On the existence of positive solutions of nonlinear second order differential equations. (English) Zbl 0857.34036

The authors establish the following criterion for the existence of a positive solution to the boundary value problem (1) \(u''(t)+ f(t,u(t))=0\), \(0<t<1\), \(a\cdot u(0)-b\cdot u'(0)=0\), \(c\cdot u(1)+d\cdot u'(1)=0\).
Theorem. Suppose that \(f\in C([0,1]\times [0,\infty);[0,\infty))\); \(a,b,c,d\geq 0\); \(cb+ac+ab>0\); \(0<M:=\min\{ {{c+4d}\over{4(c+d)}}, {{a+4b}\over{4(a+b)}}\}<1\) and assume that there exist two distinct positive constants \(\lambda\), \(\eta\) such that \[ \begin{aligned} f(t,u) &\leq\lambda \biggl(\int^1_0 k(s,s)ds\biggr)^{-1} \qquad\text{on} \quad [0,1]\times [0,\lambda],\\ \text{and} f(t,u) &\geq\eta \biggl( \int^{3\over4}_{1\over4} k({\textstyle{1\over2}},s)ds) \biggr)^{-1} \qquad\text{on}\quad [{\textstyle{{1\over4},{3\over4}}}] \times[M\eta,\eta], \end{aligned} \] where \(k(t,s)\) is the Green’s function of the differential equation \(u''(t)=0\), \(t\in(0,1)\) with respect to the same boundary conditions as (1). Then (1) has at least one positive solution \(u\) such that \(|u|:= \sup_{t\in[0,1]}|u(t)|\) between \(\lambda\) and \(\eta\).
The authors apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.


34B15 Nonlinear boundary value problems for ordinary differential equations
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