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Let $1<p<2$. Under some assumptions on $V,$ $K,$ existence of infinitely many solutions $(u,\phi) ∈ H^1(\mathbb{R}^3) \times D^{1,2}(\mathbb{R}^3)$ is proved for the Schrödinger-Poisson system $$\begin{cases} -\Delta u+V(x)u+\phi u=K(x)|u|^{p-2}u \ \ \ {\rm in} \ \mathbb{R}^3,\\ -\Delta\phi=u^2 \ \ \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ as well as for the Klein-Gordon-Maxwell system $$\begin{cases} -\Delta u+[V(x)-(\omega+e\phi)^2]u=K(x)|u|^{p-2}u \ \ \ {\rm in} \ \mathbb{R}^3,\\ -\Delta\phi+e^2u^2\phi=-e\omega u^2 \ \ \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ where $ω, e > 0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v48n3.15.07}, url = {http://global-sci.org/intro/article_detail/jms/9932.html} }Let $1<p<2$. Under some assumptions on $V,$ $K,$ existence of infinitely many solutions $(u,\phi) ∈ H^1(\mathbb{R}^3) \times D^{1,2}(\mathbb{R}^3)$ is proved for the Schrödinger-Poisson system $$\begin{cases} -\Delta u+V(x)u+\phi u=K(x)|u|^{p-2}u \ \ \ {\rm in} \ \mathbb{R}^3,\\ -\Delta\phi=u^2 \ \ \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ as well as for the Klein-Gordon-Maxwell system $$\begin{cases} -\Delta u+[V(x)-(\omega+e\phi)^2]u=K(x)|u|^{p-2}u \ \ \ {\rm in} \ \mathbb{R}^3,\\ -\Delta\phi+e^2u^2\phi=-e\omega u^2 \ \ \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ where $ω, e > 0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results.