Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be a bounded $C^2$ domain and $\mathrm{\Sigma }\subset \mathrm{\partial }\mathrm{\Omega }$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\mathrm{\Sigma }$ by $d_{\mathrm{\Sigma }}$. In this paper, we study positive solutions of the equation $(\ast )-\mathrm{\Delta }u-\mu u\mathrm{/}{d}_{\mathrm{\Sigma }}^2=g(u,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })$ in $\mathrm{\Omega }$, where $\mu \le (\frac{N-k}{2}{)}^2$ and the source term $g:\mathbb{R}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(\ast )$ with boundary measure data $u=\nu$ in two main cases, provided that the total mass of $\nu$ 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 $g(u,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })=\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q$, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that $\nu$ is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of $\mathrm{\Sigma }$, or under other equivalent conditions.