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Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term

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Abstract

We consider a nonlinear Robin problem driven by a general nonhomogeneous differential operator plus an indefinite potential term. The reaction is of generalized logistic type. Using variational tools we prove a multiplicity theorem producing three nontrivial solutions with sign information (positive, negative and nodal). In the particular case of (p, 2)-equations, employing also critical groups, we produce a second nodal solution. Our results extend earlier multiplicity results for coercive problems.

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References

  1. Ambrosetti, A., Lupo, D.E.: On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal. 8, 1145–1150 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal. 3, 635–645 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gasiński, L., Papageorgiou, N.S.: Exercises in Analysis. Part 2: Nonlinear Analysis. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  4. Gasiński, L., Papageorgiou, N.S.: Positive solutions for the Robin \(p\)-Laplacian problem with competing nonlinearities. Adv. Calc. Var. 12, 31–56 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, Volume I: Theory. Rluwer Academic Publisher, Dordrecht (1997)

    Book  MATH  Google Scholar 

  6. Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Part. Differ. Equ. 16, 311–361 (1991)

    Article  MATH  Google Scholar 

  7. Liu, Y.J., Migórski, S., Nguyen, V.T., Zeng, S.D.: Existence and convergence results for elastic frictional contact problem with nonmonotone subdifferential boundary condtions. Acta Math. Sci. 41, 1–18 (2021)

    Article  MATH  Google Scholar 

  8. Liu, Y.J., Liu, Z.H., Wen, C.-F., Yao, J.-C., Zeng, S.D.: Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems. Appl. Math. Optim. 84, 2037–2059 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, Z.H., Motreanu, D., Zeng, S.D.: Generalized penalty and regularization method for differential variational-hemivariational inequalities. SIAM J. Optim. 31, 1158–1183 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)

    Book  MATH  Google Scholar 

  11. Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Scuola Norm. Super. Pisa-Cl. Sci. 11, 729–788 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Papageorgiou, N.S., Rǎdulescu, V.D.: Robin problems with indefinite, unbounded potential and reaction of arbitrary growth. Rev. Mat. Complutense 29, 91–126 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Papageorgiou, N.S., Rǎdulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16, 737–764 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Papageorgiou, N.S., Rǎdulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69, 393–430 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Papageorgiou, N.S., Rǎdulescu, V.D., Repovš, D.D.: Nonlinear Analysis Theory and Methods. Springer, Switzerland (2019)

    Book  MATH  Google Scholar 

  16. Papageorgiou, N.S., Rǎdulescu, V.D., Repovš, D.D.: Nonlinear, nonhomogeneous Robin problems with indefinite potential and general reaction. Appl. Math. Optim. 81, 823–857 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Papageorgiou, N.S., Rǎdulescu, V.D., Repovš, D.D.: Anisotropic equations with indefinite potential and competing nonlinearities. Nonlinear Anal. 201, 111861 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Papageorgiou, N.S., Scapellato, A.: Multiple solutions for Robin \((p, q)\)-equations plus an indefinite potential and a reaction concave near the origin. Anal. Math. Phys. 11, 27 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Papageorgiou, N.S., Vetro, C., Vetro, F.: Pairs of solutions for Robin problems with an indefinite and unbounded potential, resonant at zero and infinity. Manusc. Math. 158, 487–503 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Papageorgiou, N.S., Vetro, C., Vetro, F.: Parameter dependence for the positive solutions of nonlinear, nonhomogeneous Robin problems. RACSAM 114, 29 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pucci, P., Serrin, J.B.: The Maximum Principle. Springer, Basel (2007)

    Book  MATH  Google Scholar 

  22. Struwe, M.: A note on a result of Ambrosetti and Mancini. Annali Mate. Pura Appl. 131, 107–115 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Struwe, M.: Variational Methods. Springer, Berlin (2008)

    MATH  Google Scholar 

  24. Zeng, S.D., Migórski, S., Liu, Z.H.: Nonstationary incompressible Navier–Stokes system governed by a quasilinear reaction–diffusion equation. Sci. Sin. Math. 52, 331–354 (2022). (in Chinese)

    Article  Google Scholar 

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Acknowledgements

The authors are immensely grateful to the reviewers for their careful reading and suggestions that brought substantial improvements to the manuscript.

Funding

This project was supported by the Natural Science Foundation of Guangxi Grants Nos. 2021GXNSFFA196004 and GKAD21220144, the NNSF of China Grant Nos. 12001478 and 12101143, the China Postdoctoral Science Foundation funded project No. 2022M721560, the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611.

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Authors and Affiliations

  1. School of Science, Guangxi University of Science and Technology, Liuzhou, 545006, Guangxi, People’s Republic of China

    Yunru Bai

  2. Department of Mathematics, National Technical University, Zograrou Compus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  3. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, Guangxi, People’s Republic of China

    Shengda Zeng

  4. Department of Mathematics, Nan**g University, Jiangsu, 210093, Nan**g, People’s Republic of China

    Shengda Zeng

  5. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30-348, Krakow, Poland

    Shengda Zeng

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  1. Yunru Bai
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  2. Nikolaos S. Papageorgiou
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  3. Shengda Zeng
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Correspondence to Shengda Zeng.

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Bai, Y., Papageorgiou, N.S. & Zeng, S. Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term. Results Math 78, 134 (2023). https://doi.org/10.1007/s00025-023-01907-5

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