## 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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### References:

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