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On the elliptic equation $ \Delta u=\varphi (x)u\sp \gamma$ in $ {\bf R}\sp 2$


Authors: Nichiro Kawano, Takaŝi Kusano and Manabu Naito
Journal: Proc. Amer. Math. Soc. 93 (1985), 73-78
MSC: Primary 35J60
DOI: https://doi.org/10.1090/S0002-9939-1985-0766530-X
MathSciNet review: 766530
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Abstract: The equation $ ( * )\Delta u = \phi (x){u^\gamma }$ is considered in $ {{\mathbf{R}}^2}$, where $ \gamma \ne 1$ and $ \phi (x) \geqslant 0$ is locally Hölder continuous. Sufficient conditions are obtained for $ ( * )$ to possess infinitely many positive solutions which are defined throughout $ {R^2}$ and have logarithmic growth as $ \vert x\vert \to \infty $. An extension of the main result to the higher-dimensional case is also attempted.


References [Enhancements On Off] (What's this?)

  • [1] R. E. Edwards, Functional analysis, Holt, Rinehart and Winston, New York, 1965. MR 0221256 (36:4308)
  • [2] N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984) (to appear). MR 750393 (86b:35052)
  • [3] W.-M. Ni, On the elliptic equation $ \Delta u + K(x){u^{(n + 2)/(n - 2)}} = 0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493-529. MR 662915 (84e:35049)
  • [4] -, On the elliptic equation $ \Delta u + K(x){e^{2u}} = 0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), 343-352. MR 656628 (84g:58107)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0766530-X
Keywords: Semilinear elliptic equation, positive solution, entire solution, supersolution, subsolution
Article copyright: © Copyright 1985 American Mathematical Society

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