Norm estimates for radially symmetric solutions of semilinear elliptic equations

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 $k$.