Multiple positive solutions for a superlinear problem: a topological approach

We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation $u^{\mathrm{\prime }\mathrm{\prime }}+f(x,u)=0$. We allow $x\mapsto f(x,s)$ to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that $f(x,s)\mathrm{/}s$ is below ${\lambda }_1$ as $s\to 0^{+}$ and above ${\lambda }_1$ as $s\to +\mathrm{\infty }$. In particular, we can deal with the situation in which $f(x,s)$ has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for $u^{\mathrm{\prime }\mathrm{\prime }}+a(x)g(u)=0$, where we prove the existence of $2^n-1$ positive solutions when $a(x)$ has $n$ positive humps and $a^{-}(x)$ is sufficiently large.