Effect of weights on stable solutions of a quasilinear elliptic equation

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.

arXiv commentThe original version posted on arxiv in 2011 and was not submitted for publication. This note is accepted in Nonlinear Dynamics and Systems Theory (Invited paper for a special issue)