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On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem. (English) Zbl 1212.34040
Summary: The following boundary-value problem $\begin{cases} u''+a(x)f(u)=0,\quad x_0<x<x_1,\\ u(x_0)=u(x_1)=0,\quad u'(x_0)>0,\\ u \text{ has exactly } k-1 \text{ zeros in } (x_0,x_1), \end{cases} \tag{"}$$(P_k)$$"$ is considered under the following conditions: $$k$$ is a positive integer, $$a\in C^2[x_0,x_1]$$, $$a(x)>0$$ for $$x\in [x_0,x_1]$$, $$f\in C^1(\mathbb{R})$$, $$f(s)>0$$, $$f(-s)=-f(s)$$ for $$s>0$$. It is shown that, if either $$(f(s)/s)'>0$$ for $$s>0$$ and $$([a(x)]^{-1/2})''\leq 0$$ for $$x\in [x_0,x_1]$$ or $$(f(s)/s)'<0$$ and $$([a(x)]^{-1/2})''\geq 0$$ for $$x\in [x_0,x_1]$$, then problem $$(P_{k})$$ has at most one solution. To prove the uniqueness of solutions of ($$P_{k})$$, 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.

MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 35J65 Nonlinear boundary value problems for linear elliptic equations