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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.

##### MSC:
 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
##### Keywords:
uniform convergence; blow up