Positive solutions of semilinear elliptic problems with a Hardy potential (1803.08397v1)

Abstract: Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0, \,p\ne 1$ and $\mu \in \mathbb{R},\,\mu\ne 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $\delta^{- \frac2{p-1}}$ at the boundary. In general domains we extend results of arXiv:arch-ive/1407.0288 and show that there exists a unique singular solutions $u$ such that $u/\delta^{\beta_-}\to c$ on the boundary for an arbitrary positive function $c \in C^{2+\gamma}(\partial\Omega) \, (\gamma \in (0,1)), c \ge 0$. Here $\beta_-$ is the smaller root of $\beta(\beta-1)+\mu=0$.