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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
DOI: https://doi.org/10.1090/S0002-9947-1995-1290720-4
MathSciNet review: 1290720
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Abstract: The semilinear elliptic equation $\Delta u + f(u) = 0$ in ${R^n}$ with the condition ${\lim _{|x| \to \infty }}u(x) = 0$ is studied, where $n \geqslant 2$ and $f(u)$ has a superlinear and subcritical growth at $u = \pm \infty$. For example, the functions $f(u) = |u{|^{p - 1}}u - u\;(1 < p < \infty \;{\text {if}}\;n = 2,\;1 < p < (n + 2)/(n - 2)\;{\text {if}}\;n \geqslant 3)$ and $f(u) = u\log |u|$ are treated. The ${L^2}$ and ${H^1}$ norm estimates ${C_1}{(k + 1)^{n/2}} \leqslant ||u|{|_{{L^2}}} \leqslant ||u|{|_{{H^1}}} \leqslant {C_2}{(k + 1)^{n/2}}$ are established for any radially symmetric solution $u$ which has exactly $k \geqslant 0$ zeros in the interval $0 \leqslant |x| < \infty$. Here ${C_1},\;{C_2} > 0$ are independent of $u$ and $k$.


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Keywords: Semilinear elliptic equation, radially symmetric solution, norm estimate
Article copyright: © Copyright 1995 American Mathematical Society