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Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations. (English) Zbl 1057.35041

Existence and non-existence of solutions for the nonlinear elliptic system \[ \begin{gathered} -\Delta u+ [m^2_0- (\omega+\phi)^2] u=| u|^{p-2}u,\tag{1}\\ \Delta\phi= (\omega+\phi) u^2,\tag{2}\end{gathered} \] in the critical case \(p= 2^*= 2N/(N- 2)\) are investigated. If \(| m_0|> |\omega|\) and \(p=6\) then any weak solution \((u,\phi)\in H^1(\mathbb{R}^3)\times{\mathcal D}^{1,2}(\mathbb{R}^3)\) of system (1)–(2) vanishes identically. If \(| m_0|> |\omega|\) and \(4< q< 6\) then for each \(\mu> 0\) the system \[ \begin{gathered} -\Delta u+ [m^2_0- (\omega+\phi)^2] u= \mu| u|^{q-2} u+| u|^{2^*- 2} u,\tag{3}\\ \Delta \phi = (\omega+ \phi)u^2,\tag{2}\end{gathered} \] has at least a radically symmetric (nontrivial) solution \((u,\phi)\in H^1(\mathbb{R}^3)\times{\mathcal D}^{1,2}(\mathbb{R}^3)\). Moreover, if \(q= 4\) the system (3)–(2) still possesses a nontrivial solution provided that \(\mu\) is sufficiently large. These are the main results of this work.

MSC:

35Q40 PDEs in connection with quantum mechanics
35J60 Nonlinear elliptic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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