A priori estimates for non-coercive Dirichlet problems with subquadratic gradient terms

Abstract

We deal with some quasilinear elliptic problems posed in a bounded smooth convex domain Ω⊂RN (N≥3), namely{−Δu=λu+μ(x)|∇u|q+f(x),x∈Ω,u=0,x∈∂Ω, where the data satisfyμ∈L∞(Ω),μ⪈0;f∈Lp(Ω),p>N,f⪈0 and 1<q≤2. We provide sufficient conditions on f,μ (allowing μ to vanish on ∂Ω) that yield the sharp estimate λ‖u‖L∞(Ω)≤C for any bounded solution u with λ∈(0,λ1), which is the non-coercive regime. The estimate leads to remarkable consequences such as a multiplicity result and a precise asymptotic behavior of the bounded but blowing up solutions as λ→0+.