The limiting behavior of global minimizers in non-reflexive Orlicz-Sobolev spaces

Abstract:

Let $\Omega$ be a smooth, bounded $N$-dimensional domain. For each $p>N$, let $\Phi _{p}$ be an N-function satisfying $p\Phi _{p}(t)\leq t\Phi _{p}^{\prime }(t)$ for all $t>0$, and let $I_{p}$ be the energy functional associated with the equation $-\Delta _{\Phi _{p}}u=f(u)$ in the Orlicz-Sobolev space $W_{0}^{1,\Phi _{p}}(\Omega )$. We prove that $I_{p}$ admits at least one global, nonnegative minimizer $u_{p}$ which, as $p\rightarrow \infty$, converges uniformly on $\overline {\Omega }$ to the distance function to the boundary $\partial \Omega$.