On the existence of positive solutions for some nonlinear elliptic problems in unbounded domain in R2

We are interested in the following nonlinear elliptic equation Δu+uϕ(.,u)=0 in D, where D is a smooth unbounded domain in R2. Under appropriate conditions on the nonlinearity ϕ(x,t), related to a certain Kato class, we give some existence results and asymptotic behavior for positive solutions of the above equation.