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

Abstract

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