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Two Nontrivial Solutions for Robin Problems Driven by a p–Laplacian Operator

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Differential and Difference Equations with Applications (ICDDEA 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 333))

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Abstract

By variational methods and critical point theorems, we show the existence of two nontrivial solutions for a nonlinear elliptic problem under Robin condition and when the nonlinearty satisfies the usual Ambrosetti-Rabinowitz condition.

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References

  1. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    MathSciNet  MATH  Google Scholar 

  2. Averna, D., Papageorgiou, N.S., Tornatore, E.: Positive solutions for nonlinear Robin problems. Electron. J. Differ. Equ. 2017(204), 1–25 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bonanno, G.: Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal. 1, 205–220 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Bonanno, G.: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75(5), 2992–3007 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Bonanno, G., D’Aguì, G.: Two non-zero solutions for elliptic Dirichlet problems. Z. Anal. Anwend 35, 449–464 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer, New York (2007)

    MATH  Google Scholar 

  7. Drabek, P., Schindler, I.: Positive solutions for the p-Laplacian with Robin boundary conditions on irregular domains. Appl. Math. Lett. 24, 588–591 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

    MATH  Google Scholar 

  9. Marano, S.A., Motreanu, D.: Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian. J. Differ. Equ. 182, 108–120 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Marano, S.A., Motreanu, D.: On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems. Nonlinear Anal. 48(1), 37–52 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Motreanu, D., Sciammetta. A., Tornatore, E.: A sub-supersolution approach for Robin boundary value problem with full gradient dependence. Mathematics 658(8) (2020)

    Google Scholar 

  13. Papageorgiou, N.S., Radulescu, V.D.: Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differ. Equ. 256, 2449–2479 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Papageorgiou, N.S., Radulescu, V.D.: Bifurcation near infinity for the Robin p-Laplacian. Manuscripta Math. 148, 415–433 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Papageorgiou, N.S., Radulescu, V.D., Repovs, D.: Positive solutions for nonlinear nonhomogeneous Robin problems. Forum Math. 30, 553–580 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Ricceri, B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 113(1–2), 401–410 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75(3), 220–226 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The paper is partially supported by PRIN 2017—Progetti di Ricerca di rilevante Interesse Nazionale, “Nonlinear Differential Problems via Variational, Topological and Set-valued Methods” (2017AYM8XW).

Author information

Authors and Affiliations

  1. Department of Engineering, University of Messina, 98166, Messina, Italy

    G. D’Aguì

  2. Department of Mathematics and Computer Sciences, University of Palermo, 90123, Palermo, Italy

    A. Sciammetta & E. Tornatore

Authors
  1. G. D’Aguì
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  2. A. Sciammetta
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  3. E. Tornatore
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Corresponding author

Correspondence to E. Tornatore .

Editor information

Editors and Affiliations

  1. Department of Exact Sciences and Engineering, Military Academy, Lisbon, Portugal

    Sandra Pinelas

  2. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, USA

    John R. Graef

  3. Mathematik und Didaktik, Katholische Universität Eichstätt, Eichstätt, Bayern, Germany

    Stefan Hilger

  4. Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  5. Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Xanthi, Greece

    Christos Schinas

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D’Aguì, G., Sciammetta, A., Tornatore, E. (2020). Two Nontrivial Solutions for Robin Problems Driven by a p–Laplacian Operator. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_16

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