Asymptotic Solutions for a Dirichlet Problem with an Exponential Nonlinearity
James L. Moseley · SIAM Journal on Mathematical Analysis · 1983
We consider the two-dimensional nonlinear Dirichlet problem \[ \begin{gathered} - \Delta u = \lambda e^u ,\quad y \in \Omega , \hfill \\ u = \phi ,\quad y \in \partial \Omega , \hfill \\ \end{gathered} \] where $y = (y_1 ,y_2 )$, $\Delta $ is the Laplacian operator, $\Omega $ is a simply connected region bounded by a smooth closed Jordan curve, the boundary data $\phi $ is continuous and $\lambda $ is positive. Our primary concern is with obtaining the large norm (second) solution for $\lambda $ tending to $0_ + $. This is accomplished by obtaining an asymptotic solution which is used as a first approximation for a modified Newton’s method. In this paper we examine the implicit constraints previously required for $\phi \equiv 0$ and extend the results to the case of nonzero boundary data.