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