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Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$

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  • In this paper, we are concerned with the following problem

    (P) $ -\Delta u + V(x)u+\lambda \phi (x) u =f(x,u), x\in \mathbb{R}^3$
    $ -\Delta\phi = u^2, \lim_{|x|\rightarrow +\infty}\phi(x)=0,$

    where $\lambda >0$ is a parameter, the potential $V(x)$ may not be radially symmetric, and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some simple assumptions on $V$ and $f$, we prove that the problem (P) has a positive solution for $\lambda$ small and has no any nontrivial solution for $\lambda$ large.

    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35Q35.

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