Elliptic Schrödinger equations with gradient-dependent nonlinearity and Hardy potential singular on manifolds

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in ${\mathbb{R}}^N$ of dimension $k$, $0\le k<N-2$. For $\mu \le (\frac{N-k-2}{2}{)}^2$, put $L_{\mu }:=\Delta +\mu d_{\Sigma }^{-2}$ where $d_{\Sigma }(x)=\mathrm{dist}(x,\Sigma )$. We study boundary value problems for equation $-L_{\mu }u=g(u,|\nabla u|)$ in $\Omega \setminus \Sigma$, subject to the boundary condition $u=\nu$ on $\partial \Omega \cup \Sigma$, where $g:\mathbb{R}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a continuous and nondecreasing function with $g(0,0)=0$, $\nu$ is a given nonnegative measure on $\partial \Omega \cup \Sigma$. When $g$ satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on $\nu$. If $g(u,|\nabla u|)=|u{|}^p|\nabla u{|}^q$, 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 $\nu$ for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities.

34 pages. arXiv admin note: text overlap with arXiv:2203.01328