The principal eigenvalue for a class of singular quasilinear elliptic operators and applications

Abstract

We characterize the principal eigenvalue associated to the singular quasilinear elliptic operator −∆u − µ(x) |∇u| q uq−1 in a bounded smooth domain Ω ⊂ RN with zero Dirichlet boundary conditions. Here, 1 < q ≤ 2 and 0 ≤ µ ∈ L∞(Ω). As applications we derive some existence of solutions results (as well as uniqueness, nonexistence and homogenization results) to a problem whose model is    −∆u = λu + µ(x) |∇u| q |u| q−1 + f(x) in Ω, u = 0 on ∂Ω, where λ ∈ R and f ∈ Lp(Ω) for some p > N 2 .