In this paper, we investigate the gravitational lensing for power-Maxwell charged quintessence black hole (BH) without cosmological constant in Rastall’s theory of gravity in both the strong-field limit and weak-field limit. Furthermore, we discuss the gravitational lensing by the linear electrodynamic (LE) and nonlinear electrodynamic (NLE) Maxwell field with the equation of state (EoS) parameter of the quintessence ([Formula: see text]) and compared to the case with Einstein’s gravity. We discuss the behaviors of the photon sphere radius [Formula: see text], impact parameter [Formula: see text], strong lensing coefficients [Formula: see text], [Formula: see text] and the deflection angle [Formula: see text] on the equatorial plane [Formula: see text] with the different values of Rastall parameter [Formula: see text] for the case of LE and NLE Maxwell field in the strong field limit. We also investigate the angular separation [Formula: see text], angular position [Formula: see text] and the magnification of the relativistic images. Further, we also study about the weak lensing BHs in Rastall gravity. We investigate the behaviors of weak deflection angle with the different values of Rastall parameter [Formula: see text] for the case of LE and NLE Maxwell field in the weak-field limit.

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We investigate the strong gravitational lensing on equatorial plane as well as quasi-equatorial plane by the Kerr–Newman-Nut-Quintessence (KNNQ) black hole (BH) with the equation of state (EoS) parameter of the quintessence [Formula: see text] and the quintessence density [Formula: see text]. Our results show that the strong gravitational lensing in the KNNQ black hole space–time has some distinct behaviors from those in the backgrounds of the four dimension Kerr black hole. Also, we investigate the strong gravitational lensing on equatorial plane as well as quasi-equatorial plane by the KNNQ BH with the effects of Nut charge, spin parameter and quintessence parameter. First, we calculate the null geodesic equations using the Hamilton–Jacobi separation method. Then we investigate the equatorial lensing by KNNQ black hole. We obtain the deflection angle and deflection coefficients in the equatorial plane, which is affected by EoS parameter of the quintessence [Formula: see text], quintessence density [Formula: see text], Nut parameter [Formula: see text], spin parameter [Formula: see text] and quintessence parameter [Formula: see text] [Formula: see text]. Next, we discuss the lens equation and the observables in the equatorial plane. Finally, we investigate gravitational lensing by the KNNQ black hole in the quasi-equatorial plane. In this work, the quintessence density [Formula: see text], the EoS parameter of the quintessence [Formula: see text], Nut parameter [Formula: see text], spin parameter [Formula: see text] and quintessence parameter [Formula: see text] [Formula: see text] have significant effects on the strong gravitational lensing both in equatorial plane as well as quasi-equatorial plane.

We study the gravitational lensing of acoustic charged black holes in strong and weak field limit approximations. For this purpose, we first numerically obtain the deflection limit coefficients and deflection angle in the strong field limit. We observe that the strong deflection angle α D increases with increasing magnitude of the charged parameter Q and that the strong deflection angle α D of an acoustic charged black hole with tuning parameter ξ = 4 is greater than that of a standard Reissner–Nordström black hole (ξ = 0). We also study the astrophysical consequences via strong gravitational lensing by taking the example of various supermassive black holes in the center of several galaxies and observe that the acoustic charged black hole could be quantitatively distinguished from standard Reissner–Nordström (ξ = 0) and standard Schwarzschild (ξ = 0, Q = 0) black holes. Furthermore, by using the Gauss–Bonnet theorem, we derive the weak deflection angle in the background of an acoustic charged black hole in the curved spacetime. We find that, for fixed values of the charged parameter Q and the tuning parameter (ξ = 0 or 4), the weak deflection angle σ D decreases with the impact parameter b. We also observe that the weak deflection angle σ D decreases with increasing magnitude of the charged parameter Q for a fixed value of the tuning parameter (ξ = 0 or 4). Our results suggest that the observational test for an acoustic charged black hole is indeed feasible, and it is generalized to the cases of acoustic Schwarzschild (Q = 0), standard Reissner–Nordström (ξ = 0), and standard Schwarzschild (ξ = 0, Q = 0) black holes.

In this work, we construct two new wormhole solutions in the theory dealing with non-minimal coupling between curvature and matter. We take into account an explicitly non-minimal coupling between an arbitrary function of scalar curvature [Formula: see text] and the Lagrangian density of matter. For this purpose, we discuss the Wormhole geometries inspired by non-minimal curvature coupling in [Formula: see text] gravity for linear model in [Formula: see text] as well as nonlinear model in [Formula: see text]. To derive these solutions, we choose the Gaussian and Lorentzian density distributions. To check the viability of these solutions, we plot the graphs for energy conditions and wormhole parameters. It is found that obtained wormhole solutions in both distributions satisfy the energy condition. The resulting wormhole solutions for both non-commutative distributions are determined to be physically stable when we evaluate the stability of these wormhole solutions graphically. It is concluded that wormhole solutions exist with viable physical properties in the non-minimal curvature–matter coupling of [Formula: see text] gravity with Gaussian and Lorentzian distributions.