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Quasilinear Elliptic Cooperative and Competitive Systems

Annamaria Canino, Simone Mauro

TL;DR

This work analyzes a two‑component quasilinear elliptic system on a bounded domain with a gradient‑type nonlinearity derived from a potential $G_\beta$, under Dirichlet boundary conditions and subcritical growth. Facing non‑differentiability of the natural energy, the authors employ nonsmooth critical point theory via the weak slope to obtain multiplicity results and a least energy state, addressing both cooperative ($\beta>0$) and competitive ($\beta<0$) couplings. They establish a Palais–Smale type compactness framework under spectral gap assumptions and prove the existence of infinitely many fully nontrivial weak solutions for $\beta>0$, as well as a least energy solution below the least energy of the scalar problems; in the competitive regime they adapt a Nehari‑manifold reduction and a topological linking argument to secure a nonnegative least energy solution. Overall, the results extend variational methods for subcritical quasilinear systems to nonsmooth settings, providing a rigorous pathway to multiple and minimal energy states with potential applications in nonlinear optics and related fields.

Abstract

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = λ_1 u_1 + g_{β,1}(u) & \text{in } Ω, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = λ_2 u_2 + g_{β,2}(u) & \text{in } Ω, \\[2mm] u_1 = u_2 = 0 & \text{on } \partialΩ, \end{cases} \] where $λ_1, λ_2 < μ_1$, the first Dirichlet eigenvalue of the Laplacian, and $Ω$ is a bounded domain. The nonlinearity derives from a potential $G_β$ with subcritical growth. Due to the lack of differentiability of the associated energy functional, we employ nonsmooth critical point theory and variational methods based on the concept of weak slope. We prove the existence of least energy solutions in both the cooperative ($β> 0$) and competitive ($β< 0$) regimes.

Quasilinear Elliptic Cooperative and Competitive Systems

TL;DR

This work analyzes a two‑component quasilinear elliptic system on a bounded domain with a gradient‑type nonlinearity derived from a potential , under Dirichlet boundary conditions and subcritical growth. Facing non‑differentiability of the natural energy, the authors employ nonsmooth critical point theory via the weak slope to obtain multiplicity results and a least energy state, addressing both cooperative () and competitive () couplings. They establish a Palais–Smale type compactness framework under spectral gap assumptions and prove the existence of infinitely many fully nontrivial weak solutions for , as well as a least energy solution below the least energy of the scalar problems; in the competitive regime they adapt a Nehari‑manifold reduction and a topological linking argument to secure a nonnegative least energy solution. Overall, the results extend variational methods for subcritical quasilinear systems to nonsmooth settings, providing a rigorous pathway to multiple and minimal energy states with potential applications in nonlinear optics and related fields.

Abstract

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = λ_1 u_1 + g_{β,1}(u) & \text{in } Ω, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = λ_2 u_2 + g_{β,2}(u) & \text{in } Ω, \\[2mm] u_1 = u_2 = 0 & \text{on } \partialΩ, \end{cases} \] where , the first Dirichlet eigenvalue of the Laplacian, and is a bounded domain. The nonlinearity derives from a potential with subcritical growth. Due to the lack of differentiability of the associated energy functional, we employ nonsmooth critical point theory and variational methods based on the concept of weak slope. We prove the existence of least energy solutions in both the cooperative () and competitive () regimes.
Paper Structure (6 sections, 25 theorems, 99 equations)

This paper contains 6 sections, 25 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Palais Smale condition
  4. The scalar problem
  5. Proof of Theorem \ref{['thm1.2']}
  6. Proof of Theorem \ref{['main thm competitive quasilinear']}

Key Result

Theorem 1.2

Assume that a.0-a.2 hold, $A_i(x,-s)=A_i(x,s)$. Then, there exists $\beta_1>0$ such that the problem Qb has infinitely many fully non-trivial weak solutions for every $\beta>\beta_1$ and $\lambda_1,\lambda_2<\frac{p-2-\gamma}{p-2}\nu\mu_1$. Furthermore, $e_\beta$ is achieved by some $u^*$, i.e. $u^*

Theorems & Definitions (57)

  • Example 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: nonsmooththeory1
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Equivariant Mountain Pass, nonsmooththeory1
  • Theorem 2.6
  • ...and 47 more