arXiv subjects

Zifei Shen

Publications and source records attributed to Zifei Shen.

5 recordsLinked to original sources

arXiv2023

On concentration of real solutions for fractional Helmholtz equation

This paper studies the nonlinear fractional Helmholtz equation \begin{equation}\label{main} (-Δ)^{s} u-k^{2} u=Q(x)|u|^{p-2}u, ~~\mathrm{in}~~\mathbb{R}^{N},~~N\geq3, \end{equation} where $\frac{N}{N+1} 0$ large, the existence of real-valued solutions for (\ref{main}) are proved, and in the limit $k\longrightarrow\infty$, sequence of solutions associated with ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.

math.AP↗

arXiv2023

Complex and real valued solutions for fractoinal Helmholtz equation

In this paper, we are concerned with the limiting absorption principle for the fractional Helmholtz equation, By establishing the boundedness estimate for the resolvent of fractional Helmholtz operator, we obtain the nontrivial Lq(Rn) complex valued solutions for (0.1). By setting up a dual variational framework, we also obtain the real valued solutions for (0.1) via a non-vanishing principle.

math.AP↗

arXiv2023

Stein-Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein-Weiss type inequalities with general kernels on the upper half space and study the existence of extremal functions for this inequality with the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of the positive solutions of the corresponding Euler-Lagrange integral system. As an application, we finally derive some Liouville type results for the Hartree type equations in the half space.

math.AP↗

arXiv2017

On the critical Choquard equation with potential well

In this paper we are interested in the following nonlinear Choquard equation $$ -Δu+(λV(x)-β)u =\big(|x|^{-μ}\ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $λ,β\in\mathbb{R}^+$, $0<μ 0$ is a constant such that the operator $-Δ+λV(x)-β$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $λ$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$ goes to infinity. Furthermore, for any $0<β<β_{1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$ is the first eigenvalue of $-Δ$ on $Ω$ with Dirichlet boundary condition.

math.AP↗

arXiv2015

Groundstates for nonlinear fractional Choquard equations with general nonlinearities

We study the following nonlinear Choquard equation driven by a fractional Laplacian: $$ (-Δ)^{s}u+ u =(|x|^{-μ}\ast F(u))f(u)|{4.14mm}{in}|{1.14mm} \mathbb{R}^N, $$ with $N\geq3$, $s\in(0,1)$ and $μ\in(0,N)$. By Supposing that the nonlinearities satisfy the general Berestycki-Lions type conditions \cite{BL}, we are able to prove the existence of groundstates for this equation by variational methods.

math.AP↗