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

Konstantinos T. Gkikas, Phuoc-Tai Nguyen

TL;DR

This work analyzes semilinear elliptic equations with a Hardy-type singular potential concentrated on a compact submanifold Σ of a bounded domain and with nonlinear sources depending on the solution and its gradient. It develops a comprehensive framework combining sharp two-sided Green and Martin kernel estimates, weighted weak-L^p gradient bounds, and a boundary-trace approach to address boundary data supported on ∂Ω∪Σ. Existence results are established under a subcritical integral condition for general g, and extended to the power-type nonlinearity g(u,|∇u|) = |u|^p|∇u|^q via capacities, including subcritical, critical, and capacity-precise regimes; the latter are formulated in terms of Bessel-type capacities Cap_{⋅}^{Γ}. The paper also treats the critical Hardy case μ = ${((N-2)/2)}^2$, the case Σ = {0}, and provides a unified capacity-based criterion that generalizes previous results for source-type nonlinearities, with implications for boundary data localization 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.

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

TL;DR

This work analyzes semilinear elliptic equations with a Hardy-type singular potential concentrated on a compact submanifold Σ of a bounded domain and with nonlinear sources depending on the solution and its gradient. It develops a comprehensive framework combining sharp two-sided Green and Martin kernel estimates, weighted weak-L^p gradient bounds, and a boundary-trace approach to address boundary data supported on ∂Ω∪Σ. Existence results are established under a subcritical integral condition for general g, and extended to the power-type nonlinearity g(u,|∇u|) = |u|^p|∇u|^q via capacities, including subcritical, critical, and capacity-precise regimes; the latter are formulated in terms of Bessel-type capacities Cap_{⋅}^{Γ}. The paper also treats the critical Hardy case μ = , the case Σ = {0}, and provides a unified capacity-based criterion that generalizes previous results for source-type nonlinearities, with implications for boundary data localization on manifolds.

Abstract

Let () be a bounded domain and is a compact boundaryless submanifold in of dimension , . For , put where . We study boundary value problems for equation in , subject to the boundary condition on , where is a continuous and nondecreasing function with , is a given nonnegative measure on . When satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on . If , there are ranges of , 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 for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities.
Paper Structure (16 sections, 33 theorems, 142 equations)

This paper contains 16 sections, 33 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Submanifold $\Sigma$.
  5. Estimates on the Green function and Martin kernel
  6. Weak Lebesgue estimates of the gradient of Green and Martin operators
  7. Auxiliary estimates
  8. Estimate on the gradient of Green kernel and Martin kernel
  9. Nonlinear equations with subcritical source
  10. Subcritical source
  11. Nonlinear equations with $g(u,|\nabla u|) = |u|^p|\nabla u|^q$
  12. Abstract setting
  13. Necessary and sufficient conditions for the existence
  14. The case $\Sigma=\{0\}$ and $\mu=(\frac{N-2}{2})^2$
  15. Properties of the first eigenpair of $-L_\mu$
  16. ...and 1 more sections

Key Result

Theorem 1.3

Assume $g$ satisfies for some $\tilde{p}>1$, $\tilde{q}>1$, $\tilde{k} > 0$. Then there exists $\varrho_0>0$ depending on $N,\mu,\Omega,\Lambda_g,\tilde{k}, \tilde{p}, \tilde{q}$ such that for any $\varrho \in (0,\varrho_0)$, problem sourceprobrho admits a positive weak solution $u \geq \varrho \mathbb K_\mu [\nu]$ a.e.

Theorems & Definitions (57)

  • Definition 1.1: Boundary trace
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 47 more