Constant sign and nodal solutions for singular quasilinear elliptic systems

TL;DR

This paper proves the existence of multiple solutions, including sign-changing and nodal solutions, for singular quasilinear elliptic systems using topological degree and sub-supersolution methods.

Contribution

It introduces a novel combination of topological degree, sub-supersolutions, and truncation techniques to find multiple solutions with specific sign properties.

Findings

Existence of at least three nontrivial solutions.

Two solutions have opposite constant signs.

One nodal solution with components of opposite signs.

Abstract

We consider singular quasilinear elliptic systems with homogeneous Dirichlet boundary condition. Using Leray-Schauder topological degree, combined with the sub-supersolutions method and suitable truncation arguments, we establish the existence of at least three nontrivial solutions, two of which are of opposite constant sign. The third solution is nodal and exhibits components of at least opposite constant sign. In the case of a sign-coupled system, these components are of changing and synchronized sign.