Exact multiplicity of positive solutions in semipositone problems with concave-convex type nonlinearities

J. A. Iaia, University of North Texas, U.S.A.
S. Gadam, 'Yashodha', Chitradurga, India

E. J. Qualitative Theory of Diff. Equ., No. 4. (2001), pp. 1-9.

Abstract: We study the existence, multiplicity, and stability of positive solutions to:
$$\eqalign{- u''(x) &= \lambda f(u(x)) \ \text{for} \ x \in (-1, 1), \lambda > 0, \cr
u(-1)&= 0\ = u(1) ,}$$
where $f : [0, \infty) \to \Bbb R$ is semipositone ($f(0)<0$) and superlinear ($\lim_{t \to \infty} f(t) /t = \infty)$. We consider the case when the nonlinearity $f$ is of concave-convex type having exactly one inflection point. We establish that $f$ should be appropriately concave (by establishing conditions on $f$) to allow multiple positive solutions. For any $\lambda > 0$, we obtain the exact number of positive solutions as a function of $f(t)/t$ and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in [1] by giving a complete classification of positive solutions for concave-convex type nonlinearities.

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