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Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition. (English) Zbl 1324.35021

The article concerns Klein-Gordon-Maxwell systems of the form \[ \begin{cases} -\Delta u + V(x) u + \psi u = f(u), \\ -\Delta \phi = u^2, \end{cases} \, \quad \;x \in \mathbb R^3. \tag{1} \] It is assumed that the potential \(V(x)\) is positive and periodic with respect to the lattice \(\mathbb Z^3\). First, the author proves the existence of a ground state solution provided that \(f\) is subcritical and satisfies some weaker conditions than the well-known Ambrosetti-Rabinowitz condition. Second, the author obtains a positive solution for \(f(t) = f_0(t) + \lambda g(t)\), where \(f_0\) is subcritical and \(g\) is without growth restrictions provided \(\lambda > 0\) is sufficiently small.
The proofs rely on variational methods and perturbation arguments.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
35Q40 PDEs in connection with quantum mechanics
35A15 Variational methods applied to PDEs
35B20 Perturbations in context of PDEs
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