Energy minimizers of a thin film equation with Born repulsion force (Q652001)

Let \(\Omega\subset {\mathbb R}^m\) be a bounded domain with smooth boundary and assume that \(\alpha >1\), \(\beta >0\), and \(\varepsilon >0\). This paper is concerned with the study of the nonlinear equation \[ -\Delta u=p-u^{-\alpha}\left(1-\left(\frac\varepsilon u\right)^\beta\right)\qquad\text{in \(\Omega\)}, \] subject to the Neumann boundary condition \[ \frac{\partial u}{\partial\nu}=0\qquad\text{on \(\partial\Omega\)}, \] where \(p=u^{-\alpha}-\varepsilon^\beta u^{-\alpha-\beta}-\Delta u\). The first main result in this paper concerns the existence of global energy minimizers. Next, the author proves that if \(u\) is a local energy minimizer satisfying \[ \int_\Omega u^{-\alpha -\beta}<\infty, \] then \(u\) is a classical solution with positive lower bound. The asymptotic behaviour of global energy minimizers as \(\varepsilon\rightarrow 0\) is also studied in the present paper. The proofs combine energy estimates, truncation techniques, and lower semi-continuity arguments.