Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree

TL;DR

This paper proves the existence of multiple positive periodic solutions for a class of nonlinear differential equations with super-sublinear growth, using coincidence degree theory, and extends results to subharmonic solutions and PDE boundary conditions.

Contribution

It introduces a novel application of coincidence degree theory to establish high multiplicity of positive solutions for super-sublinear indefinite problems with complex weight functions.

Findings

Existence of 3^m - 1 positive T-periodic solutions for large parameters.

Abundance of positive subharmonic solutions and symbolic dynamics.

Extension of results to Neumann and Dirichlet boundary conditions for elliptic PDEs.

Abstract

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\lambda, \mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.