论文标题
Schrödinger出生的侵袭系统的扰动方法：亚临界和关键案例中的解决方案
A perturbation approach for the Schrödinger-Born-Infeld system: solutions in the subcritical and critical case
论文作者
论文摘要
在本文中，我们研究以下具有一般非线性$$ \ left \ { \ begin {array} {ll} - \ triangle u+u+u+ϕu = f（u）+μ| u |^4u \，\，\，\ mbox {in} - \ \ textrm {div} \ displayStyle \ bigG（\ frac {\ nablaϕ} {\ sqrt {\ sqrt {1- | \ nabla ϕ |^2}}} \ bigg）= u^2＆\ mbox＆\ mbox {in} u（x）\ rightArrow0，\，\，ϕ（x）\ rightArrow0，＆\，\ text {as} \，\，\，x \ rightarrow \ rightarrow \ infty，\ end \ end {array} \ right。 $$，$μ\ geq0 $和$ f \ in C（\ r，\ r）$满足合适的假设。 该系统源于非线性schrödinger方程和出生式菲尔德理论的合适耦合。我们使用一种新的扰动方法来证明在亚临界和关键案例中上述系统的非平凡解决方案的存在和多样性。我们强调的是，我们的结果涵盖了$ f（u）= | u |^{p-1} u $ for $ p \ in（2，{5}/{2}] $和$μ= 0 $，它留在\ cite {azzollini19}中，是一个空旷的问题。
In this paper, we study the following Schrödinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+ϕu=f(u)+μ|u|^4u\,\,&\mbox{in}\,\,\R^3,\\ -\textrm{div}\displaystyle\bigg(\frac{\nablaϕ}{\sqrt{1-|\nablaϕ|^2}}\bigg)=u^2&\mbox{in}\,\,\R^3,\\ u(x)\rightarrow0,\,\,ϕ(x)\rightarrow0,&\,\text{as}\,\,x\rightarrow\infty, \end{array} \right. $$ where $μ\geq0$ and $f\in C(\R,\R)$ satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case $f(u)=|u|^{p-1}u$ for $p\in(2,{5}/{2}]$ and $μ=0$ which was left in \cite{Azzollini19} as an open problem.
