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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 18, Issue 3, 1999, pp. 517–523**

**DOI: 10.4171/ZAA/895**

Published online: 1999-09-30

On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations

Tadie^{[1]}(1) Copenhagen University, Denmark

For $f\in C((0, \infty)) \cap C^1 ((0, \infty))$ and $b > 0$, existence and uniqueness of radial solutions $u = u(r)$ of the problem $\Delta u + f(u_+) = 0$ in $\mathbb R^n (n > 2), u(0) = b$ and $u’(0) = 0$ are well known. The uniqueness for the above problem with boundary conditions $u(R) = 0$ and $u’(0) = 0$ is less known beside the cases where $lim_{r \to \infty} u(r) = 0$. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem $D_{\alpha}u + f(u_+) = 0 (r > 0), u(p) = 0$ and $u’(0) = 0$ based only on the evolution of the functions $f(t)$ and $\frac{d}{dt} \frac{f(t)}{t}$.

*Keywords: *Semilinear elliptic equations, comparison results for nonlinear differential equations

Tadie: On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations. *Z. Anal. Anwend.* 18 (1999), 517-523. doi: 10.4171/ZAA/895