On stable entire solutions of semi-linear elliptic equations with weights

Source abstract

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of − div ⁡ ( ω 1 ∇ u ) = ω 2 f ( u ) in R N , $$\begin{equation} -\operatorname {div}( \omega _1 \nabla u) = \omega _2 f(u) \qquad \text {in}\ \ \mathbb {R}^N, \end{equation}$$ with positive smooth weights ω 1 ( x ) , ω 2 ( x ) \omega _1(x),\omega _2(x) . We consider the cases f ( u ) = e u , u p f(u) = e^u, u^p where p > 1 p>1 and − u − p -u^{-p} where p > 0 p>0 . We obtain various non-existence results which depend on the dimension N N and also on p p and the behaviour of ω 1 , ω 2 \omega _1,\omega _2 near infinity. Also the monotonicity of ω 1 \omega _1 is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights ω 1 ( x ) = ( | x | 2 + 1 ) α 2 \omega _1(x) = ( |x|^2 +1)^\frac {\alpha }{2} and ω 2 ( x ) = ( | x | 2 + 1 ) β 2 g ( x ) \omega _2(x) = ( |x|^2+1)^\frac { \beta }{2} g(x) , where g ( x ) 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.