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Constant sign and nodal solutions for singular quasilinear elliptic systems

Nouredine Medjoudj, Abdelkrim Moussaoui

TL;DR

This work investigates a singular quasilinear elliptic system $$(\mathrm{P})$$ with homogeneous Dirichlet data and proves the existence of at least three nontrivial solutions: two with opposite constant signs and a third nodal solution. The authors combine sub-supersolutions with truncation and use Leray–Schauder degree theory, via a regularized problem $$(\mathrm{P}^{\varepsilon})$$, to isolate a nodal solution inside a small rectangle while controlling the singularities. A careful degree analysis on a large ball and its complement yields a nontrivial critical point; passage to the limit $\varepsilon\to 0$ produces a nodal solution distinct from the constant-sign ones. With additional sign-coupled structure, they further show that the nodal solution exhibits simultaneous sign changes of both components, extending known results to a general quasilinear singular framework. The results provide a robust multiplicity framework for singular systems lacking variational structure and generalize previous semilinear and Lane–Emden-type findings.

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.

Constant sign and nodal solutions for singular quasilinear elliptic systems

TL;DR

This work investigates a singular quasilinear elliptic system with homogeneous Dirichlet data and proves the existence of at least three nontrivial solutions: two with opposite constant signs and a third nodal solution. The authors combine sub-supersolutions with truncation and use Leray–Schauder degree theory, via a regularized problem , to isolate a nodal solution inside a small rectangle while controlling the singularities. A careful degree analysis on a large ball and its complement yields a nontrivial critical point; passage to the limit produces a nodal solution distinct from the constant-sign ones. With additional sign-coupled structure, they further show that the nodal solution exhibits simultaneous sign changes of both components, extending known results to a general quasilinear singular framework. The results provide a robust multiplicity framework for singular systems lacking variational structure and generalize previous semilinear and Lane–Emden-type findings.

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.
Paper Structure (9 sections, 5 theorems, 104 equations)

This paper contains 9 sections, 5 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Opposite constant sign solutions
  3. Nodal solutions
  4. The regularized system
  5. The degree on $\mathcal{B}_{R_{\varepsilon }}(0)$
  6. The degree on $\mathcal{B}_{R_{\varepsilon }}(0)\backslash \overline{\mathcal{B}}_{\underline{u}}(0)$.
  7. Proof of Theorem \ref{['T3']}.
  8. Proof of Theorem \ref{['T2']}.
  9. Nodal solutions with precise sign information

Key Result

Theorem 2.1

Assume that $(\mathrm{H}_{2})$ and $(\mathrm{H}_{3})$ hold. Then, problem $(\mathrm{P})$ admits two opposite constant-sign solutions $(u_{1,+},u_{2,+})$ and $(u_{1,-},u_{2,-})$ in $\mathcal{C}^{1,\tau }( \overline{\Omega })\times \mathcal{C}^{1,\tau }(\overline{\Omega }),$ for certain $\tau \in (0,1 In particular, every positive solution $(u_{1,+},u_{2,+})\in W_{0}^{1,p_{1}}(\Omega )\times W_{0}^{

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 3.1
  • proof
  • Theorem 3.4
  • proof
  • ...and 1 more