RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 55Finally in the appendix we show how some more general equations can be reduced to(3), and we explain the concept of natural dimension, introduced in [20].2. Preliminary results and autonomous caseThe main purpose of this paper is to explain the method of investigation of positivesolution of (3) which has been used in [2], [3], [4], [11], [1], [12], [13], [14], [15], [16],[17]. The advantage in the use of this method lies essentially on the fact that we canbenefit of a phase portrait, and of the use of techniques typical of dynamical systemstheory, such as invariant manifold theory and Mel’nikov functions. Moreover, restrictingourselves to the study of radial solutions, we overcome the difficulties derivingfrom the lack of compactness of the critical and supercritical case. With our methodwe can also naturally detect and classify singular solutions, which are not easily foundby variational techniques or by standard shooting arguments. The main fault of themethod is that it can just give information on radial solutions. However we wish tostress that, when the domain has radial symmetry (e.g. it is the whole R n ), G.S. andsolutions of the Dirichlet problem, if they exist, are radial in many different situations,which will be discussed in details in the following sections, see [6], [9], [42], [44].Furthermore radial solutions play a key role also for many parabolic equationsassociated to (2). In fact in many cases the ω-limit set is made up of the union of radialsolutions, see e. g. [39], [23].The first step in this analysis consists in applying the following change of coordinates(4)α l = pl−p ,β l = p(l−1)l−p− 1, γ l = β l − (n − 1), l > px l = u(r)r α ly l = u ′ (r)|u ′ (r)| p−2 r β lr = e twhere l > p is a parameter. This tool allows us to pass from (3) to the followingdynamical system:( ) ( ) ( ) ( )ẋl αl 0 xl(5)=+ y l |y l | 2−pp−1ẏ l 0 γ l y l −g(x l , t)Here and later “·” stands for d dt , and(6) g l (x l , t) := f (x l exp(−α l t), exp(t))e α l(l−1)t .This transformation was introduced by Fowler in the 30s for the case p = 2, and wegeneralized it to the case p > 1 just recently in [12], [13], [15] [14], [16], [17]. It willbe useful to embed system (5), and in general all the dynamical systems that will beintroduced in the paper, in a one-parameter family as follows:(7)( ) ( ) ( )ẋl αl 0 xl=+ẏ l 0 γ l y l()y l |y l | 2−pp−1−g l (x l , t + τ)