No abstract

By the use of a higher dimensional version of the Poincaré–Birkhoff theorem, we are able to generalize a result of Jacobowitz and Hartman, thus proving the existence of infinitely many periodic solutions for a weakly coupled superlinear system

We study existence and multiplicity of radial ground states for the scalar curvature equation u + K (|x|) u n+2 n−2 = 0, x ∈ R n , n > 2, when the function K : R + → R + is bounded above and below by two positive constants, i.e. 0 < K ≤ K (r) ≤ K for every r > 0, it is decreasing in (0, 1) and increasing in (1, +∞). Chen and Lin (Commun Partial Differ Equ 24:785-799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio K /K is smaller than some computable values. Keywords Scalar curvature equation • Ground states • Fowler transformation • Invariant manifold • Shooting method • Bubble tower solutions • Phase plane analysis • Multiplicity results Mathematics Subject Classification 35J61 • 37D10 • 34C37 F. Dalbono: Partially supported by the GNAMPA project "Dinamiche non autonome, analisi reale e applicazioni". M. Franca: Partially supported by the GNAMPA project "Sistemi dinamici, metodi topologici e applicazioni all'analisi nonlineare". A. Sfecci: Partially supported by the GNAMPA project "Problemi differenziali con peso indefinito: tra metodi topologici e aspetti dinamici".

We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight on the nonlinearity f (u, r). In particular we are interested in the case where f is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.