. Dunford, N., Schwartz, J.: Linear Operators I. Wiley-Interscience, New York (1958)

    MATH  Google Scholar 

  20. Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Filippakis, M., Kristaly, A., Papageorgiou, N.S.: Existence of five nonzero solutions with exact sign for a p-Laplacian equation. Discrete Contin. Dyn. Syst. 24, 405–440 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. de Figueiredo, D.: Positive solutions of semilinear elliptic problems. In: Differential Equations, Sao Paulo, 1981. Lecture Notes in Math., vol. 957, pp. 34–87. Springer, Berlin (1982)

    Chapter  Google Scholar 

  23. de Figueiredo, D.: Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 81. Published for the Tata Institute of Fundamental Research, Bombay; by Springer, Berlin, 1989

  24. Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  25. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Book  MATH  Google Scholar 

  26. He, C., Li, G.: The existence of a nontrivial solution to the p & q Laplacian problem with nonlinearity asymptotic to u p−1 at infinity in \({\mathbb{R}}^{N}\). Nonlinear Anal. 65, 1110–1119 (2006)

    Google Scholar 

  27. Jiu, Q., Su, J.: Existence and multiplicity results for Dirichlet problems with p-Laplacian. J. Math. Anal. Appl. 281, 587–601 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kristaly, A., Rădulescu, V., Varga, C.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  29. Ladyzhenskaya, O.A., Uraltseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  30. Lieberman, G.: The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu, S.: Remarks on multiple solutions for elliptic resonant problems. J. Math. Anal. Appl. 336, 498–505 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Liu, S., Li, S.: Critical groups at infinity, saddle point reduction and elliptic resonant problems. Commun. Contemp. Math. 5, 761–773 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Liu, J., Liu, S.: The existence of multiple solutions to quasilinear elliptic equations. Bull. Lond. Math. Soc. 37, 592–600 (2005)

    Article  MATH  Google Scholar 

  34. Liu, J., Su, J.: Remarks on multiple nontrivial solutions for quasilinear resonant problems. J. Math. Anal. Appl. 258, 209–222 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Maunder, C.R.F.: Algebraic Topology. Dover, New York (1996)

    Google Scholar 

  36. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  37. Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Existence and multiplicity of solutions for asympotically linear, noncoercive, elliptic equations. Monatshefte Math. 159, 59–80 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  39. Papageorgiou, N.S., Smyrlis, G.: On nonlinear nonhomogeneous resonant Dirichlet equations. Pac. J. Math. 264, 421–453 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  41. Rădulescu, V.: Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods. Contemporary Mathematics and Its Applications, vol. 6. Hindawi, New York (2008)

    Book  Google Scholar 

  42. Sun, M.: Multiplicity of solutions for a class of quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386, 661–668 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Thews, K.: Nontrivial solutions of elliptic equations at resonance. Proc. R. Soc. Edinb. 85A, 119–129 (1980)

    Article  MathSciNet  Google Scholar 

  44. Vazquez, J.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  45. Zhang, Z., Li, S., Liu, S., Feng, W.: On an asymptotically linear elliptic Dirichlet problem. Abstr. Appl. Anal. 7, 509–516 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the two referees for their corrections and remarks that improved the paper considerably. V. Rădulescu acknowledges the support through Grant CNCS PCE-47/2011.

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Authors and Affiliations

  1. Department of Mathematics, National Technical University, Zografou Campus, Athens, 15780, Greece

    Nikolaos S. Papageorgiou

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Vicenţiu D. Rădulescu

  3. Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

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  1. Nikolaos S. Papageorgiou
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  2. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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Papageorgiou, N.S., Rădulescu, V.D. Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance. Appl Math Optim 69, 393–430 (2014). https://doi.org/10.1007/s00245-013-9227-z

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