Abstract
The following boundary-value problem \begin{equation}\tag*{(P$_k$)} \left\{ \begin{array}{c} u'' + a(x) f(u) = 0, \quad x_0 < x < x_1, \\[1ex] u(x_0) = u(x_1) = 0, \quad u'(x_0)>0, \\[1ex] u\ \mbox{has\ exactly}\ k-1\ \mbox{zeros\ in}\ (x_0,x_1), \end{array} \right. \end{equation} is considered under the following conditions: $k$ is a positive integer, $a \in C^2[x_0,x_1]$, $a(x)>0$ for $x \in [x_0, x_1]$, $f \in C^1({\bf R})$, $f(s)>0$, $f(-s) = - f(s)$ for $s>0$. It is shown that if either $(f(s)/s)'>0$ for $s>0$ and $( [a(x)]^{-\frac{1}{2}} )'' \le 0$ for $x \in [x_0, x_1]$ or $(f(s)/s)' <0$ for $s <0$ and $( [a(x)]^{-\frac{1}{2}} )'' \ge 0$ for $x \in [x_0, x_1]$, then $(\mathrm{P}_k)$ has at most one solution. To prove the uniqueness of solutions of $(\mathrm{P}_k)$ , the shooting method is used. The results obtained here are applied to the study of radially symmetric solutions of the Dirichlet problem for semilinear elliptic equations in annular domains.
Citation
Satoshi Tanaka. "On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem." Differential Integral Equations 20 (1) 93 - 104, 2007. https://doi.org/10.57262/die/1356050282
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