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

• Main Authors: Joseph Iaia, S. Gadam
• Format: Article
• Language: English
• Published: University of Szeged 2001-01-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=80

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.