146 research outputs found
Scaling properties of functionals and existence of constrained minimizers
No full textIn this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on Hilbert spaces that are invariant under translations. Our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals. We introduce family of maps, called scaling paths, that permits to show the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional I associated to the Schrödinger-Poisson equation in R3iΨt+δΨ-(|x|-1*|Ψ|2)Ψ+|Ψ|p-2Ψ=0 when 2<3. In particular we prove that I achieves its minimum on the constraint {u∈H1(R3):||u||2=Ρ} for every sufficiently small Ρ>0. In this way we recover the case studied in Sanchez and Soler (2004) [20] for p=8/3 and we complete the case studied by the authors for 3<10/3 in Bellazzini and Siciliano (2011) [4]. © 2011 Elsevier Inc
Ground states for semi-relativiscti SPS energy
No full textWe prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy
I-alpha,I- beta(rho) = inf (u is an element of H 1/2 (R3) integral R3 vertical bar u vertical bar 2dx=rho) 1/2 parallel to u parallel to(2)(H1/2(R3)) + alpha integral integral(R3xR3) vertical bar u(x)vertical bar(2)vertical bar u(y)vertical bar(2)/vertical bar x - y vertical bar dxdy - beta integral(R3) vertical bar u vertical bar(8/3)dx alpha, beta > 0 and rho > 0 is small enough. The minimization problem is L-2 critical and in order to characterize the values alpha, beta > 0 such that I-alpha,I- beta(rho) > -infinity for every rho > 0, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant S > 0 such that
1/S parallel to phi parallel to(L8/3(R3))/parallel to phi parallel to(1/2)(H1/2(R3)) <= (integral integral(R3 x R3) vertical bar phi(x)vertical bar(2)vertical bar phi(y)vertical bar(2)/vertical bar x - y vertical bar dxdy)(1/8)
for all phi is an element of C-0(infinity)(R-3). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm parallel to u parallel to(2)(H1/2(R3)) by the homogeneous one parallel to u parallel to(H1/2(R3)) in the energy above
Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension
No full textWe consider the Cauchy problems associated with semi-relativistic NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between other results, we prove the existence and stability of ground states for sNLS in the L2 supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW, which is also established along the paper, by showing an inflation of norms phenomenon. Concerning the Cauchy theory we show, under radial symmetry assumption the following results: a local existence result in H1 for energy subcritical nonlinearity and a global existence result in the L2 subcritical regime
Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime
No full textIn this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mariş in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Mariş. The planar case turns out to be more intricate and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions
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