Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 72, pp. 1-14.

Title: New type of multi-bump solutions for Schrodinger-Poisson systems

Authors: Tao Wang (Hunan Univ. of Science and Tech., Xiangtan, Hunan, China)
         Xiaoyu Tian (Hunan Univ. of Science and Tech., Xiangtan, Hunan, China)
         Wenling He (Hunan Univ. of Science and Tech., Xiangtan, Hunan, China)

Abstract:
In this article, we study the existence of non-radial positive solutions of the Schrodinger-Poisson system
$$\displaylines{
-\Delta u+u+V(|x|)\Phi(x)u =Q(|x|) |u|^{p-1}u,
 \quad x\in \mathbb{R}^3, \cr
-\Delta\Phi=V(|x|)u^2,\quad x\in \mathbb{R}^3,
}$$
where $1< p < 5$ and $V, Q$ are radial potential functions. By developing some refined estimates, via the Lyapunov-Schmidt reduction method, we construct infinitely many multi-bump solutions when $V, Q$ have some suitable algebraical decay at infinity. The maximum points of those multi-bump solutions are located on the top and bottom circles of a cylinder.   This result not only gives a new type of multi-bump solutions but also extends the existence of multi-bump solutions to a general  class of potential functions with a relatively slow decay rate at infinity.

Submitted April 17, 2025. Published July 14, 2025.
Math Subject Classifications: 35B09, 35J05, 35J15, 35J60, 35Q55.
Key Words: Schrodinger-Poisson system; multi-bump solutions; Lyapunov-Schmidt reduction method.