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: $$-\mathrm{d}\mathrm{i}\mathrm{v}(A_k(x,u_k)\mathrm{\mid }\mathrm{\nabla }{u}_k{\mathrm{\mid }}^{p_k-2}\mathrm{\nabla }{u}_k)+{\displaystyle \frac{1}{p_k}}{D}_s{A}_k(x,u_k)\mathrm{\mid }\mathrm{\nabla }{u}_k{\mathrm{\mid }}^{p_k}=g_k(x,u)\text{in }\mathrm{\Omega },u_k=0\text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $k=1,\dots ,d$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $N\ge 2$, $\mathit{p}=(p_1,\dots ,p_d)$, $p_k>1$. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $W_0^{1,\mathit{p}}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^d)$, where $W_0^{1,\mathit{p}}(\mathrm{\Omega })=W_0^{1,p_1}(\mathrm{\Omega })\times \cdots \times W_0^{1,p_d}(\mathrm{\Omega })$.