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)=0u″+f(x,u)=0. We allow x↦f(x,s)x↦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)/sf(x,s)/s is below λ1λ1 as s→0+s→0+ and above λ1λ1 as s→+∞s→+∞. In particular, we can deal with the situation in which f(x,s)f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on topological degree which provides the multiplicity of solutions. Applications are given for u″+a(x)g(u)=0u″+a(x)g(u)=0, where we prove the existence of 2n−12n−1 positive solutions when a(x)a(x) has n   positive humps and a−(x)a−(x) is sufficiently large.