## Differential and Integral Equations

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

Satoshi Tanaka

#### Abstract

The following boundary-value problem $$\tag*{(P_k)} \left\{ \begin{array}{c} u'' + a(x) f(u) = 0, \quad x_0 < x < x_1, \\[1ex] u(x_0) = u(x_1) = 0, \quad u'(x_0)>0, \\[1ex] u\ \mbox{has\ exactly}\ k-1\ \mbox{zeros\ in}\ (x_0,x_1), \end{array} \right.$$ 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.

#### Article information

Source
Differential Integral Equations, Volume 20, Number 1 (2007), 93-104.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356050282

Mathematical Reviews number (MathSciNet)
MR2282828

Zentralblatt MATH identifier
1212.34040

Subjects
Primary: 34B15: Nonlinear boundary value problems

#### Citation

Tanaka, Satoshi. On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem. Differential Integral Equations 20 (2007), no. 1, 93--104. https://projecteuclid.org/euclid.die/1356050282