Year: 2025
Type: preprint
Abstract: We study bounded positive weak solutions of the singular quasilinear problem $-Δ_p u+\vartheta|\nabla u|^q=u^{-γ}+f(u)$ in a bounded $C^2$ domain, with zero Dirichlet boundary data, where $p>1$, $γ>0$, $01$ the leading profile is a universal power of the boundary distance; for $γ=1$ it is logarithmic, with the sharp constant $(p/(p-1))^{1/p}$. In both cases the normalized gradient converges uniformly to the inward normal multiplied by the corresponding constant. For $0<γ<1$ we prove global $C^{1,α}$ regularity and a classical Hopf boundary estimate. We also obtain the optimal Sobolev integrability threshold and the sharp logarithmic modulus of continuity. Finally, positivity of the total source on the solution range yields symmetry and monotonicity in symmetric domains strictly convex in one direction; when $N\ge2$ and $\vartheta>0$, we assume $q\ge\max\{1,p/2\}$.
Source: arXiv (Cornell University)
Author: Le, Phuong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416242002
Cites:
Cited by:
Related to: