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Volume 38, Issue 1
Multiple Positive Solutions for a Nonhomogeneous Schrödinger-Poisson System with Critical Exponent

Lijun Zhu & Hongying Li

DOI: 10.4208/jpde.v38.n1.2

J. Part. Diff. Eq., 38 (2025), pp. 21-33.

Published online: 2025-04

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  • Abstract

In this paper, we consider the following nonhomogeneous Schrödinger-Poisson system $$\begin{cases}-∆u+u+\eta \phi u=u^5+\lambda f(x), \ & x\in\mathbb{R}^3,\\ -∆\phi=u^2, \ & x\in\mathbb{R}^3, \end{cases}$$where $\eta\ne 0,$ $λ>0$ 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 $\eta >0$ has at least two positive solutions, the system with $\eta<0$ has at least one positive solution. Our result generalizes and improves some recent results in the literature.

  • Keywords

Schrödinger-Poisson system, critical exponent, variational method, positive solutions.

  • AMS Subject Headings

35B33, 35J20, 35J60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-38-21, author = {Zhu , Lijun and Li , Hongying}, title = {Multiple Positive Solutions for a Nonhomogeneous Schrödinger-Poisson System with Critical Exponent}, journal = {Journal of Partial Differential Equations}, year = {2025}, volume = {38}, number = {1}, pages = {21--33}, abstract = {

In this paper, we consider the following nonhomogeneous Schrödinger-Poisson system $$\begin{cases}-∆u+u+\eta \phi u=u^5+\lambda f(x), \ & x\in\mathbb{R}^3,\\ -∆\phi=u^2, \ & x\in\mathbb{R}^3, \end{cases}$$where $\eta\ne 0,$ $λ>0$ 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 $\eta >0$ has at least two positive solutions, the system with $\eta<0$ has at least one positive solution. Our result generalizes and improves some recent results in the literature.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v38.n1.2}, url = {http://global-sci.org/intro/article_detail/jpde/23951.html} }
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 -

In this paper, we consider the following nonhomogeneous Schrödinger-Poisson system $$\begin{cases}-∆u+u+\eta \phi u=u^5+\lambda f(x), \ & x\in\mathbb{R}^3,\\ -∆\phi=u^2, \ & x\in\mathbb{R}^3, \end{cases}$$where $\eta\ne 0,$ $λ>0$ 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 $\eta >0$ has at least two positive solutions, the system with $\eta<0$ has at least one positive solution. Our result generalizes and improves some recent results in the literature.

Zhu , Lijun and Li , Hongying. (2025). Multiple Positive Solutions for a Nonhomogeneous Schrödinger-Poisson System with Critical Exponent. Journal of Partial Differential Equations. 38 (1). 21-33. doi:10.4208/jpde.v38.n1.2
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