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Multiple positive solutions for a class of semipositone Neumann two point boundary value problems. (English) Zbl 0783.34016

Summary: We consider the two point Neumann boundary value problem \(-u''(x)= \lambda f(u(x))\), \(x\in(0,1)\), \(u'(0)=0=u'(1)\), where \(\lambda\) is a positive parameter, \(f\in C^ 2[0,\infty)\), \(f'(u)>0\) for \(u>0\), and for some \(\beta>0\), \(f(u)<0\) for \(u\in [0,\beta)\) (semipositone) and \(f(u)>0\) for \(u>\beta\). We discuss existence and multiplicity results for positive solutions. In particular, we prove that if the set \(S=(\pi^ 2 n^ 2/f'(\beta), \theta^ 2/-2F(\beta))\), where \(n\in N\), \(F(u)=\int_ 0^ u f(s)ds\) and \(\theta\) is the unique positive zero of \(F\), is nonempty, then there exist at least \(2n+1\) positive solutions for each \(\lambda\in S\). Furthermore, if \(f''>0\) on \([0,\beta)\) and \(f''<0\) on \((\beta,\infty)\), then we prove that there are exactly \(2n+1\) positive solutions for each \(\lambda\in S\). We also discuss examples to which our results apply.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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