Mathematics – Analysis of PDEs
Scientific paper
2011-08-15
Mathematics
Analysis of PDEs
To appear in Proc. Amer. Math. Soc
Scientific paper
We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \label{pop} -div(\omega_1 \nabla u) = \omega_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $ \omega_1(x),\omega_2(x)$. We consider the cases $ f(u) = e^u, u^p$ where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $ p$ and the behaviour of $ \omega_1,\omega_2$ near infinity. Also the monotonicity of $ \omega_1$ is involved in some results. Our methods here are the methods developed by Farina, \cite{f2}. We examine a specific class of weights $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x) = (|x|^2+1)^\frac{\beta}{2} g(x)$ where $ g(x)$ is a positive function with a finite limit at $ \infty$. For this class of weights non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.
Cowan Craig
Fazly Mostafa
No associations
LandOfFree
On stable entire solutions of semi-linear elliptic equations with weights does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On stable entire solutions of semi-linear elliptic equations with weights, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On stable entire solutions of semi-linear elliptic equations with weights will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-194501