Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.09221

Boundary singularities for the Lane-Emden equation

Weiwei Ao·Sho Katayama·Juncheng Wei
View original source ↗Read source document ↗Export metadata ↓

Abstract

For every $n\ge3$ and $(n+1)/(n-1)<p<(n+2)/(n-2)$, we construct positive solutions of $Δu+u^p=0$ in the half-space with zero Dirichlet data away from the origin, the homogeneous singular profile there, and Poisson-kernel decay at infinity, resolving a question of del Pino--Musso--Pacard. Under a weighted non-degeneracy condition, these half-space solutions serve as building blocks for positive very weak solutions in smooth bounded domains, singular along prescribed compact smooth boundary submanifolds of normal dimension $n$. We verify non-degeneracy in the initial stable interval and at and just beyond its threshold when this lies below $(n+2)/(n-2)$. For isolated boundary singularities, the construction applies throughout the strict subcritical interval outside a locally finite exceptional set.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Weiwei Ao, Sho Katayama, Juncheng Wei. 2026-11-06. Boundary singularities for the Lane-Emden equation. https://arxiv.org/abs/2611.09221

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Convergence of iterated restricted mean values to harmonic functions

Let $Ω\subset\mathbb{R}^n$ be a bounded domain and let $δ$ be a continuous positive function on $Ω$ with $B(x,δ(x))\subseteqΩ$ for every $x\inΩ$. Let $σ(f)(x)$ denote the average of $f$ over the ball $B(x,δ(x))$. Restricted mean value theorems give conditions under which a fixed point of $σ$ must be harmonic. We study the dynamical counterpart of this question: the behavior of the iterates $σ^k(f)$. We show that for \emph{every} bounded domain and every bounded measurable $f$ that is continuous at the points of $\partialΩ$, the iterates $σ^k(f)$ converge locally uniformly in $Ω$ to the Perron--Wiener--Brelot solution of the Dirichlet problem with boundary values $f|_{\partialΩ}$, and that the convergence is uniform on $\overlineΩ$ for all continuous $f$ if and only if $Ω$ is regular for the Dirichlet problem. This contains and extends earlier results for $C^{2,α}$ domains with $C^{2,α}$ data and for strictly convex domains with continuous data. We also give explicit rates of convergence: an algebraic rate $O(1/k)$ on domains with a uniform exterior ball condition and a geometric rate on convex domains, when $δ$ is comparable to the distance to the boundary.

math.AP↗
arXiv2026

On the regularity of entropy solutions to stochastic degenerate parabolic equations

We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does not require an assumption on the derivative of the symbol of the corresponding kinetic equation, a restriction imposed in earlier studies. This allows us to obtain regularity results for certain equations not accounted for by prior theory, albeit with reduced regularity exponents. Our approach uses a kinetic formulation with two transport equations, one of second order and one of first order, leveraging a form of "parabolic regularity" inherent in these equations that was not utilized in previous studies.

math.AP↗
arXiv2026

Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

Consider the Dirichlet-to-Neumann map $Λ_β$ associated with the Schrödinger operator $(D+β\A)^2$ with a magnetic potential in a bounded Lipschitz domain $Ω$, where $β>1$ is the field strength parameter. Assume that the magnetic field $\B=\nabla \times \A$ is of finite type. We show that if $β>β_0$, the ground state for $Λ_β$ decays exponentially away from a neighborhood of the subset of $\partialΩ$, on which $\B$ vanishes to the maximal order.

math.AP↗