Papers published in the English version of the journal
Existence and Asymptotic Behavior of Solutions of a Nonhomogeneous Quasilinear Schrödinger–Poisson System
Y. Wang,
J. Zhang Inner Mongolia Normal University, Hohhot, China
Abstract:
In this paper, we study the existence and asymptotic behaviour of solutions of the nonhomogeneous quasilinear Schrödinger–Poisson system
\begin{equation*} \begin{cases} -\Delta u +V(x)u+\lambda \phi u=f(x, u)+g(x),&x\in \mathbb{R}^3 , -\Delta \phi -\varepsilon^4 \Delta_4 \phi=\lambda u^2 ,&x \in \mathbb{R}^3, \end{cases} \end{equation*}
where
$\lambda$ and
$\varepsilon$ are positive parameters,
\begin{equation*} \Delta _4\phi =\operatorname{div}(|\nabla \phi|^2 \nabla \phi), \end{equation*}
$V$ is a continuous and coercive potential function with positive infimum, and
$f$ is a Carathéodory function defined on
$\mathbb{R}^3 \times \mathbb{R}$ and satisfying the classic Ambrosetti–Rabinowitz condition. Under some suitable assumptions on
$V(x)$,
$f(x,u)$, and
$g(x)$, we obtain the existence of two different energy nontrivial solutions by use of variational methods and truncation technique for sufficiently small
$\lambda$ and fixed
$\varepsilon$. Moreover, the asymptotic behaviour of these solutions is studied whenever
$\varepsilon$ and
$\lambda$, respectively, tend to zero.
Keywords:
variational method, nonhomogeneous, quasilinear Schrödinger–Poisson system. Received: 12.06.2024
Revised: 27.03.2025
Language: English
