Uniform Convergence of Global Least Energy Solutions to Dirichlet Systems in Non-reflexive Orlicz–Sobolev Spaces

We prove that for each \(p\in (1,\infty )\) $p\in (1,\infty )$ the energy functional associated with the Dirichlet system \(\begin{aligned} \left\{ \begin{array}{lll} -{\text {div}}(\phi _{p}(\left| \nabla u\right| )\nabla u)=\partial _{1}F(u,v) & \textrm{in} & \Omega ,\\ -{\text {div}}(\phi _{p}(\left| \nabla v\right| )\nabla v)=\partial _{2}F(u,v) & \textrm{in} & \Omega ,\\ u=v=0 & \textrm{on} & \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}-\text{div}({\varphi }_p(\left|\mathrm{\nabla },u\right|)\mathrm{\nabla }u)={\partial }_{1}F(u,v)\hfill & \text{in}\hfill & \mathrm{\Omega },\hfill \\ -\text{div}({\varphi }_p(\left|\mathrm{\nabla },v\right|)\mathrm{\nabla }v)={\partial }_{2}F(u,v)\hfill & \text{in}\hfill & \mathrm{\Omega },\hfill \\ u=v=0\hfill & \text{on}\hfill & \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ admits at least one global, nonnegative minimizer \((u_{p},v_{p})\in W_{0}^{\Phi _{p}}(\Omega )\times W_{0}^{\Phi _{p}}(\Omega )\) $(u_p,v_p)\in W_0^{{\mathrm{\Phi }}_p}(\mathrm{\Omega })\times W_0^{{\mathrm{\Phi }}_p}(\mathrm{\Omega })$ which converges uniformly on \(\overline{\Omega }\) $\overline{\mathrm{\Omega }}$ to \((d_{\Omega },d_{\Omega }),\) $(d_{\mathrm{\Omega }},d_{\mathrm{\Omega }}),$ as \(p\rightarrow \infty \) $p\to \infty$ . Here \(\Phi _{p}(t):=\int _{0}^{t}s\phi _{p}(\left| s\right| )\textrm{d}s\) ${\mathrm{\Phi }}_p(t):={\int }_0^{t}s{\varphi }_p(\left|s\right|)\text{d}s$ and \(d_{\Omega }\) $d_{\mathrm{\Omega }}$ stands for the distance function to the boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ .