Separable infinity harmonic functions in cones

  • Marie-Françoise Bidaut-Véron
  • Marta Garcia-Huidobro
  • Laurent Véron


We study the existence of separable infinity harmonic functions in any cone of \(\mathbb R^N\) vanishing on its boundary under the form \(u(r,\sigma )=r^{-\beta }\psi (\sigma )\). We prove that such solutions exist, the spherical part \(\psi \) satisfies a nonlinear eigenvalue problem on a subdomain of the sphere \(S^{N-1}\) and that the exponents \(\beta =\beta _+>0\) and \(\beta =\beta _-<0\) are uniquely determined if the domain is smooth. We extend some of our results to non-smooth domains.

Mathematics Subject Classification

35D40 35J70 35J62 



This article has been prepared with the support of the collaboration programs ECOS C14E08 and FONDECYT grant 1160540 for the three authors. The authors are grateful to the referee for a careful reading of their work.


  1. 1.
    Arronson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arronson, G.: On the equation \(u_xu_{xx}+2u_{xy}u_xu_y+u_yu_{yy}=0\). Ark. Mat. 7, 395–425 (1968)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandle, C., Marcus, M.: Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curvature of the boundary. Complex Var. 49, 555–570 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Bhattacharya, T.: A note on non-negative singular infinity-harmonic functions in the half-space. Rev. Mat. Complut. 18, 377–385 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bhattacharya, T.: A boundary Harnack principle for infinity-Laplacian and some related results. Bound. Val. Probl. 2007, 1–17 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Evans, E.C., Smart, ChK: Everywhere differentiability of infinity harmonic functions. Calc. Var. Part. Differ. Equ. 42, 289–299 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gkikas, K., Véron, L.: The spherical p-harmonic Eigenvalue problem in non-smooth domains. J. Funct. Anal. 274, 1155–1176 (2018). MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Krol, I.N.: The behaviour of the solutions of a certain quasilinear equation near zero cusps of the boundary. Proc. Steklov Inst. Math. 125, 140–146 (1973)Google Scholar
  9. 9.
    Katzourakis, N.: An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in \(L^\infty \). Springer Briefs in Mathematics. Springer, Berlin (2015)MATHGoogle Scholar
  10. 10.
    Kichenassamy, S., Véron, L.: Singular solutions of the \(p\)-Laplace equation. Math. Ann. 275, 599–615 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Juutinen, P.: Principal eigenvalue of a very badly degenerate operator and applications. J. Differ. Equ. 236, 532–550 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Juutinen, P.: The boundary Harnack inequality for infinity harmonic functions in Lipschitz domains satisfying the interior ball condition. Nonlinear Anal. 69, 1941–1944 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lasry, J.M., Lions, P.L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283, 583–630 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lewis, J., Nyström, K.: Boundary behavior and the Martin boundary problem for \(p\)-harmonic functions in Lipschitz domains. Ann. Math. 172, 1907–1948 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Porretta, A., Véron, L.: Separable p-harmonic functions in a cone and related quasilinear equations on manifolds. J. Eur. Math. Sci. 11, 1285–1305 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Tolksdorf, P.: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Part. Differ. Equ. 8, 773–817 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Marie-Françoise Bidaut-Véron
    • 1
  • Marta Garcia-Huidobro
    • 2
  • Laurent Véron
    • 1
  1. 1.Laboratoire de Mathématiques et Physique ThéoriqueUniversité Francois RabelaisToursFrance
  2. 2.Departamento de MathematicàPontificia Università CatolicaSantiagoChile

Personalised recommendations