Orbital stability of standing waves for fractional Hartree equation with unbounded potentials

Abstract

We prove the existence of the set of ground states in a suitable energy space Σ s = {u :2 ) for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set of standing waves. The main ingredient is the observation that Σ s is compactly embedded in L 2 . This enables us to apply the concentration compactness argument in the works of Cazenave-Lions and Zhang, namely, relative compactness for any minimizing sequence in the energy space.2010 Mathematics Subject Classification. Primary 35Q55, 35B35, Secondary 35C08.

Qualitative analysis of solutions to a nonlocal Choquard–Kirchhoff diffusion equations in ℝN$$ {\mathbb{R}}^N $$

“…Moreover, the term $$$ \left({\mathcal{K}}_{\mu}\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u $$$ is also known as Hardy potential, and this type of potential is important in analyzing many aspects of physical phenomena with singular poles (at origin); see, for example, previous works 27–32 . There are a lot of studies of evolution equations with this type of potentials, for example, parabolic equations, 33–39 wave equations, 40–44 and Schrödinger equations 45–50 …”

“…Moreover, the term (  𝜇 * |u| q ) |u| q−2 u is also known as Hardy potential, and this type of potential is important in analyzing many aspects of physical phenomena with singular poles (at origin); see, for example, previous works. [27][28][29][30][31][32] There are a lot of studies of evolution equations with this type of potentials, for example, parabolic equations, [33][34][35][36][37][38][39] wave equations, [40][41][42][43][44] and Schrödinger equations. [45][46][47][48][49][50] For problem (1.1), the well-posedness of nonnegative solutions, the asymptotic behavior of global solutions, and the finite time blow-up of solutions were considered recently.…”

Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

“…These results were significantly extended by Grillakis, Shatah and Strauss in [19] for general Hamiltonian systems that are invariant under a group of transformations. Recently, these arguments have been applied to the study of the orbital stability for nonlinear Schrödinger type equations with potentials, see [14,17,37,40,44].…”

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

“…For example, the Efimov states (the circumstances that the two-particle attraction is so weak that any two bosons can not form a pair, but the three bosons can be stable bound states): see e.g., [16]; effects on dipole-bound anions in polar molecules: see e.g., [4,7,28]; capture of matter by black holes (via near-horizon limits): see e.g., [9,19]; the motions of cold neutral atoms interacting with thin charged wires (falling in the singularity or scattering): see e.g., [5,12]; the renormalization group of limit cycle in nonrelativistic quantum mechanics: see e.g., [6,8]; and so on. The are a lot of studies of evolution equations with this type of potentials, see, for example, [1,2,3,45,32,50] for parabolic equation, [11,49,52,39] for wave equations, and [24,22,23,42,47,48,53] for Schrödinger equation. However, as far as we know, there seems little studies of fourth-order plate equation with Hardy-Hénon potentials.…”