In this paper, we study the following Schrödinger-Born-Infeld system with a general nonlinearity$\{\begin{array}{cc}-\mathrm{\Delta }u+u+\varphi u=f(u)+\mu |u{|}^{4}u\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ -\mathrm{\text{div}}\left(\frac{\mathrm{\nabla }\varphi }{\sqrt{1-|\mathrm{\nabla }\varphi {|}^2}}\right)=u^2\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ u(x)\to 0,\varphi (x)\to 0,\hfill & \text{as}x\to \infty ,\hfill \end{array}$ where $\mu \ge 0$ and $f\in C(\mathbb{R},\mathbb{R})$ 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 $\mu =0$ which was left in [2] as an open problem.

在本文中，我们研究以下具有一般非线性的Schrödinger-Born-Infeld系统$\{\begin{array}{cc}-\mathrm{\Delta }ü+ü+\varphi ü=F（ü）+\mu |ü{|}^4ü\hfill & \text{在}{\mathbb{[R}}^3，\hfill \\ -\mathrm{\text{股利}}（\frac{\mathrm{\nabla }\varphi }{\sqrt{\mathrm{1个}-|\mathrm{\nabla }\varphi {|}^{\mathrm{2个}}}}）={ü}^{\mathrm{2个}}\hfill & \text{在}{\mathbb{[R}}^3，\hfill \\ ü（X）\to 0，\varphi （X）\to 0，\hfill & \text{作为}X\to \infty ，\hfill \end{array}$ 在哪里 $\mu \ge 0$ 和 $F\in C（\mathbb{[R}，\mathbb{[R}）$满足适当的假设。该系统源于非线性Schrödinger方程和Born-Infeld理论的适当耦合。在亚临界和临界情况下，我们使用一种新的摄动方法来证明上述系统的非平凡解的存在性和多重性。我们强调，我们的结果涵盖了案例$F（ü）=|ü{|}^{p-\mathrm{1个}}ü$ 为了 $p\in （\mathrm{2个}，5/\mathrm{2个}]$ 和 $\mu =0$ 在[2]中作为未解决的问题保留下来。