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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.

##### MSC:
 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