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Some uniqueness results in quasilinear subhomogeneous problems

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Abstract

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in Bonheure et al. (Trans Amer Math Soc 370(10):7081–7127, 2018). We apply it to generalized p-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called (p, r)-Laplacian are also treated.

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  • 22 February 2021

    The author’s family name was wrongly published and it is corrected now

Notes

  1. This property, known as (generalized) hidden convexity, plays an important role in several uniqueness results [5, 6, 8, 22], and has connections with Hardy and Picone inequalities, as shown in [9].

References

  1. Alama, S.: Semilinear elliptic equations with sublinear indefinite nonlinearities. Adv. Differ. Equ. 4, 813–842 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids (French, with English summary). C. R. Acad. Sci. Paris Ser. I Math. 305(16), 725–728 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Bandle, C., Pozio, M., Tesei, A.: The asymptotic behavior of the solutions of degenerate parabolic equations. Trans. Amer. Math. Soc. 303, 487–501 (1987)

    Article  MathSciNet  Google Scholar 

  4. Bandle, C., Pozio, M., Tesei, A.: Existence and uniqueness of solutions of nonlinear Neumann problems. Math. Z. 199, 257–278 (1988)

    Article  MathSciNet  Google Scholar 

  5. Belloni, M., Kawohl, B.: A direct uniqueness proof for equations involving the \(p\)-Laplace operator. Manuscr. Math. 109, 229–231 (2002)

    Article  MathSciNet  Google Scholar 

  6. Benguria, R., Brezis, H., Lieb, E.H.: The Thomas–Fermi–von Weizsäcker theory of atoms and molecules. Comm. Math. Phys. 79, 167–180 (1981)

    Article  MathSciNet  Google Scholar 

  7. Bobkov, V., Tanaka, M.: Remarks on minimizers for \((p,q)\)-Laplace equations with two parameters. Commun. Pure App. Anal. 17(3), 1219–1253 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bonheure, D., Földes, J., Moreira dos Santos, E., Saldaña, A., Tavares, H.: Paths to uniqueness of critical points and applications to partial differential equations. Trans. Amer. Math. Soc. 370(10), 7081–7127 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)

    Article  MathSciNet  Google Scholar 

  11. Delgado, M., Suárez, A.: On the uniqueness of positive solution of an elliptic equation. Appl. Math. Lett. 18, 1089–1093 (2005)

    Article  MathSciNet  Google Scholar 

  12. Díaz, J.I.: New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems. J. Pure Appl. Funct. Anal. 5(4), 925–949 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Díaz, J.I.: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations. Research Notes in Mathematics, 106. Pitman, Boston, MA (1985)

  14. Díaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires (French) [Existence and uniqueness of positive solutions of some quasilinear elliptic equations]. C. R. Acad. Sci. Paris Sér. I Math. 305, 521–524 (1987)

    MathSciNet  MATH  Google Scholar 

  15. DiBenedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)

    Article  MathSciNet  Google Scholar 

  16. Kajikiya, R., Sim, I., Tanaka, S.: A complete classification of bifurcation diagrams for a class of \((p,q)\)-Laplace equations. J. Math. Anal. Appl. 462(2), 1178–1194 (2018)

    Article  MathSciNet  Google Scholar 

  17. Kaufmann, U., Ramos Quoirin, H., Umezu, K.: Positivity results for indefinite sublinear elliptic problems via a continuity argument. J. Differ. Equ. 263, 4481–4502 (2017)

    Article  MathSciNet  Google Scholar 

  18. Kaufmann, U., Ramos Quoirin, H., Umezu, K.: Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity. NoDEA Nonlinear Differential Equations Appl. 25, Art. 12 (2018)

  19. Kaufmann, U., Ramos Quoirin, H., Umezu, K.: A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete Contin. Dyn. Syst. 40, 617–645 (2020)

    Article  MathSciNet  Google Scholar 

  20. Kaufmann, U., Ramos Quoirin, H., Umezu, K.: Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. arXiv:2001.11318 (2020)

  21. Kaufmann, U., Ramos Quoirin, H., Umezu, K.: Uniqueness and positivity issues in a quasilinear indefinite problem. arXiv:2007.09498 (2020)

  22. Kawohl, B., Krömer, S.: Uniqueness and symmetry of minimizers of Hartree type equations with external Coulomb potential. Adv. Calc. Var. 5, 427–432 (2012)

    Article  MathSciNet  Google Scholar 

  23. Kawohl, B., Lucia, M., Prashanth, S.: Simplicity of the principal eigenvalue for indefinite quasilinear problems. Adv. Differ. Equ. 12, 407–434 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    Article  MathSciNet  Google Scholar 

  25. Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhen–Skaya and Uraltseva for elliptic equations. Comm. Partial Differential Equations 16, 311–361 (1991)

    Article  MathSciNet  Google Scholar 

  26. Marano, S.A., Papageorgiou, N.S.: Constant-sign and nodal solutions of coercive \((p,q)\)-Laplacian problems. Nonlinear Anal. Theory Methods Appl. 77, 118–129 (2013)

    Article  MathSciNet  Google Scholar 

  27. Morales-Rodrigo, C., Suárez, A.: Uniqueness of solution for elliptic problems with non-linear boundary conditions. Comm. Appl. Nonlinear Anal. 13, 69–78 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Nazarov, A.I.: On the symmetry of extremals in the weight embedding theorem. Function theory and mathematical analysis. J. Math. Sci. (N.Y.) 107(3), 3841–3859 (2001)

    Article  Google Scholar 

  29. Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)

    Book  Google Scholar 

  30. Tanaka, M.: Uniqueness of a positive solution and existence of a sign-changing solution for \((p, q)\)-Laplace equation. J. Nonlinear Funct. Anal. 2014, 1–15 (2014)

    Google Scholar 

  31. Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)

    Article  MathSciNet  Google Scholar 

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  1. CIEM-FaMAF, Universidad Nacional de Córdoba, 5000, Córdoba, Argentina

    Humberto Ramos Quoirin

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Ramos Quoirin, H. Some uniqueness results in quasilinear subhomogeneous problems. Arch. Math. 116, 433–444 (2021). https://doi.org/10.1007/s00013-020-01560-2

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