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The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field. (English) Zbl 1448.35171

The paper is concerned with the Schrödinger-Bopp-Podolsky system in \(\mathbb{R}^3\) \[\left\{\begin{array}{l} -\Delta u+\omega u+q^2 \phi u=|u|^{p-2}u,\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2, \end{array}\right.\tag{1}\] where \(\omega>0\), \(a\ge 0\), \(p\in (2,3]\) and \(q>0\). By using the fibering method, the authors prove that system (1) has no nontrivial solutions for \(q\) large, and it has two radial solutions for \(q\) small. Some qualitative properties for the energy level of solutions and variational characterizations of the extremal values of \(q\) are also presented.

MSC:

35J47 Second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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References:

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