Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities

TL;DR

This paper proves the existence of multiple positive solutions for a class of nonlinear Sturm-Liouville boundary value problems with conflicting nonlinearities, using topological degree theory, and extends results to radially symmetric solutions of elliptic PDEs.

Contribution

It introduces new conditions under which multiple positive solutions exist for nonlinear Sturm-Liouville problems with conflicting nonlinearities, employing Leray-Schauder degree theory.

Findings

Existence of at least 2^m - 1 positive solutions when parameters are large.

Applicable to classical superlinear equations with growth conditions.

Extension to radially symmetric solutions of elliptic PDEs.

Abstract

We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0,L\mathclose{]}$, and the nonlinearities $g_{i}(s), k_{j}(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u^{p}=0$, with $p>1$. When the positive parameters $\beta_{j}$ are sufficiently large, we prove the existence of at least $2^{m}-1$ positive solutions for the Sturm-Liouville boundary value problems associated with the equation. The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets. Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.