In their seminal paper from 1983, Berestycki and Lions study and analyze equations of the following form: $$ -\Delta u = f(u), \quad u:\mathbb{R}^d\to \mathbb{R} \,.$$ where $f:\mathbb{R}\to \mathbb{R}$.

Their analysis is quite full for $d=1,3$- They give existence conditions, and describs the structure of the $C^2$ localized solutions. In the end of the paper, they explain why $d=2$ is harder, and there are no known similar results for it.

My Question: More then 30 years have passed - has anyone made a similar analysis for $d=2$?

In their seminal paper from 1983, Berestycki and Lions study and analyze equations of the following form: $$ -\Delta u = f(u), \quad u:\mathbb{R}^d\to \mathbb{R} \,.$$ where $f:\mathbb{R}\to \mathbb{R}$.

Their analysis is quite full for $d=1,3$- They give existence conditions, and describe the structure of the $C^2$ localized solutions. In the end of the paper, they explain why $d=2$ is harder, and there are no known similar results for it.

My Question: More then 30 years have passed - has anyone made a similar analysis for $d=2$?

In their seminal paper from 1983, Berestycki and Lions study and analyze equations of the sort $$ -\Delta u = f(u), \quad u:\mathbb{R}^d\to \mathbb{R} \,.$$ where $f:\mathbb{R}\to \mathbb{R}$.

Their analysis is quite full for $d=1,3$, gives existence conditions, and describes the structure of the $C^2$ localized solutions. In the end of the paper, they explain why $d=2$ is harder, and there are no known similar results for it.

My Question: More then 30 years have passed - has anyone made a similar analysis for $d=2$?

In their seminal paper from 1983, Berestycki and Lions study and analyze equations of the following form: $$ -\Delta u = f(u), \quad u:\mathbb{R}^d\to \mathbb{R} \,.$$ where $f:\mathbb{R}\to \mathbb{R}$.

Their analysis is quite full for $d=1,3$- They give existence conditions, and describs the structure of the $C^2$ localized solutions. In the end of the paper, they explain why $d=2$ is harder, and there are no known similar results for it.

My Question: More then 30 years have passed - has anyone made a similar analysis for $d=2$?