Maximal solutions for the Infinity-eigenvalue problem

In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v,\, |\nabla v|-\lambda_{1, \infty}(\Omega) v \} & = & 0 & \text{in} & \Omega v & = & 0 & \text{on} & \partial \Omega, \end{array} \right. $$ has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as $\ell \nearrow 1$ of concave problems of the form $$ \left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v_{\ell},\, |\nabla v_{\ell}|-\lambda_{1, \infty}(\Omega) v_{\ell}^{\ell} \} & = & 0 & \text{in} & \Omega v_{\ell} & = & 0 & \text{on} & \partial \Omega. \end{array} \right. $$ In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the concave problems as happens for the usual eigenvalue problem for the $p-$Laplacian for a fixed $1<p<\infty$.