2025/09/01 by Tapia, José Carmona, Malanchini, Paolo, Aparicio, Antonio J. Martínez +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.01355
We investigate the existence and nonexistence of solutions to the Dirichlet problem \ \beginalignedat2 -Δp u + g(u) |∇ u|p amp;= λf(u) amp;amp;in Ω,
u amp;= 0 amp;amp;on ∂Ω, \endalignedat . where Ω⊂ ℝN is a smooth bounded domain, p∈ (1,∞), λ>0 and g∈ C(ℝ). Our main assumption is that :f ℝ→ ℝ is a continuous function such that f(s)>0 for all s∈ (α,β), where 0<α<β are two zeros of f.
If f(0)≥ 0, we show that an area condition involving f and g is both sufficient and necessary in order to have a pair (λ,u)∈ ℝ+× C01(Ω), with u≥ 0 and ‖u‖C(Ω)∈ (α,β], solving~\eqrefpba.
We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative g is, the stronger its regularizing effect on~\eqrefpba. We prove that, regardless of the shape of f, for any fixed λ, there always exists a function g such that~\eqrefpba admits a nonnegative solution with maximum in (α,β].