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Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian

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Abstract

In this paper we extend two nowadays classical results to a nonlinear Dirichlet problem to equations involving the fractional p-Laplacian. The first result is an existence in a non-resonant range more specific between the first and second eigenvalue of the fractional p-Laplacian. The second result is the anti-maximum principle for the fractional p-Laplacian.

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References

  1. Adams, R.A.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)

    Google Scholar 

  2. Amghibech, S.: On the discrete version of Picone’s identity. Discrete Appl. Math. 156(1), 1–10 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anane, A.: Simplicité et isolation de la première valeur propre du \(p\)-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305(16), 725–728 (1987)

    MATH  MathSciNet  Google Scholar 

  4. Anane, A., Gossez, J.: Strongly nonlinear elliptic problems near resonance: a variational approach. Commun. Partial Differ. Equ. 15(8), 1141–1159 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anane, A., Tsouli, N.: On a nonresonance condition between the first and the second eigenvalues for the \(p\) -Laplacian. Int. J. Math. Math. Sci. 26(10), 625–634 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anane, A., Tsouli, N.: On a nonresonance condition between the first and the second eigenvalue for the \(p\)-Laplacian. Math. Rech. Appl. 6, 101–114 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Arcoya, D., Colorado, E., Leonori, T.: Asymptotically linear problems and antimaximum principle for the square root of the Laplacian. Adv. Nonlinear Stud. 12(4), 683–701 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Arcoya, D., Gámez, J.L.: Bifurcation theory and related problems: anti-maximum principle and resonance. Commun. Partial Differ. Equ. 26(9–10), 1879–1911 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Arcoya, D., Orsina, L.: Landesman-Lazer conditions and quasilinear elliptic equations. Nonlinear Anal. 28(10), 1623–1632 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Arias, M., Campos, J., Gossez, J.-P.: On the antimaximum principle and the Fučik spectrum for the Neumann \(p\)-Laplacian. Differ. Integral Equ. 13(1–3), 217–226 (2000)

    MATH  Google Scholar 

  11. Armstrong, S.N.: Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations. J. Differ. Equ. 246(7), 2958–2987 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Birindelli, I.: Hopf’s lemma and anti-maximum principle in general domains. J. Differ. Equ. 119(2), 450–472 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Boccardo, L., Drábek, P., Giachetti, D., Kučera, M.: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 10, 1083–1103 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bourgain, J., Brezis, H., Mironescu, P.: Another look at sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (Eds.) Optimal Control and Partial Differential Equations, A Volume in Honour of A. Bensoussan’s 60th Birthday, pp. 439–455. IOS Press (2001)

  15. Brasco, L., Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. 37(3), 769–799 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brasco, L., Parini, E.: The second eigenvalue of the fractional \(p\)-Laplacian. Adv. Calc. Var. 9(4), 323–355 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  17. Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional \(p\)-Laplacian. Discrete Contin. Dyn. Syst. 36(4), 1813–1845 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  18. Clément, Ph, Peletier, L.A.: An anti-maximum principle for second-order elliptic operators. J. Differ. Equ. 34(2), 218–229 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cuesta, M., de Figueiredo, D., Gossez, J.-P.: The beginning of the Fučik spectrum for the \(p\)-Laplacian. J. Differ. Equ. 159(1), 212–238 (1999)

    Article  MATH  Google Scholar 

  20. Del Pezzo, L.M., Fernández Bonder, J., López Ros, L.: An optimization problem for the first eigenvalue of the p-fractional Laplacian. arXiv:1601.03019

  21. Del Pezzo, L.M., Quaas, A.: Global bifurcation for fractional \(p\)-Laplacian and application. Z. Anal. Anwend. 35(4), 411–447 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  22. Del Pezzo, L.M., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional \(p\)-Laplacian. arXiv:1609.04725

  23. del Pino, M., Drábek, P., Manásevich, R.: The Fredholm alternative at the first eigenvalue for the one-dimensional \(p\)-Laplacian. J. Differ. Equ. 151(2), 386–419 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  25. Demengel, F., Demengel, G.: Functional spaces for the theory of elliptic partial differential equations. Universitext, Springer, London (2012) (Translated from the 2007 French original by Reinie Erné)

  26. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Dolph, C.L.: Nonlinear integral equations of the Hammerstein type. Trans. Am. Math. Soc. 66, 289–307 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  28. Drábek, P.: The \(p\)-Laplacian–Mascot of nonlinear analysis. Acta Math. Univ. Comenian. (N.S.) 76(1), 85–98 (2007)

    MATH  MathSciNet  Google Scholar 

  29. 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)

    Article  MATH  MathSciNet  Google Scholar 

  30. Drábek, P., Girg, P., Manásevich, R.: Generic Fredholm alternative-type results for the one dimensional \(p\)-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 8(3), 285–298 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Drábek, P., Takáč, P.: A counterexample to the Fredholm alternative for the \(p\)-Laplacian. Proc. Am. Math. Soc. 127(4), 1079–1087 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fleckinger, J., Gossez, J.-P., Takáč, P.: François de Thélin, existence, nonexistence et principe de l’antimaximum pour le \(p\)-Laplacien. C. R. Acad. Sci. Paris Sér. I Math. 321(6), 731–734 (1995)

    MathSciNet  Google Scholar 

  33. Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Math. Univ. Parma (N.S.) 5(2), 373–386 (2014)

    MATH  MathSciNet  Google Scholar 

  34. García-Melián, J., Rossi, J.D.: Maximum and antimaximum principles for some nonlocal diffusion operators. Nonlinear Anal. 71(12), 6116–6121 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Godoy, T., Gossez, J.-P., Paczka, S.: On the antimaximum principle for the \(p\)-Laplacian with indefinite weight. Nonlinear Anal. 51(3), 449–467 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Gossez, J.-P.: Sur le principe de l’antimaximum. Cahiers Centre Études Rech. Opér. 36, 183–187 (1994) (Hommage à Simone Huyberechts)

  37. Greco, A., Servadei, R.: Hopf’s lemma and constrained radial symmetry for the fractional Laplacian. (English summary). Math. Res. Lett. 23(3), 863–885 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  38. Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

    MATH  Google Scholar 

  39. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Revista Matemática Iberoamericana (2014) (To appear)

  40. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49(1–2), 795–826 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  41. Lindqvist, P.: On the equation \({\rm div}\,(\vert \nabla u\vert ^{p-2}\nabla u)+\lambda \vert u\vert ^{p-2}u=0\). Proc. Am. Math. Soc. 109(1), 157–164 (1990)

    MATH  MathSciNet  Google Scholar 

  42. Lindqvist, P.: Addendum: “On the equation \({\rm div}(\vert \nabla u\vert ^{p-2}\nabla u)+\lambda \vert u\vert ^{p-2}u=0\)”. Proc. Am. Math. Soc. 116(2), 583–584 (1992)

    MATH  Google Scholar 

  43. Manásevich, R., Takáč, P.: On the Fredholm alternative for the \(p\)-Laplacian in one dimension. Proc. Lond. Math. Soc. (3) 84(2), 324–342 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  44. Parini, E.: Continuity of the variational eigenvalues of the \(p\)-Laplacian with respect to \(p\). Bull. Aust. Math. Soc. 83(3), 376–381 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  45. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures et Appl. (9) 101(3), 275–302 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  46. Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)

    MATH  MathSciNet  Google Scholar 

  47. Servadei, R., Valdinoci, E.: A Brezis–Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12(6), 2445–2464 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. Sect. A 144(4), 831–855 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  49. Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58(1), 133–154 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  50. Takáč, P.: On the Fredholm alternative for the \(p\)-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51(1), 187–237 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  51. Takáč, P.: A variational approach to the Fredholm alternative for the \(p\)-Laplacian near the first eigenvalue. J. Dyn. Differ. Equ. 18(3), 693–765 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

L. M. Del Pezzo was partially supported by CONICET PIP 5478/1438 (Argentina) and A. Quaas was partially supported by Fondecyt Grant No. 1110210, Millennium Nucleus Center for Analysis of PDE NC130017 and Basal CMM UChile.

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Authors and Affiliations

  1. CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina

    Leandro M. Del Pezzo

  2. Departamento de Matemática, Universidad Técnica Federico Santa María Casilla V-110, Avda. España, 1680, Valparaiso, Chile

    Alexander Quaas

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  1. Leandro M. Del Pezzo
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  2. Alexander Quaas
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Correspondence to Alexander Quaas.

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Dedicated to Paul H. Rabinowitz.

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Del Pezzo, L.M., Quaas, A. Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian. J. Fixed Point Theory Appl. 19, 939–958 (2017). https://doi.org/10.1007/s11784-017-0405-5

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