Abstract

We consider a Dirichlet problem driven by a weighted (p,q)-Laplacian with a reaction that involves a critical term and a locally defined perturbation. Using variational tools and cut-off techniques, we show that the problem has a sequence of arbitrarily small nodal solutions.

Author information

Contact details are reproduced from the original publication and may be historical.

Zhenhai Liu

Guangxi Colleges and Universities, Key Laboratory of Complex System Optimization, Yulin Normal University, Yulin 537000, P. R. China
and: Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, Guangxi 530006, P. R. China

zhhliu@hotmail.com

Nikolaos S. Papageorgiou

Department of Mathematics, National Technical University, Athens, Greece

npapg@math.ntua.gr

Keywords

  • Weighted (p
  • q)-Laplacian
  • critical term
  • locally defined perturbation
  • nonlinear regularity
  • extremal constant sign solutions
  • nodal solutions
  • cut-off function

Mathematics Subject Classification

35J20, 35J60

References

  1. J. I. Diaz, J. E. Saa: Existence et unicité de solutions positives pour certaines equations elliptiques quasilinéarires, C. R. Acad. Sci. Paris 305 (1987) 521–524.Google Scholar
  2. L. Gasiński, N. S. Papageorgiou: Nonlinear Analysis, Chapman & Hall, Boca Raton (2006).Google Scholar
  3. L. Gasiński, N. S. Papageorgiou: Nodal solutions for nonlinear nonhomogeneous Robin problems with an indefinite potential, Proc. Edinburgh Math. Soc. 61 (2018) 943–959.https://doi.org/10.1017/S0013091518000044Google Scholar
  4. L. Gasiński, N. S. Papageorgiou: On a nonlinear parametric Robin problem with a locally defined redction, Nonlinear Analysis 185 (2019) 374–387.https://doi.org/10.1016/j.na.2019.03.019Google Scholar
  5. S. Hu, N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publishers, Dordrecht (1997).https://doi.org/10.1007/978-1-4615-6359-4Google Scholar
  6. R. Kajikiya: A critical point theorem related to the symmetric mountain pass theorem and its applications to elliptic equations, J. Funct. Analysis 225 (2005) 352–370.https://doi.org/10.1016/j.jfa.2005.04.005Google Scholar
  7. O. A. Ladyzhenskaya, N. N. Uraltseva: Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).Google Scholar
  8. Z. Li, Z. Q. Wang: Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Physik 56 (2005) 609–629https://doi.org/10.1007/s00033-005-3115-6Google Scholar
  9. G. Lieberman: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Diff. Equations 16/2-3 (1991) 311–361.https://doi.org/10.1080/03605309108820761Google Scholar
  10. N. S. Papageorgiou, V. D. Rădulescu, D. Repovš: Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discr. Cont. Dyn. Systems A 37 (2017) 2589–2618.https://doi.org/10.3934/dcds.2017111Google Scholar
  11. N. S. Papageorgiou, V. D. Rădulescu, D. Repovš: Double phase problems with reaction of arbitrary growth, Z. Angew. Math. Physik 69 (2018), art. no. 108, 21 pp.https://doi.org/10.1007/s00033-018-1001-2Google Scholar
  12. N. S. Papageorgiou, V. D. Rădulescu, D. Repovš: Nonlinear Analysis-Theory and Methods, Springer, Cham (2019).https://doi.org/10.1007/978-3-030-03430-6Google Scholar
  13. N. S. Papageorgiou, V. D. Rădulescu, D. Repovš: Nonlinear nonhomogeneous Robin problems with a indefinite potential and general reaction, Appl. Math. Optimization 81 (2020) 823–857.https://doi.org/10.1007/s00245-018-9521-xGoogle Scholar
  14. N. S. Papageorgiou, C. Vetro, F. Vetro: Multiple solutions for parametric double phase Dirichlet problems, Comm. Contemp. Math. 23/4 (2021), art. no. 2050006, 18 p.https://doi.org/10.1142/S0219199720500066Google Scholar
  15. P. Pucci, J. Serrin: The Maximum Principle, Birkhäuser, Basel (2007).https://doi.org/10.1007/978-3-7643-8145-5Google Scholar
  16. Z. Q. Wang: Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Diff. Equation Appl. 6 (2001) 15–33.https://doi.org/10.1007/PL00001436Google Scholar

Suggested citation

Z. Liu, N. S. Papageorgiou. “Nodal Solutions for a Weighted (p,q)-Equation.” Journal of Convex Analysis 29 (2022), No. 2, 559–570. https://doi.org/10.68381/jca29032

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.