Effect of weights on stable solutions of a quasilinear elliptic equation

TL;DR

This paper investigates how weights influence the existence of stable solutions in quasilinear elliptic equations, revealing that positive lower bounds on weights limit solutions, while nonnegative weights affect critical dimensions.

Contribution

It establishes Liouville theorems for stable solutions with weights in quasilinear elliptic equations, including specific nonlinearities like exponential and power functions, highlighting the impact of weight functions.

Findings

Liouville theorems for bounded radial stable solutions in certain dimensions

Impact of weight bounds on the existence of solutions

Extension to specific nonlinearities like exponential and power functions

Abstract

In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\Delta_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in C^1(\mathbb{R})$. We refer to $f(x)$ as {\it weight} and to $F(u)$ as {\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\le f$ will push certain critical dimensions. This analytical observation has potential to be applied in various models to push certain well-known critical dimensions. For a general nonlinearity $F\in C^1(\mathbb{R})$ and $f(x)=|x|^\alpha$, we prove Liouville theorems in dimensions $n\le \frac{4(p+\alpha)}{p-1}+p$, for bounded radial stable solutions. For specific nonlinearities $F(u)=e^u$, $u^q$ where $q>p-1$ and $-u^{q}$ where $q<0$, known as the Gelfand, the Lane-Emden and the negative exponent nonlinearities, respectively, we prove Liouville theorems for both radial finite Morse index (not necessarily bounded) and stable (not necessarily radial nor bounded) solutions.