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

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

2012-04-09

arxiv.org/abs/1204.1825v1

Mathematics

Analysis of PDEs

23pages

Scientific paper

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $\lambda\geqslant0$, $\alpha\in[0,8)$ and $\max\{2,\frac{2+\alpha}{2}\}

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