Multiple positive solutions of some boundary value problems. (English) Zbl 0805.34021

The authors consider the second order boundary value problem (1) \(-u'' = f(t,u)\), \(0<t<1\), \(\alpha u(0) - \beta u'(0) = 0\), \(\gamma u(1) + \delta u'(1) = 0\), where \(f\) is continuous and \(f(t,u) \geq 0\) for \(t \in[0,1]\) and \(u \geq 0\), \(\alpha, \beta, \gamma, \delta \geq 0\) and \(\alpha \beta + \alpha \gamma + \alpha \delta>0\). They prove the existence of two positive solutions of (1) provided \(f(t,u)\) is superlinear at one end (zero or infinitely) and sublinear at the other. It is shown that these results also imply the existence of multiple positive radial solutions of certain semilinear elliptic boundary value problems. The proofs are based on the fixed point arguments.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
35J15 Second-order elliptic equations
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