19 research outputs found
Sign-preserving solutions for a class of asymptotically linear systems of second-order ordinary differential equations
No full textWe study multiplicity of solutions to an asymptotically linear Dirichlet problem associated with a planar system of second order ordinary differential equations. The existence of two sign-preserving component-wise solutions is guaranteed when the Morse indexes of the linearizations at zero and at infinity do not coincide, and one of the asymptotic problems has zero-index. The proof is developed in the framework of topological and shooting methods and it is based on a detailed analysis and characterization of the phase angles in a two-dimensional setting
Morse-Smale index theorems for elliptic boundary deformation problems
No full textAbstractMorse-type index theorems for self-adjoint elliptic second order boundary value problems arise as the second variation of an energy functional corresponding to some variational problem. The celebrated Morse index theorem establishes a precise relation between the Morse index of a geodesic (as critical point of the geodesic action functional) and the number of conjugate points along the curve. Generalization of this theorem to linear elliptic boundary value problems appeared since seventies. (See, for instance, Smale (1965) [12], Uhlenbeck (1973) [15] and Simons (1968) [11] among others.) The aim of this paper is to prove a Morse–Smale index theorem for a second order self-adjoint elliptic boundary value problem in divergence form on a star-shaped domain of the N-dimensional Euclidean space with Dirichlet and Neumann boundary conditions. This result will be achieved by generalizing a recent new idea introduced by authors in Deng and Jones (2011) [5], based on the idea of shrinking the boundary
Multiplicity results for asymmetric boundary value problems with indefinite weights
No full textWe prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the form u″+f(t,u)=0, u(0)=u(T)=0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues
Risultati di molteplicità per problemi ai limiti superlineari e asintoticamente lineari
No full textNo abstract availabl
Multiplicity of ground states for the scalar curvature equation
No full textWe study existence and multiplicity of radial 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, 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 ̄/K̲ is smaller than some computable values
Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry
No full textWe study existence and multiplicity of positive ground states for the scalar curvature equation , x in R^n, when the function is bounded above and below by two positive constants, i.e. for every positive r, it is decreasing in (0,R) and increasing in for a certain positive constant R.
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 which guarantees the existence of a large number of ground states with fast decay,
i.e. such that as , 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 uniqu
Multiplicity of solutions for asymptotically linear n-th order boundary value problems
No full textIn this paper we investigate existence and multiplicity of solutions, with
prescribed nodal properties, to a two-point boundary value problem of
asymptotically linear -th order equations. The proof follows a shooting
approach and it is based on the weighted eigenvalue theory for linear -th
order boundary value problem
A multiplicity result for a class of strongly indefinite asymptotically linear second order systems
No full textWe prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearit
Nodal Solutions for Supercritical Laplace Equations
No full textIn this paper we study radial solutions for the following equation where , 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 . 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 . 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
