On the existence of global solutions of second-order quasilinear elliptic inequalities

TL;DR

This work addresses the existence of global positive solutions to the second-order quasilinear elliptic inequality $- \operatorname{div} A(x,u,\nabla u) \ge f(u)$ in ${\mathbb R}^n$, under Carathéodory-type structure and $p$-growth with $p>1$. The authors prove existence under a sharp integral condition on $f$, namely $\int_0^{\varepsilon} f(t) t^{-\left(1+\frac{n(p-1)}{n-p}\right)} dt < \infty$, and analyze a radial source-term scenario $F(|x|)$ via a nonincreasing density with $\int_0^{\infty} r^{\frac{n(p-1)}{p}} F(r) dr < \infty$, yielding global positive solutions with $\operatorname{ess\,inf}_{\mathbb R^n} u = 0$. The results show that, in the regime $n>p$, such solutions exist and can be positive almost everywhere under mild nonvanishing conditions on $F$, with positivity established through a weak Harnack framework. Collectively, the paper extends existence theory for nonradial, general quasilinear inequalities beyond standard radial reductions and provides decay-type estimates for the constructed solutions.

Abstract

We study the existence of global positive solutions of the differential inequalities $$ - \operatorname{div} A (x, u, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n \ge 2$ and $A$ is a Carathéodory function such that $$ (A (x, s, ζ) - A (x, s, ξ))(ζ- ξ) \ge 0, $$ $$ C_1 |ξ|^p \le ξ A (x, s, ξ), \quad |A (x, s, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 > 0, \; p > 1, $$ for almost all $x \in {\mathbb R}^n$ and for all $s \in {\mathbb R}$ and $ζ, ξ\in {\mathbb R}^n$.

Theorems & Definitions (5)

• Theorem 2.1
• Theorem 2.2
• Remark 2.1
• proof : Proof of Theorem \ref{['T2.2']}
• proof : Proof of Theorem \ref{['T2.1']}