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 - <p style="text-align: justify;">Let $1&lt;p&lt;2$. 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 $$\begin{cases} -\Delta u+V(x)u+\phi u=K(x)|u|^{p-2}u \&nbsp; \&nbsp; \ {\rm in}&nbsp; \ \mathbb{R}^3,\\ -\Delta\phi=u^2 \&nbsp; \&nbsp; \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ as well as for the Klein-Gordon-Maxwell system&nbsp;<span style="text-align: justify;">$$\begin{cases} -\Delta u+[V(x)-(\omega+e\phi)^2]u=K(x)|u|^{p-2}u \&nbsp; \&nbsp; \ {\rm in}&nbsp; \ \mathbb{R}^3,\\ -\Delta\phi+e^2u^2\phi=-e\omega u^2 \&nbsp; \&nbsp; \ {\rm in} \ \mathbb{R}^3 \end{cases}$$</span>&nbsp;where $ω, e &gt; 0.$ This is in sharp contrast to D&#39;Aprile and Mugnai&#39;s non-existence results.</p>