Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. (English) Zbl 0726.35011

In this interesting paper the asymptotic behaviour of solutions \(for\)
-\(\Delta\) u\(=\lambda f(u)\), \(u>0\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), \(\lambda\to 0\), is studied, where f(u) is an exponentially dominated nonlinear function. Especially the proved theorems deal with cases of uniform convergence, m-point blow up, entire blow up and with relations for blow up points. Some corollaries concerning special cases (\(\Omega\) star-shaped or convex) are added. In some proofs complex analysis is widely used.


35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations