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Quasilinear elliptic equations with boundary blow-up

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Abstract

Assume that Ω is a bounded domain in ℝ N withN ≥2, which has aC 2-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δp u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].

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References

  1. C. Bandle and M. Essén,On the solutions of quasilinear elliptic problems with boundary blow-up, inPartial Differential Equations of Elliptic Type (A. Alvino, E. Fabes and G. Talenti, eds.), Symposia Mathematica35, Cambridge University Press, 1994, pp. 93–111.

  2. C. Bandle and M. Marcus,Sur les solutions maximales de problèmes elliptique non-linéaires, C.R. Acad. Sci. Paris311 (1990), 91–93.

    MATH  MathSciNet  Google Scholar 

  3. C. Bandle and M. Marcus,Large solutions of semilinear elliptic equations with singular coefficients, inOptimization and Nonlinear Analysis (A. Ioffe, M. Marcus and S. Reich, eds.), Pitman, London, 1992, pp. 25–38.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Bandle and M. Marcus,Asymptotic behaviour of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary, Ann. Inst. Henri Poincaré12 (1995), 155–171.

    MATH  MathSciNet  Google Scholar 

  6. C. Bandle and H. Leutwiler,On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformation, Aequationes Math.42 (1991), 166–181.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Bieberbach,δu = e u und die Automorphen Funktionen, Math. Ann.77 (1916), 173–212.

    Article  MathSciNet  Google Scholar 

  8. G. Diaz and R. Letelier,Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Analysis. Theory, Methods and Applications20 (1993), 97–125.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. I. Diaz,Nonlinear partial differential equations and free boundaries I: Elliptic equations, Research Notes in Math.106, Pitman, London, 1993.

    Google Scholar 

  10. J. B. Keller,On solutions of δu = f(u), Comm. Pure Appl. Math.10 (1957), 503–510.

    Article  MATH  MathSciNet  Google Scholar 

  11. V. A. Kondrat’ev and V. A. Nikishkin,Asymptotics near the boundary of a solution of a singular boundary value problem for a semilinear elliptic equation (Engl. transl), Differential Equations26 (1990), 345–348.

    MATH  MathSciNet  Google Scholar 

  12. A. C. Lazer and P. J. McKenna,On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc.111 (1991), 721–730.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. C. Lazer and P. J. McKenna,On a singular nonlinear elliptic boundary value problem II: Boundary blowup, preprint, 1993.

  14. A. C. Lazer and P. J. McKenna,On a problem of Bieberbach and Rademacher, preprint.

  15. C. Loewner and L. Nirenberg,Partial differential equations invariant under conformal or projective transformations, inContribution to Analysis (L. Ahlfors, ed.), Academic Press, New York, 1974.

    Google Scholar 

  16. M. Marcus,On solutions with blow-up at the boundary for a class of semilinear elliptic equations, inDevelopments in PDE and Applications (G. Buttazzo, G. Galdi and L. Zanghirati, eds.), Plenum Press, New York, 1992, pp. 65–78.

    Google Scholar 

  17. M. Marcus and L. Véron,Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C.R. Acad. Sci. Paris, Série I317 (1993), 559–563.

    MATH  Google Scholar 

  18. M. Marcus and L. Veron,Uniqueness and asymptotic behaviour of solutions with boundary blowup for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincaré, to appear.

  19. J. Matero,Boundary-blow-up problems in a fractal domain, Z. Anal. Anw.15 (1996), 1–26.

    MathSciNet  Google Scholar 

  20. J. Matero,Boundary-blow-up problems in non-tangentially accessible domains, preprint (1995).

  21. A. B. Muravnik and L. A. Peletier,On self-similar solutions of a semilinear elliptic equation, preprint.

  22. R. Osserman,On the inequality δu ≥f(u), Pacific J. Math.7 (1957), 1641–1647.

    MATH  MathSciNet  Google Scholar 

  23. S. L. Pohozaev,The Dirichlet problem for the equation δu = u 2, Sov. Math. Dokl.1 (1961), 1143–1146.

    MathSciNet  Google Scholar 

  24. H. Rademacher,Einige besondere Probleme der partiellen Differentialgleichungen, inDie Differential- und Integralgleichungen der Mechanik und Physik, 2. ed. (P. Frank und R. von Mises, eds.), Rosenberg, New York, 1943.

    Google Scholar 

  25. L. Véron,Semilinear elliptic equations with uniform blow-up on the boundary, J. Analyse Math.58 (1992), 94–102.

    Google Scholar 

  26. W. Walter,über ganze Lösungen der Differentialgleichung δu = f(u), J. Ber. Deutsch. Math. Vereinigung57 (1955), 94–102.

    MATH  Google Scholar 

  27. H. Wittich,Ganze Lösungen der Differentialgleichung δu = eu, Math. Z.49 (1944), 579–582.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Uppsala University, Box 480, S-75106, Uppsala, Sweden

    Jerk Matero

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Matero, J. Quasilinear elliptic equations with boundary blow-up. J. Anal. Math. 69, 229–247 (1996). https://doi.org/10.1007/BF02787108

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  • DOI: https://doi.org/10.1007/BF02787108

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