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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
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