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

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 $W_0^{1,\boldsymbol p}(\Omega)=W_0^{1,p_1}(\Omega)\times\dots\times W_0^{1,p_d}(\Omega)$.