RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 75more details. The results can be summed up as follows. Let us introduce the followinghypotheses:M 1 there exists ρ > 0 such that k(ρ) > 0 is a non degenerate maximum and k(r) isuniformly positive and monotone increasing for 0 ≤ r ≤ ρ.M 2 there exists R > 0 such that k(R) > 0 is a non degenerate maximum and k(r) > 0is uniformly positive and monotone decreasing for r ≥ R.O 1 k(r) is oscillatory as r → 0 and admits infinitely many positive non degeneratecritical points.O 2 k(r) is oscillatory as r → ∞ and admits infinitely many positive non degeneratecritical points.Then we have the following result:THEOREM 11. Consider equation (3) and assume that k(r) = K(r ǫ ) is bounded.Then, for ǫ > 0 small enough, we have at least as many G.S. with fast decay as the nondegenerate critical points of k(r). Moreover1. Assume that either M 2 or O 2 is satisfied. Then the there are uncountably manyG.S. with slow decay and uncountably many crossing solutions.2. Assume that either M 1 or O 1 is satisfied. Then there are uncountably manyS.G.S. with fast decay and uncountably many solutions v(r) of Dirichlet problemin the exterior of a ball.3. Assume that both Hypotheses 1 and 2 are satisfied. Then the positive solutionsof equation (3) have a structure of type C.Furthermore, if k(r) is uniformly positive, then G.S. and S.G.S. are decreasing.REMARK 6. Note that when k(r) is decreasing for r small and increasing for rlarge, we are not able to state the existence of S.G.S. and of G.S. with slow decay. Thisis due to the fact that, in such a case it is not possible to construct a set B(τ) which iscontained in R 2 + for any τ, so our argument fails. However also in this case we are ableto prove the existence of G.S. with fast decay.Following [14] we can easily obtain an analogous result for the regularly perturbedproblem. The difference lies in the fact that the Melnikov condition is a bit morecomplicated, so we have to replace the assumption that k(r) has a positive critical pointby the condition that ¯M(τ) = 0 and ¯M ′ (τ) ̸= 0.Now we want to extend some of these results to the “in the large” case, so wewant to see what happens when ǫ → 1. This in fact will shed some light on the reasonfor which positive solutions exhibit the same structure, under two completely differenttypes of perturbation. The idea is to use our knowledge of the autonomous case tounderstand the non-autonomous one, replacing the Melnikov function by the energyfunction H. We will discuss the following Hypotheses