Volume 48, Issue 3
Elliptic Systems with a Partially Sublinear Local Term

J. Math. Study, 48 (2015), pp. 290-305.

Published online: 2015-09

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• Abstract

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.

• Keywords

Schrödinger-Poisson system, Klein-Gordon-Maxwell system, infinitely many solutions.

35A15, 35J50

jing@cnu.edu.cn (Yongtao Jing)

zliu@cnu.edu.cn (Zhaoli Liu)

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@Article{JMS-48-290, author = {Jing , Yongtao and Liu , Zhaoli}, title = {Elliptic Systems with a Partially Sublinear Local Term}, journal = {Journal of Mathematical Study}, year = {2015}, volume = {48}, number = {3}, pages = {290--305}, abstract = {

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} }
TY - JOUR T1 - Elliptic Systems with a Partially Sublinear Local Term AU - Jing , Yongtao AU - Liu , Zhaoli JO - Journal of Mathematical Study VL - 3 SP - 290 EP - 305 PY - 2015 DA - 2015/09 SN - 48 DO - http://doi.org/10.4208/jms.v48n3.15.07 UR - https://global-sci.org/intro/article_detail/jms/9932.html KW - Schrödinger-Poisson system, Klein-Gordon-Maxwell system, infinitely many solutions. AB -

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.

Yongtao Jing & Zhaoli Liu. (2019). Elliptic Systems with a Partially Sublinear Local Term. Journal of Mathematical Study. 48 (3). 290-305. doi:10.4208/jms.v48n3.15.07
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