no. 1
On the planar Schrödinger–Poisson system
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 169-197.

We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system

{−Δu+a(x)u+γwu=0,Δw=u2in R2
with a positive function a∈L∞(R2) and γ>0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z2-translations and therefore fails to satisfy a global Palais–Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u,w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u>0 in R2 and w(x)→−∞ as |x|→∞ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.

DOI : 10.1016/j.anihpc.2014.09.008
Classification : 35J50, 35Q40
Mots clés : Schrödinger–Poisson system, Logarithmic convolution potential, Standing wave solutions 
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Cingolani, Silvia; Weth, Tobias. On the planar Schrödinger–Poisson system. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 169-197. doi : 10.1016/j.anihpc.2014.09.008. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.008/

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