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

TL;DR

The paper studies a class of coupled quasilinear elliptic systems with Dirichlet boundary conditions by formulating an energy functional on $W_0^{1,\boldsymbol p}(\Omega)$ and employing nonsmooth critical point theory. By introducing the weak slope, Concrete Palais–Smale condition, and an equivariant Mountain Pass framework, it proves the existence of infinitely many weak solutions that are uniformly bounded in $L^{\infty}$ space. A key contribution is establishing $L^{\infty}$ regularity of solutions and tailoring the compactness arguments to the nonsmooth setting. The results extend multiplicity theory to gradient-like systems with subcritical nonlinearities and Carathéodory coefficients.

Abstract

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } Ω,\quad u_k = 0 \quad \text{on } \partialΩ, \] where $k=1,\dots,d$, $ Ω\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}}(Ω) \cap L^\infty(Ω; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(Ω)=W_0^{1,p_1}(Ω)\times\dots\times W_0^{1,p_d}(Ω)$.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2: nonsmooththeory1
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Equivariant Mountain Pass, nonsmooththeory1
• Definition 2.6
• Theorem 2.7
• proof
• Corollary 2.8