2025/01/05 by Gkikas, Konstantinos T., Nguyen, Phuoc-Tai · 1 citation
#35J10 #35J25 #35J61 #35J75 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.02605
Let Ω⊂ ℝN (N ≥ 3) be a C2 bounded domain and Σ⊂ Ω is a C2 compact boundaryless submanifold in ℝN of dimension k, 0≤ k < N-2. For μ≤ ((N-k-2)/(2))2, put Lμ:= Δ+ μdΣ-2 where dΣ(x) = dist(x,Σ). We study boundary value problems for equation -Lμu = g(u,|∇ u|) in Ω∖ Σ, subject to the boundary condition u=ν on ∂ Ω∪ Σ, where g: ℝ × ℝ+ → ℝ+ is a continuous and nondecreasing function with g(0,0)=0, ν is a given nonnegative measure on ∂ Ω∪ Σ. When g satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on ν. If g(u,|∇ u|) = |u|p|∇ u|q, there are ranges of p,q, called subcritical ranges, for which the subcritical integral condition is satisfied, hence the problem admits a solution. Beyond these ranges, where the subcritical integral condition may be violated, we establish various criteria on ν for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities.