ON STABLE ENTIRE SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS WITH WEIGHTS

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of (0.1) -div(omega(1)del u) = omega(2)f(u) in R-N, 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. We examine a specific class of weights omega(1)(x) = (vertical bar x vertical bar(2) + 1)(alpha/2) and omega(2)(x) = (vertical bar x vertical bar(2) + 1)(beta/2)g(x), where g(x) is a positive function with a finite limit at infinity. For this class of weights, non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.