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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Norm estimates for radially symmetric solutions of semilinear elliptic equations


Author: Ryuji Kajikiya
Journal: Trans. Amer. Math. Soc. 347 (1995), 1163-1199
MSC: Primary 35J60; Secondary 34B15, 35B05, 35B45
MathSciNet review: 1290720
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Abstract: The semilinear elliptic equation $ \Delta u + f(u) = 0$ in $ {R^n}$ with the condition $ {\lim _{\vert x\vert \to \infty }}u(x) = 0$ is studied, where $ n \geqslant 2$ and $ f(u)$ has a superlinear and subcritical growth at $ u = \pm \infty $. For example, the functions $ f(u) = \vert u{\vert^{p - 1}}u - u\;(1 < p < \infty \;{\text{if}}\;n = 2,\;1 < p < (n + 2)/(n - 2)\;{\text{if}}\;n \geqslant 3)$ and $ f(u) = u\log \vert u\vert$ are treated. The $ {L^2}$ and $ {H^1}$ norm estimates $ {C_1}{(k + 1)^{n/2}} \leqslant \vert\vert u\vert{\vert _{{L^2}}} \leqslant \vert\vert u\vert{\vert _{{H^1}}} \leqslant {C_2}{(k + 1)^{n/2}}$ are established for any radially symmetric solution $ u$ which has exactly $ k \geqslant 0$ zeros in the interval $ 0 \leqslant \vert x\vert < \infty $. Here $ {C_1},\;{C_2} > 0$ are independent of $ u$ and $ k$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1290720-4
PII: S 0002-9947(1995)1290720-4
Keywords: Semilinear elliptic equation, radially symmetric solution, norm estimate
Article copyright: © Copyright 1995 American Mathematical Society