Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents

In this paper, we consider the existence of a non-trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space. Firstly, we get the existence of a local Palais–Smale sequence by verifying the geometric conditions of the Mountain Pass Lemma. Secondly, we study the concentration properties of the Palais–Smale sequence of a zero weak limit. Thirdly, we deduce by contradiction to eliminate the possibility of a zero weak limit case. Lastly, applying a monotonic inequality, we shall prove that the nontrivial weak limit of the Palais–Smale sequence is indeed a weak solution.