Schrödinger-Poisson equations with singular potentials in $R^3$

Abstract

The existence and $L^{\infty }$ estimate of positive solutions are discussed for the following Schrödinger-Poisson system {ll} -Δu +(λ+\frac{1}{|y|^α})u+ϕ(x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -Δϕ= u^2,\ \lim\limits_{|x|\rightarrow +\infty}ϕ(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $\lambda ⩾0$, $\alpha \in [0,8)$ and $\max \{2,\frac{2+\alpha }{2}\}<p<5$.

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