A new dynamical approach of emden-fowler equations and systems

Abstract

We give a new approach to general Emden-Fowler equations and systems of the form (Eε) -δpu = -div(|∇u|p-2 ∇u) = ε|x|a uQ, (G) { -δpu = -div(|∇u|p-2 ∇u) = ε1 |x|a usvδ, -δqv = -div(|∇v|p-2 ∇u) = ε2 |x|b uμvm, where p, q, Q, δ, μ, s, m, a, b are real parameters, p, q ≠ 1, and ε, ε1, ε2 = ±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 (Eε) with source (ε = 1) or absorption (ε = -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 ε1 = ε2 = 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 s = m > 0; that improves the former ones. It is obtained by introducing a new type of energy function. In the nonradial case we solve a conjecture of nonexistence of ground states for the system with p = q = 2, δ = m + 1 and μ = s + 1.