Positive solutions of the prescribed mean curvature equation with exponential critical growth (Q2041501)

Given a bounded domain \(\Omega \subset \mathbb{R}^2\) with smooth boundary \(\partial\Omega\), the authors study the prescribed mean curvature equation given by \begin{align*} -\text{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=f(u) \quad\text{in }\Omega, \quad u=0 \quad\text{on }\partial\Omega, \end{align*} where \(f\colon\mathbb{R}\to\mathbb{R}\) is a superlinear continuous function with critical exponential growth. Based on an auxiliary problem along with the Nehari manifold by using Moser's iteration method and Stampacchia's estimates, the existence of a positive solution of the problem above is shown.