On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem

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.