6 papers
Two Positive Normalized Solutions on Star-shaped Bounded Domains to the Brézis-Nirenberg Problem, I: Existence
Linjie Song, Wenming Zou
We develop a new framework to prove the existence of two positive solutions with prescribed mass on star-shaped bounded domains: one is the normalized ground state and another is o…
On the Number of Normalized Ground State Solutions for a class of Elliptic Equations with general nonlinearities and potentials
Hichem Hajaiej, Eliot Pacherie, Linjie Song
We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$ -Δu - λu + V (| x |) u - f (| x |, u) = 0,\quad\text…
On radial positive normalized solutions of the Nonlinear Schrödinger equation in an annulus
Jian Liang, Linjie Song
We are interested in the following semilinear elliptic problem: \begin{equation*} \begin{cases} -Δu + λu = u^{p-1} \ \text{in} \ T,\\ u > 0, u = 0 \ \text{on} \ \partial T,\\ \int_…
Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities
Xiaojun Chang, Hichem Hajaiej, Zhouji Ma +1
We prove the existence of normalized ground state solutions for the biharmonic Schrödinger equation with combined nonlinearities and show that all ground states correspond to the l…
Positive solution for an elliptic system with critical exponent and logarithmic terms
Hichem Hajaiej, Tianhao Liu, Linjie Song +1
In this paper, we study the existence and nonexistence of positive solutions for a coupled elliptic system with critical exponent and logarithmic terms. The presence of the the log…
On the Eigenvalues of the Fractional Laplacian
Sabri Bahrouni, Hichem Hajaiej, Linjie Song
We consider the eigenvalue problem for the fractional Laplacian \begin{equation} \left\{\begin{aligned} (- Δ)_p^{s}\, u + μ(- Δ)_q^{s}\, u+ |u|^{p-2}u+μ|u|^{q-2}u=λ V(x)|u…