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.


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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