Mathematics – Analysis of PDEs
Scientific paper
2011-09-23
Mathematics
Analysis of PDEs
Submitted in July 2011 and 21 pages
Scientific paper
We establish Liouville-type theorems for stable and finite Morse index weak solutions of $-\Delta_p u= f(x) F(u) $ in $\mathbb{R}^N$. For a general non-linearity $F\in C^1(\mathbb{R})$ and $f(x)=|x|^\alpha$, we prove such theorems in dimensions $N\le \frac{4(p+\alpha)}{p-1}+p$, for bounded radial stable solutions. Then, we give some point-wise estimates for not necessarily bounded solutions. Also, similar theorems will be proved for both radial finite Morse index (not necessarily bounded) and stable (not necessarily radial nor bounded) solutions with three different non-linearities $F(u)=e^u$; $u^q, q>p-1$ and $-u^{q}, q<0$, known as the Gelfand, the Lane-Emden and the negative exponent non-linearities, respectively. The remarkable fact is that the power profile $f(x)\simeq |x|^\alpha$ will push the critical dimension. This paper is a continuation of \cite{cf,mostafagh}.
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