Positive Solutions for a Class of Semipositone Problems

Abstract

Here we consider the autonomous two point boundary value problem: $$ \hspace{3.5 cm} - u^{\prime \prime}(x)=\lambda f(u(x)) \hspace{0.3 cm} ; \hspace{1 cm} x\in (-1,1),$$ $$ u(-1)=0=u(1),$$ where $\lambda > 0$ and $f:[0,\infty)\to R$ is monotonically increasing and concave ($f^{\prime\prime} < 0$) with $f(0) < 0$ (semipositone), $f(t) > 0$ for some $t > 0$. We obtain the exact number of positive solutions. Keywords: Semipositone; Two Point Boundary Value Problem; Positive Solutions. 2010 Mathematics Subject Classification: 34B15, 34B18