On the Emden-Fowler equation type involving double critical growth

In this article, we investigate a class of nonlinear elliptic equations driven by the p-Laplacian operator in the entire space $R^N$, known as the Emden-Fowler equation type. The complexity of the problem arises from the interplay of two distinct critical growth phenomena, characterized by both Sobolev and Hardy senses. We explore the existence of positive radial solutions, with the proof relying on variational methods. Due to multiple critical nonlinearities, the Mountain Pass Lemma does not yield critical points but only Palais-Smale sequences. The primary challenge lies in the asymptotic competition among the energies carried by these multiple critical nonlinearities.