Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term

TL;DR

The paper addresses monotonicity of positive weak solutions to $-\Delta_p u=\frac{1}{u^{\gamma}}+f(u)$ in the half-space $\mathbb{R}^N_+$ with zero Dirichlet data on the boundary. It combines a priori estimates, barrier constructions, and a refined moving-plane method to prove that solutions are monotone nondecreasing in the direction normal to the boundary, even in the presence of a singular term. A key part of the analysis is the boundary behavior, showing sharp power-like bounds $u(x)\sim x_N^{\beta}$ with $\beta=\frac{p}{\gamma+p-1}$ and establishing continuity up to the boundary. The proof handles lack of compactness by rescaling and applying Liouville-type classification results, with separate compactness arguments for $1<p<2$ and $p>2$, ultimately yielding $\partial_{x_N}u\ge0$ in the whole half-space.

Abstract

We consider positive solutions to $\displaystyle -Δ_p u=\frac{1}{u^γ}+f(u)$ under zero Dirichlet condition in the half space. Exploiting a prio-ri estimates and the moving plane technique, we prove that any solution is monotone increasing in the direction orthogonal to the boundary.

Figures (1)

• Figure 1:

Theorems & Definitions (16)

• Definition 1
• Remark 2
• Theorem 3
• Definition 4
• Lemma 5
• Remark 6
• proof
• Theorem 7
• proof
• Proposition 8