Elliptic Schr\"odinger equations with gradient-dependent nonlinearity and Hardy potential singular on manifolds

TL;DR

This paper investigates boundary value problems for elliptic Schr"odinger equations with gradient-dependent nonlinearities and Hardy potential singularities on manifolds, establishing existence results under subcritical conditions and capacity criteria.

Contribution

It introduces new existence criteria for solutions to elliptic equations with singular potentials and gradient nonlinearities, extending previous results to manifold settings.

Findings

Existence of solutions under subcritical integral conditions.

Characterization of solutions using Bessel capacities.

Identification of subcritical ranges for parameters p and q.

Abstract

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, 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.