Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

We study existence and multiplicity of positive ground states for the scalar curvature equation Δu+K(|x|)un+2n-2=0,x∈Rn,n>2,when the function K: R+→ R+ is bounded above and below by two positive constants, i.e. 0 0 , it is decreasing in (0 , R) and increasing in (R, + ∞) for a certain R> 0. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio K¯/K̲ which guarantees the existence of a large number of ground states with fast decay, i.e. such that u(| x|) ∼ | x| 2-n as | x| → + ∞, which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique

Multiplicity of ground states for the scalar curvature equation

We study existence and multiplicity of radial ground states for the scalar curvature equation \u394u+K(|x|)un+2n-2=0,x 08Rn,n>2,when the function K: R+\u2192 R+ is bounded above and below by two positive constants, i.e. 0 0 , it is decreasing in (0,\ua01) and increasing in (1 , + 1e). Chen and Lin (Commun Partial Differ Equ 24:785\u2013799, 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\uaf/K\u332 is smaller than some computable values

Multiplicity results for asymptotically linear equations, using the rotation number approach

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\u2033 + 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. \ua9 2007 Birkh\ue4user Verlag Basel/Switzerland

Multiplicity results for systems of asymptotically linear second order equations

We prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solutio

Nodal Solutions for Supercritical Laplace Equations

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