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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 $d=2$?

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  • $\begingroup$ Did you try to contact Berestycki and Lions asking them if there has been any progress in this case? $\endgroup$
    – Alan
    Commented Aug 7, 2016 at 18:44
  • $\begingroup$ No, I haven't even though about it, actually. $\endgroup$
    – Amir Sagiv
    Commented Aug 7, 2016 at 19:03

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Depending on what precisely you are interested in, one place to start may be

H. Berestycki, T. Gallouët, O. Kavian. Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 5, 307–310. (link to paper at Gallouët's website)

(also depending on the specific questions of interest, you might find it useful to look at, e.g., C. Alves, M. Souto, M. Montenegro. Existence of a ground state solution for a nonlinear scalar field equation with critical growth. Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 537–554).

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  • $\begingroup$ Thanks! Can you please link to the first paper? I can't find it in GoogleScholar, neither in French or in English $\endgroup$
    – Amir Sagiv
    Commented Jul 8, 2017 at 8:49

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