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The \(\infty \)-eigenvalue problem with a sign-changing weight

  • Uriel Kaufmann
  • Julio D. Rossi
  • Joana TerraEmail author
Article
  • 25 Downloads

Abstract

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a smooth bounded domain and \(m\in C(\overline{\Omega })\) be a sign-changing weight function. For \(1<p<\infty \), consider the eigenvalue problem
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u=\lambda m(x)|u|^{p-2}u &{}\quad \text {in}\;\; \Omega ,\\ u=0 &{}\quad \text {on}\;\; \partial \Omega , \end{array} \right. \end{aligned}$$
where \(\Delta _{p}u\) is the usual p-Laplacian. Our purpose in this article is to study the limit as \(p\rightarrow \infty \) for the eigenvalues \(\lambda _{k,p}\left( m\right) \) of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when \(k=1\).

Keywords

Infinity Laplacian Eigenvalues Sign-changing weight Viscosity solutions 

Mathematics Subject Classification

35P15 35P30 35J60 

Notes

Acknowledgements

The research of UK was partially funded by Secyt-UNC 33620180100016CB (Argentina). The research of JDR was partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by MINECO MTM2015-70227-P (Spain). JT was partially supported by ANPCyT grant PICT 2016-1054 (Argentina) and by Secyt-UNC 33620180100016CB (Argentina).

References

  1. 1.
    Arias, M., Campos, J., Cuesta, M., Gossez, J.-P.: Asymmetric elliptic problems with indefinite weights. Ann. Inst. H. Poincaré Anal. Non Linéaire 19, 581–616 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anane, A.: Simplicité et isolation de la première valeur propre du \(p\)-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305, 725–728 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional p-Laplacian. Discrete Contin. Dyn. Syst. 36, 1813–1845 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Belloni, M., Kawohl, B.: The pseudo-\(p\)-Laplace eigenvalue problem and viscosity solutions as \(p\rightarrow \infty \). ESAIM Control Optim. Calc. Var. 10, 28–52 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Belloni, M., Kawohl, B., Juutinen, P.: The \(p\)-Laplace eigenvalue problem as \(p\rightarrow \infty \) in a Finsler metric. J. Eur. Math. Soc. 8, 123–138 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brasco, L., Franzina, G.: A pathological example in nonlinear spectral theory. Adv. Nonlinear Anal. 8(1), 707–714 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Champion, T., De Pascale, L., Jimenez, C.: The \(\infty \)-eigenvalue problem and a problem of optimal transportation. Commun. Appl. Anal. 13, 547–565 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Crasta, G., Fragalà, I.: Rigidity results for variational infinity ground states. arXiv:1702.01043v
  10. 10.
    Cuesta, M.: Eigenvalue problems for the p-Laplacian with indefinite weights. Electron. J. Differ. Equ. 2001(33), 9 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cuesta, M., Ramos Quoirin, H.: A weighted eigenvalue problem for the \(p\)-Laplacian plus a potential. NoDEA Nonlinear Differ. Equ. Appl. 16, 469–491 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Franzina, G., Lamberti, P.D.: Existence and uniqueness for a \(p\)-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Equ. 2010(26), 1–10 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Math. Univ. Parma (N.S.) 5, 373–386 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    García Azorero, J.P., Peral Alonso, I.: Existence and nonuniqueness for the \(p\)-Laplacian: nonlinear eigenvalues, Comm. Partial Differ. Equ. 12, 1389–1430 (1987)zbMATHGoogle Scholar
  15. 15.
    Hynd, R., Smart, C.K., Yu, Y.: Nonuniqueness of infinity ground states. Calc. Var. Partial Differ. Equ. 48, 545–554 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Juutinen, P., Lindqvist, P.: On the higher eigenvalues for the \(\infty \)-eigenvalue problem. Calc. Var. Partial Differ. Equ. 23, 169–192 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Juutinen, P., Lindqvist, P., Manfredi, J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148, 89–105 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Juutinen, P.-, Lindqvist, P., Manfredi, J.J.: On the equivalence of viscosity solutions and weak solutions for a quasilinear equation. SIAM J. Math. Anal. 33, 699–717 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kawohl, B., Lindqvist, P.: Positive eigenfunctions for the \(p\)-Laplace operator revisited. Analysis 26, 539–544 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lindgren, E.: The \(\infty \)-harmonic potential is not always an \(\infty \)-eigenfunction. PreprintGoogle Scholar
  21. 21.
    Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49, 795–826 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lindqvist, P.: On the equation \({{\rm div}} (|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109, 157–164 (1990)zbMATHGoogle Scholar
  23. 23.
    Manfredi, J.J., Rossi, J.D., Urbano, J.M.: \(p(x)\)-Harmonic functions with unbounded exponent in a subdomain. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2581–2595 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Navarro, J.C., Rossi, J.D., San Antolin, A., Saintier, N.: The dependence of the first eigenvalue of the infinity Laplacian with respect to the domain. Glasg. Math. J. 56, 241–249 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  26. 26.
    Smets, D.: A concentration-compactness lemma with applications to singular eigenvalue problems. J. Funct. Anal. 167, 463–480 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Szulkin, A., Willem, M.: Eigenvalue problems with indefinite weight. Stud. Math. 135, 191–201 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yu, Y.: Some properties of the ground states of the infinity Laplacian. Indiana Univ. Math. J. 56, 947–964 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Depto. de Matemática FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.FaMAFUniversidad Nacional de CórdobaCórdobaArgentina

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