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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1651–1673 | Cite as

On the Fredholm-type theorems and sign properties of solutions for (pq)-Laplace equations with two parameters

  • Vladimir BobkovEmail author
  • Mieko Tanaka
Article

Abstract

We consider the Dirichlet problem for the nonhomogeneous equation \(-\Delta _p u -\Delta _q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)\) in a bounded domain, where \(p \ne q\), and \(\alpha , \beta \in \mathbb {R}\) are parameters. We explore assumptions on \(\alpha \) and \(\beta \) that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the \((\alpha ,\beta )\)-plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided f is of a constant sign.

Keywords

(p, q)-Laplacian Fredholm alternative Existence of solutions Positive solutions Maximum principle Linking method 

Mathematics Subject Classification

35J62 35J20 35P30 35B50 

Notes

Acknowledgements

V. Bobkov was supported by the Grant 18-03253S of the Grant Agency of the Czech Republic and by the project LO1506 of the Czech Ministry of Education, Youth and Sports. M. Tanaka was supported by JSPS KAKENHI Grant Number 15K17577. The authors would like to thank the anonymous referee for valuable remarks and suggestions which helped to improve the manuscript.

References

  1. 1.
    Alama, S., Tarantello, G.: Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141(1), 159–215 (1996).  https://doi.org/10.1006/jfan.1996.0125 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allegretto, W., Huang, Y.: A Picone’s identity for the \(p\)-Laplacian and applications. Nonlinear Anal. Theory Methods Appl 32(7), 819–830 (1998).  https://doi.org/10.1016/S0362-546X(97)00530-0 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994).  https://doi.org/10.1006/jfan.1994.1078 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anane, A.: Simplicité et isolation de la premiere valeur propre du \(p\)-laplacien avec poids. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 305(16), 725-728 (1987). http://gallica.bnf.fr/ark:/12148/bpt6k57447681/f27
  5. 5.
    Averna, D., Motreanu, D., Tornatore, E.: Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence. Appl. Math. Lett. 61, 102–107 (2016).  https://doi.org/10.1016/j.aml.2016.05.009 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bobkov, V., Tanaka, M.: On positive solutions for \((p, q)\)-Laplace equations with two parameters. Calc. Var. Partial Differ. Equ. 54(3), 3277–3301 (2015).  https://doi.org/10.1007/s00526-015-0903-5 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bobkov, V., Tanaka, M.: On sign-changing solutions for \((p, q)\)-Laplace equations with two parameters. Adv. Nonlinear Anal. (2016).  https://doi.org/10.1515/anona-2016-0172 CrossRefzbMATHGoogle Scholar
  8. 8.
    Bobkov, V., Tanaka, M.: Remarks on minimizers for \((p, q)\)-Laplace equations with two parameters. Commun. Pure Appl. Anal. 17(3), 1219–1253 (2018).  https://doi.org/10.3934/cpaa.2018059 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, K.C.: Infinite dimensional morse theory and multiple solution problems. Birkhäuser (1993).  https://doi.org/10.1007/978-1-4612-0385-8 CrossRefzbMATHGoogle Scholar
  10. 10.
    Chaves, M.F., Ercole, G., Miyagaki, O.H.: Existence of a nontrivial solution for the \((p, q)\)-Laplacian in \(\mathbb{R}^N\) without the Ambrosetti–Rabinowitz condition. Nonlinear Anal. Theory Methods Appl. 114, 133–141 (2015).  https://doi.org/10.1016/j.na.2014.11.010 CrossRefzbMATHGoogle Scholar
  11. 11.
    Clément, P., Peletier, L.A.: An anti-maximum principle for second-order elliptic operators. J. Differ. Equ. 34(2), 218–229 (1979).  https://doi.org/10.1016/0022-0396(79)90006-8 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Drábek, P.: Geometry of the energy functional and the Fredholm alternative for the \(p\)-Laplacian in higher dimensions. In: Electronic Journal of Differential Equations, Conference 08, 103-120. (2002) https://ejde.math.txstate.edu/conf-proc/08/d1/drabek.pdf
  13. 13.
    Drábek, P., Girg, P., Takáč, P., Ulm, M.: The Fredholm alternative for the \(p\)-Laplacian: bifurcation from infinity, existence and multiplicity. Indiana Univ. Math. J. 53(2), 433–482. (2004) http://www.jstor.org/stable/24903516 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Drábek, P., Robinson, S.B.: Resonance problems for the \(p\)-Laplacian. J. Funct. Anal. 169(1), 189–200 (1999).  https://doi.org/10.1006/jfan.1999.3501 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dugundji, J.: An extension of Tietze’s theorem. Pac. J. Math. 1(3), 353–367 (1951). https://projecteuclid.org/euclid.pjm/1103052106 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Il’yasov, Y.: On positive solutions of indefinite elliptic equations. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 333(6), 533–538 (2001).  https://doi.org/10.1016/S0764-4442(01)01924-3 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Il’yasov, Y.S.: Bifurcation calculus by the extended functional method. Funct. Anal. Appl. 41(1), 18–30 (2007).  https://doi.org/10.1007/s10688-007-0002-2 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Filippakis, M.E., Papageorgiou, N.S.: Resonant \((p,q)\)-equations with Robin boundary condition. Electron. J. Differ. Equ. 2018(1), 1–24 (2018). https://ejde.math.txstate.edu/Volumes/2018/01/filippakis.pdf
  19. 19.
    Fleckinger, J., Gossez, J.-P., Takáč, P., & de Thélin, F.: Existence, nonexistence et principe de l’antimaximum pour le \(p\)-laplacien. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 321(6), 731–734 (1995) http://gallica.bnf.fr/ark:/12148/bpt6k62037127/f81
  20. 20.
    Fleckinger-Pellé J., Takáč, P.: An improved Poincaré inequality and the \(p\)-Laplacian at resonance for \(p>2\). Adv. Differ. Equ. 7(8), 951–971. http://projecteuclid.org/euclid.ade/1356651685
  21. 21.
    Fučík, S., Nečas, J., Souček, J., Souček, V.: Spectral Analysis of Nonlinear Operators, vol. 346. Springer, New York (2006).  https://doi.org/10.1007/BFb0059360 CrossRefzbMATHGoogle Scholar
  22. 22.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. Theory Methods Appl. 12(11), 1203–1219 (1988).  https://doi.org/10.1016/0362-546X(88)90053-3 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lieberman, G.M.: The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equ. 16(2–3), 311–361 (1991).  https://doi.org/10.1080/03605309108820761 CrossRefzbMATHGoogle Scholar
  24. 24.
    Marano, S., Mosconi, S.: Some recent results on the Dirichlet problem for \((p, q)\)-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 11(2), 279–291 (2017).  https://doi.org/10.3934/dcdss.2018015 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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).  https://doi.org/10.1016/j.jfa.2011.11.028 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Motreanu, D., Tanaka, M.: On a positive solution for \((p,q)\)-Laplace equation with indefinite weight. Minimax Theory Appl. 1, 1–20 (2016). http://www.heldermann-verlag.de/mta/mta01/mta0001-b.pdf
  27. 27.
    Pucci, P., Serrin, J.B.: The Maximum Principle, vol. 73. Springer, New York (2007).  https://doi.org/10.1007/978-3-7643-8145-5 CrossRefzbMATHGoogle Scholar
  28. 28.
    Takáč, P.: On the Fredholm alternative for the \(p\)-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51(1), 187–238 (2002).  https://doi.org/10.1512/iumj.2002.51.2156 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tanaka, M.: Generalized eigenvalue problems for \((p, q)\)-Laplacian with indefinite weight. J. Math. Anal. Appl. 419(2), 1181–1192 (2014).  https://doi.org/10.1016/j.jmaa.2014.05.044 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and NTIS, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.Department of MathematicsTokyo University of ScienceShinjyuku-kuJapan

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