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

Abstract

We prove that for each \(p\in (1,\infty )\) the energy functional associated with the Dirichlet system

$$\begin{aligned} \left\{ \begin{array}{lll} -{\text {div}}(\phi _{p}(\left| \nabla u\right| )\nabla u)=\partial _{1}F(u,v) & \textrm{in} & \Omega ,\\ -{\text {div}}(\phi _{p}(\left| \nabla v\right| )\nabla v)=\partial _{2}F(u,v) & \textrm{in} & \Omega ,\\ u=v=0 & \textrm{on} & \partial \Omega , \end{array} \right. \end{aligned}$$

admits at least one global, nonnegative minimizer \((u_{p},v_{p})\in W_{0}^{\Phi _{p}}(\Omega )\times W_{0}^{\Phi _{p}}(\Omega )\) which converges uniformly on \(\overline{\Omega }\) to \((d_{\Omega },d_{\Omega }),\) as \(p\rightarrow \infty \). Here \(\Phi _{p}(t):=\int _{0}^{t}s\phi _{p}(\left| s\right| )\textrm{d}s\) and \(d_{\Omega }\) stands for the distance function to the boundary \(\partial \Omega \).