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 $u''+f(x,u)=0$. 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 $f(x,s)/s$ is below $\lambda_{1}$ as $s\to 0^{+}$ and above $\lambda_{1}$ as $s\to +\infty$. 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 $u'' + a(x) g(u) = 0$, where we prove the existence of $2^{n}-1$ positive solutions when $a(x)$ has $n$ positive humps and $a^{-}(x)$ is sufficiently large.