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.


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