Existence and nonexistence of least energy nodal solution for a class of   elliptic problem in $\mathbb{R}^{2}$
  In this work, we prove the existence of least energy nodal solution for a class of elliptic problem in both cases, bounded and unbounded domain, when the nonlinearity has exponential critical growth in $\mathbb{R}^2$. Moreover, we also prove a nonexistence result of least energy nodal solution for the autonomous case in whole $\mathbb{R}^{2}$. 
Claudianor O. Alves and Denilson S. Pereira
math.AP