Semilinear Schr\"odinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities

Let ΩN (N≥ 3) be a bounded C² domain and Σ⊂∂Ω be a compact C² submanifold of dimension k. Denote the distance from Σ by d_Σ. In this paper, we study positive solutions of the equation (*)\, -Δ u -μ u/d_Σ² = g(u,|∇ u|) in Ω, where μ≤ ( N-k/2 )² and the source term g:R₊ → R₊ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data u=ν in two main cases, provided that the total mass of ν is small. In the first case g satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity g(u,|∇ u|) = |u|ᵖ|∇ u|q, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that ν is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of Σ, or under other equivalent conditions.