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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0766530-X
PII: S 0002-9939(1985)0766530-X
Keywords: Semilinear elliptic equation, positive solution, entire solution, supersolution, subsolution
Article copyright: © Copyright 1985 American Mathematical Society