Infinitely many solutions for Schrödinger-Newton equations

Abstract

We prove the existence of infinitely many non-radial positive solutions for the Schrödinger-Newton system $$\{\begin{array}{ll}\Delta u-V(|x|)u+\Psi u=0,& x\in {\mathbb{R}}^3,\\ \Delta \Psi +\frac{1}{2}{u}^2=0,& x\in {\mathbb{R}}^3,\end{array}$$ provided that $V(r)$ has the following behavior at infinity: $$ V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+θ}}\right) \quad\mbox{ as } r\rightarrow\infty, $$ where $\frac{1}{2}\le m<1$ and $a,V_0,\theta$ are some positive constants. In particular, for any $s$ large we use a reduction method to construct $s-$bump solutions lying on a circle of radius $r\sim (s\log s{)}^{\frac{1}{1-m}}$.

18 pages

References in corpus (1)

• Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem