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Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II
Author:
Martin Dindos
Journal:
Trans. Amer. Math. Soc. 355 (2003), 1365-1399
MSC (2000):
Primary 35J65, 35B65; Secondary 46E35, 42B20
Posted:
December 2, 2002
MathSciNet review:
1946396
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Abstract: Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem  in  . We improve our previous results by studying more general nonlinear terms  with polynomial (and in some cases exponential) growth in the variable  . We also study the case of nonnegative solutions.
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- J. Bergh and J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York, 1976.MR 58:2349
- 2.
- Z.-Q. Chen, R. J. Williams and Z. Zhao, On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions, Math. Ann. 315 (1999), 735-769.MR 2001a:35061
- 3.
- M. Dindos, Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds, Comm. Partial Differential Equations 27 (2002), 219-281.
- 4.
- M. Dindos, Hardy spaces and potential theory on domains in Riemannian manifolds, Preprint (2001).
- 5.
- M. Dindos and M. Mitrea, Semilinear Poisson Problem in Sobolev-Besov spaces on Lipschitz domains, Publicacions Matemàtiques 46 (2002), 353-403.
- 6.
- B. Dahlberg, On the Poisson integral for Lipschitz and
 domains, Studia Math. LXVI. (1979), 13-26.MR 81g:31007
- 7.
- B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in
 for Laplace's equation in Lipschitz domains, Ann. Math. 125 (1987), 437-465.MR 88d:35044
- 8.
- B. Dahlberg, C. Kenig and G. Verchota, Boundary value problems for the system of elastostatics on Lipschitz domains, Duke Math. J. 57 (1988), 795-818.MR 90d:35259
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- E. Fabes, M. Jodeit, Jr. and N. Rivère, Potential techniques for boundary value problems on
 -domains, Acta Math. 141 (1978), 165-186.MR 80b:31006
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- E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769-793.MR 90d:35258
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- Z. Jin, Solvability of Dirichlet problem for semilinear elliptic equations on certain domains, Pacific J. Math. 176 (1996), 117-128.MR 97m:35082
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- D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in
 and applications to electromagnetic scattering, J. Fourier Anal. and Appl. 3 (1997), 131-192.MR 99e:31009
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 , Hardy and Hölder space results, Comm. Anal. Geom. 9 (2001), no. 2, 369-421. MR 2002f:31012
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Additional Information
Martin Dindos
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email:
dindos@math.cornell.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03210-5
PII:
S 0002-9947(02)03210-5
Keywords:
Nonlinear equations,
semilinear elliptic problems,
Dirichlet boundary problems,
Lipschitz domains,
Riemannian manifolds
Received by editor(s):
September 11, 2001
Received by editor(s) in revised form:
July 24, 2002
Posted:
December 2, 2002
Article copyright:
© Copyright 2002 American Mathematical Society
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