The principal eigenvalue for a class of singular quasilinear elliptic operators and applications
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2020Resumen
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
.