A NEW DYNAMICAL APPROACH OF EMDEN-FOWLER EQUATIONS AND SYSTEMS

We give a new approach to general Emden-Fowler equations and systems of the form (E-epsilon)-Delta(p)u=-div(vertical bar del u vertical bar(p-2)del(u))=epsilon vertical bar x vertical bar(a)u(Q), (G){-Delta(p)u=-div(vertical bar del u vertical bar(p-2)del(u))=epsilon(1)vertical bar x vertical bar(a)u(s)v(delta), -Delta(p)u=-div(vertical bar del u vertical bar(p-2)del(u))=epsilon(2)vertical bar x vertical bar(a)u(mu)v(m), where p,q,Q,delta,mu,s,m,a,b are real parameters, p, q not equal 1, and epsilon,epsilon(1),epsilon(2) = +/-1. In the radial case we reduce the problem (G) to a quadratic system of four coupled first-order autonomous equations of Kolmogorov type. In the scalar case the two equations (E-epsilon) with source (epsilon = 1) or absorption (E = 1) are reduced to a unique system of order 2. The reduction of system (G) allows us to obtain new local and global existence or nonexistence results. We consider in particular the case epsilon(1) = epsilon(2) = 1. We describe the behaviour of the ground states when the system is variational. We give a result of existence of ground states for a nonvariational system with p = q = 2 and 8 = m > 0, that improves the former ones. It is obtained by introducing a new type of energy function. In the non-radial case we solve a conjecture of nonexistence of ground states for the system with p = q = 2, delta = m + 1 and mu = s + 1.