A priori estimates for solutions to nonlinear elliptic equations (Q1261958)

The paper treats the Dirichlet problem for a two-dimensional semilinear elliptic partial differential equation involving the Laplacian. The domain is assumed to be bounded and smooth. In the case of a nonlinear term with exponential growth the authors establish interior and boundary \(L^ \infty\)-a priori estimates. The interior estimates are obtained by following a ``blow-up'' idea of Brezis and Merle whereas the boundary estimates are proved by using the method of moving planes. The results give a partial answer to a problem posed by Brezis and Merle.