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On the existence and boundary behavior of solutions to a class of nonlinear Dirichlet problems


Authors: Carlos E. Kenig and Wei-Ming Ni
Journal: Proc. Amer. Math. Soc. 89 (1983), 254-258
MSC: Primary 35J65; Secondary 35J67
DOI: https://doi.org/10.1090/S0002-9939-1983-0712633-3
MathSciNet review: 712633
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Abstract: In this paper, we first extend the well-known method of super- and sub-solutions for elliptic boundary value problems to $ {L^\infty }$-boundary functions. Then we apply this method to investigate the solvability and the boundary behavior of solutions to some nonlinear elliptic equations, some Fatou-type results are obtained.


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  • [G, N, N] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 544879 (80h:35043)
  • [J, K] D. S. Jerison and C. E. Kenig, The Dirichlet problem in non-smooth domains, Ann. of Math. (2) 113 (1981), 367-382. MR 607897 (84j:35076)
  • [J, L] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241-269. MR 0340701 (49:5452)
  • [K, W] J. L. Kazdan and F. W. Warner, Curvature functions for open $ 2$-manifolds, Ann. of Math. (2) 99 (1974), 203-219. MR 0343206 (49:7950)
  • [M] C. B. Morrey, Second order elliptic equations in several variables and Hölder continuity, Math. Z. 72 (1959), 146-164. MR 0120446 (22:11200)
  • [N] W.-M. Ni, On the elliptic equation $ \Delta u + K(x){u^{(n + 2)/(u - 2)}} = 0$, its generalizations and applications to geometry, Indiana Univ. Math. J. 31 (1982), 493-529. MR 662915 (84e:35049)
  • [S] D. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., vol. 309, SpringerVerlag, Berlin and New York, 1973. MR 0463624 (57:3569)
  • [W] K. O. Widman, On the boundary behavior of solutions to a class of elliptic partial differential equations, Ark. Mat. 6 (1966), 485-533. MR 0219875 (36:2949)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0712633-3
Article copyright: © Copyright 1983 American Mathematical Society

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