Multiple positive solutions for a superlinear problem: a topological approach

TL;DR

This paper establishes the existence of multiple positive solutions for a class of superlinear boundary value problems with indefinite weights using a topological degree approach, extending previous results to more general nonlinearities.

Contribution

It introduces a new topological degree method to prove multiple positive solutions for superlinear problems with indefinite weights, covering cases with superlinear growth at zero and infinity.

Findings

Proves the existence of multiple positive solutions under certain conditions.

Shows that the number of solutions can be at least $2^{n}-1$ when the weight function has $n$ positive humps.

Applies the method to equations with indefinite weights and superlinear nonlinearities.

Abstract

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.