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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, αu(0)-βu ' (0)=0, γu(1)+δu ' (1)=0, where f is continuous and f(t,u)0 for t[0,1] and u0, α,β,γ,δ0 and αβ+αγ+αδ>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.

MSC:
34B15Nonlinear boundary value problems for ODE
34C11Qualitative theory of solutions of ODE: growth, boundedness
35J15Second order elliptic equations, general