Elliptic equations in R 2 with one-sided exponential growth ∗

We consider elliptic equations in bounded domains Ω ⊂ R 2 with nonlinearities which have exponential growth at + ∞ (subcritical and critical growth respectively) and linear growth λ at −∞, with λ> λ1, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ < λ1 is considered