RENDICONTI DEL SEMINARIO MATEMATICO

54 M. Francawhere F(u,|x|) = ∫ u0f (s,|x|)ds. The p-Laplace operator arises naturally when wewant to extend this functional to W 1,p (R n ) functions. In fact (1) is the Euler equationfor the functional E p : R × W 1,p (R n ) → R,∫E p (x, u,∇u) =(|∇u| p− F(u,|x|) ) dx.pWe will focus our attention mainly on radial solutions, hence we will reduce (1) to thefollowing singular O.D.E.(3)(u ′ |u ′ | p−2 r n−1 ) ′ + f (u, r)r n−1 = 0where r = |x| and we commit the following abuse of notation: we write u(r) foru(x) when |x| = r and u has radial symmetry; here and later ′ denotes derivation withrespect to r. Observe that (3) is singular when r = 0 and when u ′ = 0, unless p = 2.We introduce now some notation that will be in force throughout all the paper.We will use the term “regular solution” to refer to a solution u(r) of Eq. (3) satisfyingu(0) = u 0 > 0 and u ′ (0) = 0. We will use the term “singular solution” to refer to asolution v(r) of Eq. (3) such that lim r→0 v(r) = +∞.A basic question in this kind of PDE is the existence and the asymptotic behaviourof ground states (G.S.), that are solutions u(x) of (1) which are nonnegative forany x ∈ R n and such that lim |x|→∞ u(x) = 0. We are also interested in detecting singularground states (S.G.S.), that is solutions v(x) which are well defined and nonnegativefor any x ∈ R n \{0} and such that lim |x|→∞ v(x) = 0 and lim |x|→0 u(x) = +∞. Otherinteresting family of solutions for the radial equation (3) is the one of crossing solutions,that is regular solutions u(r) which are positive for r smaller than a certain valueR > 0 and become null with nonzero slope at r = R. So they can also be regardedas solutions of the Dirichlet problem in the ball of radius R. Finally we individuatesolutions u(r) of the Dirichlet problem in the exterior of the ball of radius R, that isu(R) = 0, u(r) > 0 for r > R, and u(r) has fast decay. We say that a positive solutionu(r) of (3) has fast decay if lim r→∞ u(r)r (n−p)/(p−1) < +∞ and that it has slow decayif lim r→∞ u(r)r (n−p)/(p−1) = +∞.This article has the following structure: in section 1 we introduce the generalizedFowler transformation, and we apply it to a toy example, mainly for illustrativepurpose. In sections 2 and 3 we introduce the Pohozaev function, that is one of the maintool for the analysis of equation of type (1), and we consider the case where respectivelyf (u, r) = k(r)u|u| q−1 and f (u, r) = k 1 (r)u|u| q 1−1 + k 2 (r)u|u| q 2−1 where q > p,q 2 > q 1 > p, the functions k(r), k 1 (r), k 2 (r) are positive and continuous for r > 0.In both the cases we assume that the corresponding Pohozaev functions have constantsign. In section 4 we discuss the case f (u, r) = k(r)u|u| q−1 when the Pohozaev functionchanges sign, stressing in particular the case q = p ∗ . In section 5 we explainbriefly few results concerning Eq. (2) when f (u, r) = u|u| q 1−1 + u|u| q 2−1 , whenp ∗ < q 1 < p ∗ < q 2 and p = 2. We remark that in this case there are still many openproblems. In section 6 we discuss the case f (u, r) = −k 1 (r)u|u| q 1−1 + k 2 (r)u|u| q 2−1 ,where q 1 < q 2 , and the functions k 1 and k 2 are positive and continuous for r > 0.