Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems

Abstract:

We consider the nonlinear eigenvalue problem \[ -\Delta u + g(u) = \lambda \sin u \quad \mathrm {in} \quad \Omega , \quad u > 0\quad \mathrm {in} \quad \Omega , \quad u = 0 \quad \mathrm {on} \quad \partial \Omega , \] 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 $u_\lambda (x) \quad (x \in \Omega )$ when $\lambda \gg 1$, which is explicitly represented by $g$.