On the planar Schrödinger-Poisson system

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• 作者：Silvia Cingolani^a ; ^{silvia.cingolani@poliba.it" class="auth_mail" title="E-mail the corresponding author} ; Tobias Weth^b ; ^{weth@math.uni-frankfurt.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J50 ; 35Q40
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：January-February 2016
• 年：2016
• 卷：33
• 期：1
• 页码：169-197
• 全文大小：501 K

文摘

We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system with a positive function a∈L^∞(R²)$a\in L^{\infty }({\mathbb{R}}^2)$ and γ>0$\gamma >0$. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z²${\mathbb{Z}}^2$-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)$(u,w)$ 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$u>0$ in R²${\mathbb{R}}^2$ and w(x)→−∞$w(x)\to -\infty$ as |x|→∞$|x|\to \infty$ 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.