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Uniqueness of the positive solution for singular nonlinear boundary value problems. (English) Zbl 0789.34025
Summary: This paper is concerned with the boundary value problems \[ {1 \over p(t)} (p(t)u')'+ f(u)=0,\;t>0,\;u'(0)=0,\;\lim_{t \to+\infty} u(t)=0, \] where \(f(0)=0\). Such problems arise in the study of semilinear elliptic differential equations in \(R^ n\). It is shown that the problem has at most one positive solution under appropriate conditions on \(f\) and \(p\). Our result can include the important case that \(p(t)=t^{n-1}\) and \(f(u)=u^ p-u\), where \(n>1\), \(p>1\) are some given constants.

34B15 Nonlinear boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
35J15 Second-order elliptic equations