关于具有不定势的薛定谔 - 博普 - 波多尔斯基系统：基态、多重性和指数衰减
On the Schrödinger--Bopp--Podolsky system with indefinite potential: ground states, multiplicity and exponential decay
AI总结：
研究具有不定势的薛定谔 - 博普 - 波多尔斯基系统，通过局部环绕论证、莫尔斯理论、极小化技术及对称山路定理，在合适假设下证明非平凡解存在，解在无穷远处指数衰减，还得到基态解及无界解序列。
AI中文摘要：
本文研究薛定谔 - 博普 - 波多尔斯基系统。考虑势\(V\)不定，使得薛定谔算子\(-\Delta + V\)有有限维负空间的情况。在对势\(V\)和非线性项\(f(x,u)\)的合适假设下，通过局部环绕论证和莫尔斯理论证明非平凡解的存在性，且这些解在无穷远处指数衰减。还通过极小化技术得到一个基态解。若\(f(x,u)\)关于\(u\)为奇函数，利用对称山路定理得到一个无界解序列。
英文摘要：
In this paper, we study the Schrödinger--Bopp--Podolsky system \begin{equation*} \begin{cases} -Δu + V(x)u + ϕu = f(x,u), & \text{in } \mathbb{R}^3, -Δϕ+ a^2 Δ^2 ϕ= 4πu^2, & \text{in } \mathbb{R}^3. \end{cases} \end{equation*} We consider the case where the potential \(V\) is indefinite so that the Schrödinger operator \(-Δ+ V\) has a finite-dimensional negative space. Under suitable assumptions on the potential \(V\) and nonlinearity $f(x,u)$, we prove the existence of nontrivial solutions via a local linking argument and Morse theory. Moreover, these solutions are shown to decay exponentially at infinity. Additionally, a ground state solution is obtained by minimization techniques. Finally, if \(f(x,u)\) is odd with respect to \(u\), we obtain an unbounded sequence of solutions using the symmetric mountain pass theorem.
