Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity

Abstract

In this paper, we consider the nonlinear elliptic problem

$$ - \Delta u + {\left| u \right|^{p - 1}}u + {\left| {\nabla u} \right|^q} = f$$

in ℝN, where p > 1 and q > 0. We show that if fL rloc (ℝN) for suitable r ≥ 1, then there exists a distributional solution of the equation, independently of the behavior of f at infinity. We also analyze the uniqueness of this solution in some cases.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    S. Alarcón, J. García-Melián, A. Quaas, Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations 252 (2012), 886–914.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), 779–790.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    L. Boccardo, T. Gallouët, and J. L. Vázquez, Nonlinear elliptic equations inN without growth restrictions on the data, J. Differential Equations 105 (1993), 334–363.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    L. Boccardo, T. Gallouët, and J. L. Vázquez, Solutions of nonlinear parabolic equations without growth restrictions on the data, Electron. J. Differential Equations 2001 no. 60, (2001), 1–20.

    Google Scholar 

  6. [6]

    H. Brezis, Semilinear equations inN without condition at infinity, Appl. Math. Optim. 12 (1984), 271–282.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    M. J. Esteban, P. Felmer, and A. Quaas, Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data, Proc. Edinburgh Math. Soc. 53 (2010), 125–141.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    P. Felmer and A. Quaas, On the strong maximum principle for quasilinear elliptic equations and systems, Adv. Differential Equations 7 (2002), 25–46.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

  10. [10]

    O. A. Ladyženskaja and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.

    Google Scholar 

  11. [11]

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

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    F. Leoni, Nonlinear elliptic equations inN with “absorbing” zero order terms, Adv. Differential Equations 5 (2000), 681–722.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    F. Leoni and B. Pellacci, Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data, J. Evol. Equ. 6 (2006), 113–144.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlin. Stud. 7 (2007), 237–269.

    MathSciNet  MATH  Google Scholar 

  15. [15]

    P. L. Lions, Résolution des probl`emes elliptiques quasilinéaires, Arch. Rational Mech. Anal. 74 (1980), 336–353.

    Article  Google Scholar 

  16. [16]

    P. L. Lions, Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre, J. Analyse Math. 45 (1985), 234–254.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    A. Porretta, Some uniqueness results for elliptic equations without condition at infinity, Commun. Contemporary Mathematics 5 (2003), 705–717.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    J. Schoenenberger-Deuel and P. Hess, A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 49–54 (1976).

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Salomón Alarcón.

Additional information

S. A. was supported by USM Grant # 121002.

A. Q. was partially supported by Fondecyt Grant # 1110210 and CAPDE anillo ACT-125.

All three authors were partially supported by Programa Basal CMM, U. de Chile, and Ministerio de Ciencia e Innovación and FEDER under grant MTM2008-05824 (Spain).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alarcón, S., García-Melián, J. & Quaas, A. Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity. JAMA 118, 83–104 (2012). https://doi.org/10.1007/s11854-012-0030-6

Download citation

Keywords

  • Nonnegative Solution
  • Nonlinear Elliptic Equation
  • Distributional Solution
  • Nonlinear Parabolic Equation
  • Quasilinear Elliptic Equation