Abstract:In this paper we study radial solutions for the following equationwhere x ∈ R n , n > 2, f is subcritical for r small and u large and supercritical for r 3 large and u small, with respect to the Sobolev critical exponent 2 * = 2n n−2 . The solutions 4 are classified and characterized by their asymptotic behaviour and nodal properties. 5In an appropriate super-linear setting, we give an asymptotic condition sufficient to 6 guarantee the existence of at least one ground state with fast decay with exactly j zeroes 7 for any j ≥ 0. Under the same assumptions, we also find uncountably many ground 8 states with slow decay, singular ground states with fast decay and singular ground states 9 with slow decay, all of them with exactly j zeroes. Our approach, based on Fowler 10 transformation and invariant manifold theory, enables us to deal with a wide family 11 of potentials allowing spatial inhomogeneity and a quite general dependence on u. In 12 particular, for the Matukuma-type potential, we show a kind of structural stability.

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".

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u + f (t, u) = 0, u(0) = u(T ) = 0, under suitable assumptions on f (t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Mathematics Subject Classification (2000). Primary 34B15; Secondary 37E40.

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