RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 85a stable set ˜W s (τ) through Proposition 2. We choose τ < −M, so that ˜W u (τ) crossesthe y negative semi-axis, in view of Corollary 5. We denote by B(τ) the bounded setenclosed by ˜W u (τ) and the y negative semi-axis. We choose one of the uncountablymany points in ˜W s (τ) ∩ B(τ), say Q s , and we follow backwards in t the trajectoriesx τ q (Qs , t) where Q s ∈ ˜W s (τ). We show that x τ q (Qs , t) is forced to stay in B(t + τ)for any t < 0 and we conclude that x τ q (Qs , t) is uniformly positive as t → −∞.Then, from Proposition 3 we deduce that the corresponding solutions v(r) of (3) aremonotone decreasing S.G.S.7. Appendix: reduction of div(g(|x|)∇u|∇u| p−2 ) + f (u,|x|) = 0 and natural dimensionIn this subsection we want to show how we can pass from the analysis of radial solutionsof an equation of the following class(24) div(g(|x|)∇u|∇u| p−2 ) + ¯ f (u,|x|) = 0to the analysis of solutions of an equation of the form (3). Here again x ∈ R n andg(|x|) ≥ 0 for |x| ≥ 0.We repeat the argument developed in Appendix B of [17]. In fact we exploithere an idea already used in [33] and [20], and we follow quite closely the latter paper,in which the concept of natural dimension is introduced. First of all observe that aradial solutions u(r) of (24) satisfy the following ODE:(25) (r n−1 g(r)u ′ |u ′ | p−2 ) ′ + r n−1 ¯ f (u, r) = 0.Set a(r) = r n−1 g(r) and assume that one of the Hypotheses below is satisfiedH1 a −1/(p−1) ∈ L 1 [1,∞]\ L 1 [0, 1]H2 a −1/(p−1) ∈ L 1 [0, 1]\L 1 [1,∞)Then we make the following change of variables borrowed from [20]. Let N > p be aconstant and assume that Hyp. H1 is satisfied; we define s(r) =(∫ ∞ra(τ) −1/(p−1) dτ ) −p+1N−p. Obviously s : R + 0→ R+ 0, s(0) = 0, s(∞) = ∞ ands(r) is a diffeomorphism of R + 0into itself with inverse r = r(s) for s ≥ 0. If u(r) is asolution of (25), v(s) = u(r(s)) is a solution of the following transformed equation(26) (s N−1 v s |v s | p−2 ) s + s N−1 h(s) f (v, s) = 0,where f (v, s) = ¯ f (v, r(s)) andh(s) =( ) ( )N − pp p/(p−1)g(r(s)) 1/p r(s) n−1p − 1s N−1 .If we replace Hyp. H1 by Hyp. H2 we can define s(r) as follows s(r) =(∫ r0 a(τ)−1/(p−1) dτ ) p−1N−pand obtain again (26) from (25), with the same expression for