Effect of weights on stable solutions of a quasilinear elliptic equation

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}$)$\mathrm{.}Wereferto$f(x)$asweightandto$F(u)$ 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 $f(x)=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$, 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.