TY - JOUR
T1 - Multiple Positive Solutions for a Nonhomogeneous Schrödinger-Poisson System with Critical Exponent
AU - Zhu , Lijun
AU - Li , Hongying
JO - Journal of Partial Differential Equations
VL - 1
SP - 21
EP - 33
PY - 2025
DA - 2025/04
SN - 38
DO - http://doi.org/10.4208/jpde.v38.n1.2
UR - https://global-sci.org/intro/article_detail/jpde/23951.html
KW - Schrödinger-Poisson system, critical exponent, variational method, positive solutions.
AB - <p style="text-align: justify;">In this paper, we consider the following nonhomogeneous Schrödinger-Poisson system $$\begin{cases}-∆u+u+\eta \phi u=u^5+\lambda f(x), \ &amp; x\in\mathbb{R}^3,\\ -∆\phi=u^2, \ &amp; x\in\mathbb{R}^3, \end{cases}$$where $\eta\ne 0,$ $λ&gt;0$ is a real parameter and $f∈L^{\frac{6}{5}}(\mathbb{R}^3)$ is a nonzero nonnegative function.
By using the Mountain Pass theorem and variational method, for $λ$ small, we show that
the system with $\eta &gt;0$ has at least two positive solutions, the system with $\eta&lt;0$ has at
least one positive solution. Our result generalizes and improves some recent results in
the literature.</p>