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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.

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