| Abstract: |
| Let $\Omega\subset R^N$ ($N\geq 3$) be a bounded $C^2$ domain and $\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study positive solutions of the equation $(*)\, -\Delta u -\mu u/d_\Sigma^2 = g(u,|\nabla u|)$ in $\Omega$, where $\mu\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g: R\times R_+ \rightarrow R_+$ 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 $u=\nu$ in the case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions, provided that the total mass of $\nu$ is small. This presentation is based on a joint work with Konstantinos Gkikas. |
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