On the planar Schrödinger–Poisson system

Cingolani, Silvia; Weth, Tobias

doi:10.1016/j.anihpc.2014.09.008

Cingolani, Silvia ; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 169-197

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Résumé

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

{\begin{matrix} - Δ u + a (x) u + γ w u = 0, \\ Δ w = u^{2} \end{matrix} in R^{2}

with a positive function

a \in L^{\infty} (R^{2})

and

γ > 0

. In particular, we deal with the periodic setting where the corresponding functional is invariant under

Z^{2}

-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

R^{2}

and

w (x) \to - \infty

as

| x | \to \infty

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.

MR Zbl

DOI : 10.1016/j.anihpc.2014.09.008

Classification : 35J50, 35Q40
Keywords: Schrödinger–Poisson system, Logarithmic convolution potential, Standing wave solutions

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     author = {Cingolani, Silvia and Weth, Tobias},
     title = {On the planar {Schr\"odinger{\textendash}Poisson} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {169--197},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.09.008},
     mrnumber = {3436430},
     zbl = {1331.35126},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.09.008/}
}

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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

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