Large viscosity solutions for some fully nonlinear equations

Abstract

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem

$$\left\{\begin{array}{ll}-F(D^{2} u) + \beta (u) = f \quad {\rm in} \, \Omega, \\ u = + \infty \quad \quad \quad \quad \quad \quad \,\,\,\, {\rm on}\, \partial \Omega, \end{array} \right.\quad \quad \quad \quad \quad {\rm (P)}$$

where Ω is a bounded smooth domain in \({{\mathbb R}^N, N >1 , F}\) is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to \({f \in L^\infty_{loc}(\Omega)}\) or f having only local integrability properties where viscosity solutions are well defined, i.e. \({f \in L^N_{loc}(\Omega)}\). Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions \({f \in \mathcal{C}(\Omega)}\). Based in this behavior, we also prove uniqueness.

References

  1. 1

    Bandle C., Essén M.: On the solutions of quasilinear elliptic problems with boundary blow-up. Symp. Math. 35, 93–111 (1994)

    Google Scholar 

  2. 2

    Bandle C., Marcus M.: Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour. J. Anal. Math. 58, 9–24 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3

    Bandle C., Marcus M.: Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 155–171 (1995)

    MathSciNet  MATH  Google Scholar 

  4. 4

    Bieberbach L.: \({\Delta u = e^u}\) und die automorphen funktionen. Math. Ann. 77, 173–212 (1916)

    MathSciNet  Article  Google Scholar 

  5. 5

    Caffarelli L.A.: Interior a priori estimates for solutions of fully non-linear equations. Ann. Math. 2nd Ser. 130, 189–213 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6

    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, 1st edn, vol. 43. American Mathematical Society, Colloquium Publications, Providence (1995, Printed in United States of America)

  7. 7

    Caffarelli L.A., Crandall M.G., Kocan M., Swiech A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49, 365–397 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Crandall M.G., Kocan M., Lions P.L., Swiech A.: Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. Electron. J. Differ. Equ. 24, 1–20 (1999)

    MathSciNet  Google Scholar 

  9. 9

    Da Lio F., Sirakov B.: Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations. J. Eur. Math. Soc. 9, 317–330 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Davila G., Felmer P., Quaas A.: Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. Partial Differ. Equ. 39, 557–578 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11

    Díaz G., Letelier R.: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Anal. 20, 97–125 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Dynkin, E.B.: Diffusions, Superdiffusions and Partial Differential Equations, vol. 50, American Mathematical Society, Colloquium Publications, American Mathematical Society, Providence (2002)

  13. 13

    Esteban M., Felmer P., Quaas A.: Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data. Proc. Edinb. Math. Soc. 53(2), 125–141 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14

    Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions, 2nd edn. In: Stochastic Modelling and Probability, vol. 25, Springer, New York (2006)

  15. 15

    García-Melián J.: Uniqueness of positive solutions for a boundary blow-up problem. J. Math. Anal. Appl. 360, 530–536 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. In: Classics in Mathematics, Springer, Berlin (2001)

  17. 17

    Juutinen P., Rossi J.: Large solutions for the infinity Laplacian. Adv. Calc. Var. 1, 271–289 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Keller J.B.: On solutions of \({\Delta u = f(u)}\). Commun. Pure Appl. Math. 10, 503–510 (1957)

    MATH  Article  Google Scholar 

  19. 19

    Labutin D.: Wiener regularity for large solutions of nonlinear equations. Ark. Mat. 41, 307–339 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20

    Lasry J.M., Lions P.L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. Math. Ann. 283, 583–630 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21

    Matero J.: Quasilinear elliptic equations with boundary blow-up. J. Anal. Math. 96, 229–247 (1996)

    MathSciNet  Article  Google Scholar 

  22. 22

    Osserman R.: On the inequality \({\Delta u \geq f(u)}\). Pac. J. Math. 7, 1641–1647 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23

    Radulescu, V.: Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities. In: Michel Chipot (ed.) Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4, pp. 483–591 (2007)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Alarcón.

Additional information

S.A. was partially supported by Fondecyt Grant # 11110482 and USM Grant No. 121210, and A.Q. was partially supported by Fondecyt Grant # 1070264 and CAPDE, Anillo ACT-125. Also, both authors were supported by Programa Basal, CMM, U. de Chile.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alarcón, S., Quaas, A. Large viscosity solutions for some fully nonlinear equations. Nonlinear Differ. Equ. Appl. 20, 1453–1472 (2013). https://doi.org/10.1007/s00030-012-0217-7

Download citation

Mathematics Subject Classification (2000)

  • 35J60
  • 35B40
  • 35B44
  • 35J67
  • 49L25

Keywords

  • Boundary blow-up
  • Fully nonlinear operator
  • Keller–Osserman condition
  • Asymptotic behavior
  • Uniqueness