Entire solutions to a quasilinear purely critical competitive system

TL;DR

This paper studies purely critical, competitive systems driven by the $p$-Laplacian in $\mathbb{R}^N$, introducing a symmetric pinwheel framework to produce fully nontrivial, nonnegative vector solutions and to obtain infinitely many nodal solutions for the scalar equation. It first proves a nonexistence result for ground state solutions of the system on any domain via dilation invariance and maximum principles, then develops a symmetric variational setting with a $G$-action and a cyclic coupling to realize least-energy pinwheel solutions; the construction extends from balls or half-spaces to $\mathbb{R}^N$ despite the lack of a general unique continuation for $p\neq 2$. It then proves that the purely critical scalar equation $-\Delta_p w=|w|^{p^*-2}w$ has infinitely many nonradial sign-changing solutions for $N\ge 4$ and $p\in(1,N)$ by a group-action argument, producing infinitely many nodal solutions. Overall, the work extends known $p=2$ multiplicity results to the quasilinear setting, providing a concrete variational strategy for managing critical growth and symmetry in coupled $p$-Laplacian systems.

Abstract

We establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the $p$-Laplacian in $\mathbb{R}^N$ whose nonlinear terms are purely critical. We also show that the purely critical equation for the $p$-Laplacian in $\mathbb{R}^N$ has infinitely many nodal solutions.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• proof
• proof : Proof of Theorem \ref{['thm:nonexistence']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof