Maximal solutions for the ∞-eigenvalue problem
AbstractIn this article we prove that the first eigenvalue of the {\infty}-Laplacian\left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v,|\nabla v|-\lambda% _{1,\infty}(\Omega)v\}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\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{aligned} \displaystyle\min\{-\Delta_{\infty}v_{\ell},|\nabla v_{% \ell}|-\lambda_{1,\infty}(\Omega)v_{\ell}^{\ell}\}&\displaystyle=0&&% \displaystyle\text{in }\Omega,\\ \displaystyle v_{\ell}&\displaystyle=0&&\displaystyle\text{on }\partial\Omega.% \end{aligned}\right.In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed {1<p<\infty}.