On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems

TL;DR

This paper proves the existence of infinitely many weak solutions for a class of coupled quasilinear elliptic systems using nonsmooth critical point theory, advancing understanding of such systems' solution multiplicity.

Contribution

It introduces a novel application of nonsmooth critical point theory to establish multiple solutions for coupled quasilinear elliptic systems.

Findings

Infinitely many weak solutions exist in the specified function space.

Solutions are regular and bounded within the domain.

Method extends previous techniques to coupled systems.

Abstract

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } \Omega,\quad u_k = 0 \quad \text{on } \partial\Omega, \] where $k=1,\dots,d$, $ \Omega \subset \mathbb{R}^N $ is a bounded domain with $ N \geq 2 $, $ \boldsymbol{p} = (p_1, \dots, p_d) $, $ p_k > 1 $. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $ W_0^{1,\boldsymbol{p}}(\Omega) \cap L^\infty(\Omega; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(\Omega)=W_0^{1,p_1}(\Omega)\times\dots\times W_0^{1,p_d}(\Omega)$.