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A two-dimensional Dirichlet problem with an exponential nonlinearity. (English) Zbl 0543.35036
The two-dimensional nonlinear Dirichlet problem \((\lambda>0) -\Delta u=\lambda(e^ u+\psi e^{-u})\) \(y\in \Omega\); \(u=\phi\), \(y\in \partial \Omega\) 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 \(\psi\) is analytic in \(\Omega\) and continuous on \({\bar \Omega}\), is considered. A large norm solution for \(\lambda\) tending to 0 is obtained using techniques similar to those in the author’s earlier work [ibid. 14, 719-735 (1983)] for the case where \(\psi \equiv 0\), \(\phi \equiv 0\). Specifically, a third-order asymptotic solution in \(\lambda\) is obtained and used as a first approximation in a modified Newton’s method which is then shown to converge.

35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A35 Theoretical approximation in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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