## 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.

### MSC:

 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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