Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem

where $\Omega \subset {\mathbf{R}}^N (N \ge 2)$ is an appropriately smooth bounded domain and $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution $u_\lambda$ is almost flat and almost equal to $\pi$ inside $\Omega$. We establish an asymptotic expansion of $u_\lambda(x) \enskip (x \in \Omega)$ when $\lambda \gg 1$, which is explicitly represented by $g$.