Anna Maria Micheletti

Solitons for the nonlinear Klein-Gordon equation

Abstract In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.

On the existence of the fundamental Eigenvalue of an elliptic problem in R N

Abstract We study an eigenvalue problem for functions in ℝN and we find sufficient conditions for the existence of the fundamental eigenvalue. This result can be applied to the study of the orbital stability of the standing waves of the nonlinear Schrödinger equation.

On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth

This paper deals with the existence of sign changing solutions of the problem where Ω is a bounded regular domain in , N ≥ 4, e > 0, p = (N + 2)/(N − 2), q ≥ 1, q ≠ p and .

Existence of blowing-up solutions for a slightly subcritical or a slightly supercritical non-linear elliptic equation on R n

The main purpose of this paper is to construct a family of positive solutions for both the slightly subcritical and slightly supercritical equations -Δu + V(x)u = n(n - 2)(u+)(n+2)/(n-2)±e in Rn, which blow-up and concentrate at a single point as e goes to 0, under certain conditions on the potential V.

Three solutions of a fourth order elliptic problem via variational theorems of mixed type

We prove the existence of two nontrivial solutions for the fourth order problem and u = 0 on ∂ω when λ1≥c>λi+1 and either b<λk(λk-c) and b is close to λk(λk-c) where 2≤k≤i or b>λj(λj-c) and b is close to λj(λj-c) where j≥i+1. (Here (λi)i≥1 is the sequence of the eigenvalues of –Δin H1 0(ω)). Moreover if c>λ1, c is close to λ1, b>λj(λj-c) and b is close to λj(λj-c) where j≥2 we get three non trivial solutions

Solutions in Exterior Domains of Null Mass Nonlinear Field Equations

Abstract In this paper, we are concerned with the existence of solutions of the problem: where Ω ⊂ ℝN is an exterior domain and f″(0) = 0.

Nodal solutions for a singularly perturbed nonlinear elliptic problem in a Riemannian manifold

Abstract Let (M, g) be a smooth compact Riemannian N−manifold, N ≥ 2. We show that if the scalar curvature of g is not constant, the problem −ε2∆gu + u = up−1 in M has a positive solution with two positive peaks ξ1ε and ξ2ε, and a sign changing solution with one positive peak η1ε and one negative peak η2ε, such that as ε goes to zero Scalg(ξ1ε), Scalg(η1ε) → Scalg(ξ) and Scalg(ξ2ε), Scalg(η2ε) → Scalg(ξ). Here p > 2 if N = 2 and

A note on the resonance set for a semilinear elliptic equations and an application to jumping nonlinearities

where Ω is a bounded smooth domain, u + =m ax(u, 0) and u − = − min(u, 0). The study of Σ turns out to be difficult except when Ω is an interval in R. Therefore it is interesting to have some information about the resonance set, as precise as possible. In [GK] the authors showed that if λk is a simple eigenvalue of − ∆t hen Σ∩]λk−1 ,λ k+1[ 2 coincides with two continuous curves through the point (λk ,λ k). In [DeFG] the authors characterized a curve γ through the point (λ2 ,λ 2 )w hich belongs to Σ such that Σ∩{(α, β) ∈ R 2 | λ1 λ 1} = ∅. Finally, in [MMP] and [M] the following result was shown: if k ≥ 2 is such that λk <λ k+1 then there exist two continuous curves (α, ϕk+1(α)), through (λk+1 ,λ k+1), and (α, ψk(α)), through (λk ,λ k), which respectively lie in the sets Σ ∩ ]λk, +∞[ 2 and

On the Dynamics of Solitons in the Nonlinear Schrödinger Equation

AbstractWe study the behavior of the soliton solutions of the equation