Schrödinger-Poisson systems with radial potentials and discontinuous quasilinear nonlinearity

Anmin Mao, Hejie Chang



We consider the following Schrödinger-Poisson system: $$ \begin{cases} -\triangle u+V(|x|)u+\phi u= Q(|x|) f(u) &\hbox{in } \mathbb{R}^3,\\ -\triangle \phi=u^{2} & \hbox{in } \mathbb{R}^3, \end{cases} $$ \noindent with more general radial potentials $V,Q$ and discontinuous nonlinearity $f$. The Lagrange functional may be locally {\it Lipschitz}. Using nonsmooth critical point theorem, we obtain the multiplicity results of radial solutions, we also show concentration properties of the solutions. This is in contrast with some recent papers concerning similar problems by using the classical Sobolev embedding theorems.


Schrödinger-Poisson equation; radial potentials; weighted Sobolev embedding

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