具有指数和对数非线性的非均质薛定谔-泊松系统解的存在性和渐近行为

IF 1.4 3区 数学 Q1 MATHEMATICS Journal of Fixed Point Theory and Applications Pub Date : 2024-08-06 DOI:10.1007/s11784-024-01122-x
Xiaoli Lu, Jing Zhang
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引用次数: 0

摘要

在本文中,我们考虑以下具有指数和对数非线性的非均质准线性薛定谔-泊松系统 $$\begin{aligned}-Delta u+phi u =|u|^{p-2}u\log |u|^2 +\lambda f(u) +h(x),&{}\textrm{in}\hspace{5.0pt}\Omega ,\ -\Delta \phi -\varepsilon ^4 \Delta _4 \phi =u^2,&{}\Omega ,\ u=\phi =0,&{}\textrm{on}\hspace{5.0pt}\partial\Omega ,\\end{array}.\right。\end{aligned}$where (4<p<+\infty ,\,\varepsilon,\,\lambda >;0)都是参数,((mathrm{Delta _4 \phi = div(|\nabla \phi |^2 \nabla \phi )}),((Omega \subset {mathbb {R}}^2\ )是一个有界域,并且 f 具有指数临界增长。首先,利用还原论证、截断技术、埃克兰变分原理和山口定理,我们得到当 \(\lambda \) 足够大且 \(\varepsilon \) 固定时,上述系统至少有两个能量不同的解。最后,我们研究了解相对于参数 \(\varepsilon \)的渐近行为。
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Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\phi u =|u|^{p-2}u\log |u|^2 +\lambda f(u) +h(x),&{} \textrm{in} \hspace{5.0pt}\Omega ,\\ -\Delta \phi -\varepsilon ^4 \Delta _4 \phi =u^2,&{} \textrm{in}\hspace{5.0pt}\Omega ,\\ u=\phi =0,&{} \textrm{on}\hspace{5.0pt}\partial \Omega ,\\ \end{array} \right. \end{aligned}$$

where \(4<p<+\infty ,\,\varepsilon ,\,\lambda >0\) are parameters, \(\mathrm{\Delta _4 \phi = div(|\nabla \phi |^2 \nabla \phi )}\), \(\Omega \subset {\mathbb {R}}^2\) is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for \(\lambda \) large enough and \(\varepsilon \) fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters \(\varepsilon \).

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来源期刊
CiteScore
3.10
自引率
5.60%
发文量
68
审稿时长
>12 weeks
期刊介绍: The Journal of Fixed Point Theory and Applications (JFPTA) provides a publication forum for an important research in all disciplines in which the use of tools of fixed point theory plays an essential role. Research topics include but are not limited to: (i) New developments in fixed point theory as well as in related topological methods, in particular: Degree and fixed point index for various types of maps, Algebraic topology methods in the context of the Leray-Schauder theory, Lefschetz and Nielsen theories, Borsuk-Ulam type results, Vietoris fractions and fixed points for set-valued maps. (ii) Ramifications to global analysis, dynamical systems and symplectic topology, in particular: Degree and Conley Index in the study of non-linear phenomena, Lusternik-Schnirelmann and Morse theoretic methods, Floer Homology and Hamiltonian Systems, Elliptic complexes and the Atiyah-Bott fixed point theorem, Symplectic fixed point theorems and results related to the Arnold Conjecture. (iii) Significant applications in nonlinear analysis, mathematical economics and computation theory, in particular: Bifurcation theory and non-linear PDE-s, Convex analysis and variational inequalities, KKM-maps, theory of games and economics, Fixed point algorithms for computing fixed points. (iv) Contributions to important problems in geometry, fluid dynamics and mathematical physics, in particular: Global Riemannian geometry, Nonlinear problems in fluid mechanics.
期刊最新文献
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