We consider an action for an abelian gauge field for which the density is given by a power of the Maxwell Lagrangian. In d spacetime dimensions this action is shown to enjoy the conformal invariance if the power is chosen as d/4. We take advantage of this conformal invariance to derive black hole solutions electrically charged with a purely radial electric field. Because of considering power of the Maxwell density, the black hole solutions exist only for dimensions which are multiples of four. The expression of the electric field does not depend on the dimension and corresponds to the four-dimensional Reissner-Nordström field. Using the Hamiltonian action we identify the mass and the electric charge of these black hole solutions.

We show that three-dimensional massive gravity admits Lifshitz metrics with generic values of the dynamical exponent z as exact solutions. At the point z = 3, exact black hole solutions which are asymptotically Lifshitz arise. These spacetimes are three-dimensional analogues of those that were recently proposed as gravity duals for anisotropic scale invariant fixed points.The enormous success of gauge-gravity duality [1] has triggered the interest in generalizing the holographic techniques to other areas of physics. Recently, the attempts to generalize AdS/CFT correspondence to nonrelativistic condensed matter physics have received considerable attention. Besides being an active line of research, this has given raise to very interesting new ideas; see Ref.[2] and references therein for a review.Recently, candidates to be gravity duals for nonrelativistic scale invariant theories, both exhibiting Galilean invariance or not, have been proposed. In Refs. [3,4], spacetimes whose isometry group is the so-called Schrödinger group were proposed to be gravity duals for Galilean and scale invariant systems. In Ref.[5], the scale invariant fixed points that do not exhibit Galilean symmetry were also analyzed, and the metric of the corresponding gravity duals were introduced (see Eq.(2) below). These metrics manifestly exhibit the anisotropic scale invariance

We obtain electrically charged black hole solutions of the Einstein equations in arbitrary dimensions with a nonlinear electrodynamics source. The matter source is deriving from a Lagrangian given by an arbitrary power of the Maxwell invariant. The form of the general solution suggests a natural partition for the different ranges of this power. For a particular range, we exhibit a class of solutions whose behavior resemble to the standard Reissner-Nordström black holes. There also exists a range for which the black hole solutions approach asymptotically the Minkowski spacetime slower than the Schwarzschild spacetime. We have also found a family of not asymptotically flat black hole solutions with an asymptotic behavior growing slower than the Schwarzschild (anti) de Sitter spacetime. In odd dimensions, there exists a critical value of the exponent for which the metric involves a logarithmic dependence. This critical value corresponds to the transition between the standard behavior and the solution decaying to Minkowski slower than the Schwarzschild spacetime.