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We investigate a 3D nonlinear focusing Schr\"odinger equations with perturbed energy critical inhomogeneous non-linearity in the radial regime, i ₜ u + u =|x|^-a |u|^p-2 u - |x|^-b |u|^4-2bu in \, \, Rₜ Rₓ³, where 0<a, b<2 and 2+4-2a3 p 6-2a. First, we prove the existence/nonexistence of ground states. Subsequently, we establish the scattering versus blowup below the ground-state energy threshold. Our methodology hinges on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. An intriguing aspect of our study is the absence of scaling invariance within the equation, attributed to the interplay of competing nonlinearities. Moreover, the presence of singular weights serves as a barrier to translation invariance in the spatial variable, adding more complexity to the analysis. To the best of our knowledge, this study represents the inaugural exploration of the inhomogeneous nonlinear Schr\"odinger equation with a leading-order focusing energy critical nonlinearity and a defocusing perturbation.
Gou et al. (Mon,) studied this question.