Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_\alpha$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $\alpha \to 0$ or $\alpha \to N$. This leads to the uniqueness and nondegeneracy of ground states when $\alpha$ is sufficiently close to $0$ or close to $N$.Comment: 18 pages, revised versio

Infinitely many solutions for semilinear nonlocal elliptic equations under noncompact settings

In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.Comment: 29 pages, Typos are fixed, Intro and refereces are extende

Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

We are concerned with the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs \cite{LeLe}. In Jang and Seok \cite{JS}, fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $\hbar$ goes to the zero, the $\hbar$-parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint