## 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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