We are concerned with the classification of positive radial solutions for the system ∆u = v p , ∆v = f (|∇u|), where p > 0 and f ∈ C 1 [0, ∞) is a nondecreasing function such that f (t) > 0 for all t > 0. We show that in the case where the system is posed in the whole space R N such solutions exist if and only ifThis is the counterpart of the Keller-Osserman condition for the case of single semilinear equation. Similar optimal conditions are derived in case where the system is posed in a ball of R N . If f (t) = t q , q > 1, using dynamical system techniques we are able to describe the behaviour of solutions at infinity (in case where the system is posed in the whole R N ) or around the boundary (in case of a ball).

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the formWe first classify all the positive radial solutions in case Ω is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if 0 ≤ α < p − 1 and mq < (p − 1 − α)(p − 1 − β). Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.Keywords: Radial symmetric solutions, p-Laplace operator; asymptotic behavior, cooperative and irreducible dynamical systems 2010 AMS MSC: 35J47, 35J92, 35B40, 70G60

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequalityHere I α stands for the Riesz potential of order α ∈ (0, N ), p > 0 and q ∈ R. For a large class of operators L A (which includes the m-Laplace and the m-mean curvature operator) we obtain optimal ranges of exponents p, q and α for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.Keywords: Quasilinear elliptic inequalities; m-Laplace operator; m-mean curvature operator; existence and nonexistence of positive solutions.Example 1.4. The m-mean curvature operator given by• A is S-m-C if M −1 t m−2 ≤ A(x, u, t) ≤ M t m−2 for all t > 0, (1.3)for some constant M > 1.

We study the equationFirst, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

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