RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 69where ǫ > 0 is small enough and ρ = n − 2, such that (2) admits no G.S. with fastdecay.In the same article Bianchi has also proved that a potential satisfying condition(18), with ρ > n − 2 and ǫ arbitrary chosen, and some further sufficient conditionsnecessarily admits a G.S. with fast decay. This result shows how sensitive to smallchanges in the potential k the behaviour of positive solutions is.Now we turn again to the case p ̸= 2, and we focus our attention on radialsolutions. In order to find a G.S. with fast decay we need to find a balance betweenthe gain of energy, due to the values for which k(r)r p∗ −q is increasing, and the loss ofenergy, due to the values for which it is decreasing. A first important result concerningthe structure of positive solutions is the following:THEOREM 5. Consider (3) and assume that there is R > 0 such that one of thefollowing conditions is satisfiedJ + (r) ≥ 0 for any 0 ≤ r ≤ R and it is decreasing for r ≥ RJ − (r) ≤ 0 for any r ≥ R and it is increasing for 0 ≤ r ≤ R.Then regular solutions have one of the following structure.1. They are all crossing solutions2. They are all G.S. with slow decay3. There is D > 0 such that u(d, r) is a crossing solution for d > D, it is a G.S.with slow decay for d < D, and a G.S. with fast decay for d = D.The result concerning J + (r) has been proved in [35], evaluating the Pohozaevfunction on regular solutions. The part concerning J − (r) is not explicitly stated in[35], however it can be easily obtained as follows, see also [18]. We can construct astable set ˜W s (τ) through Proposition 2, and then deduce the existence of solutions withfast decay. Then, applying the argument of [35] to these solutions, we conclude. Wethink that one could easily reach a classification result also for S.G.S. in this situation,combining the argument in [35] with a dynamical argument. In fact, if we restrictto regular solutions, structure A and B give back structure 1 and 2 respectively, andstructure 3 is a special case of C.We have already seen that, when either J + (r) or J − (r) are positive for anyr > 0 we have structure A (so we are in the first case), while when they are negativewe have structure B (so we are in the second case). In order to derive a sufficientcondition for structure C to exist, we start from the case p = 2 and following [43] weintroduce the function:(19) Z(t) := e − (n−2)p2 t J + (t) − e (n−2)p2 t J − (t)Then we define ρ + = inf{r ∈ (0,∞)| J + (r) < 0}, and ρ − = sup{r ∈ (0,∞)| J − (r) 0 and ρ − = 0 if J − (r) ≥ 0 for any