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Semilinear elliptic equations with uniform blow-up on the boundary

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Abstract

We prove the existence and the uniqueness of a solutionu of−Lu+h|u| α-1u=f in some open domain ℝd, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC 2 compact hypersurface, lim x→∂G (dist(x, ∂G))2α/(α-1) f(x)=0, and lim x→∂G u(x)=∞.

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References

  1. S. Agmon, A. Douglis and L. Nirenberg,Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math.12 (1959), 623–727.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. D. Aleksandrov,Uniqueness conditions and estimates for the solution of the Dirichlet problem, Vestnik Leningrad Univ.18 (1963), 5–29.

    Google Scholar 

  3. P. Aviles,A study of isolated singularities of solutions of a class of non-linear elliptic partial differential equations, Commun. Partial Differ. Equ.7 (1982), 609–643.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Bandle and M. Marcus,Sur les solutions maximales de problèmes elliptiques non linéaires: bornes isopérimétriques et comportement asymptotique, C. R. Acad. Sci. Paris311 Ser. I (1990), 91–93.

    MATH  MathSciNet  Google Scholar 

  5. E. B. Dynkin,A probabilistic approach to one class of nonlinear differential equations, to appear.

  6. D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlerhen 224, Springer-Verlag, Berlin, 1982.

    Google Scholar 

  7. I. Iscoe,On the support of measure-valued critical branching Brownian motion, Ann. Probab.16 (1988), 200–221.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. N. V. Krylov,Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht, 1987.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Loewner and L. Nirenberg,Partial differential equations invariant under conformal or projective transformations, inContributions to Analysis (L. Ahlfors et al., eds.), 1974, pp. 245–272.

  12. R. Osserman,On the inequality Δu≧f(u), Pacific J. Math.7 (1957), 1641–1647.

    MATH  MathSciNet  Google Scholar 

  13. L. Veron,Comportement asymptotique des solutionsd'équations elliptiques semi-linéaires dans R N, Ann. Mat. Pura. Appl.127 (1981), 25–50.

    Article  MATH  MathSciNet  Google Scholar 

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Veron, L. Semilinear elliptic equations with uniform blow-up on the boundary. J. Anal. Math. 59, 231–250 (1992). https://doi.org/10.1007/BF02790229

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  • DOI: https://doi.org/10.1007/BF02790229

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