Asymptotic Behavior and Classification of Solutions to Hartree Type Equations with Exponential Nonlinearity

In this paper, we are mainly concerned with the following Hartree type equations with exponential nonlinearity (-Delta)u(x) = (1/|x|(sigma) * e(pu))e(pu(x)), in R-2 where p is an element of(0, +infinity), u may change sign. We first prove the equivalence between the PDEs and the corresponding integral equations, further to get the exact asymptotic behavior of solutions to the above PDEs equation. Finally, we classify all classical solutions to the integral equations via the method of moving spheres in integral form. Consequently, we obtain the classification results of classical solutions for the PDEs.