On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ1λ1. Furthermore, because of the isolation of λ1λ1, we prove the existence of the second eigenvalue λ2λ2. Then, using the Trudinger–Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0<λ<λ10<λ<λ1 by the Mountain Pass Lemma, and in the case of λ1≤λ<λ2λ1≤λ<λ2 by the Linking Argument Theorem.