arXiv subjects

Linjie Song

Publications and source records attributed to Linjie Song.

23 records · Page 2Linked to original sources

arXiv2023

A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS

We first give an abstract framework to show the uniqueness of Ground State Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all the existing results in the literature only addressed particular cases. Moreover, our self-contained approach offers a general framework to study the existence/non-existence and multiplicity of normalized solutions. We will exhibit concrete examples to which our method applies, and verify all the assumptions we need. Our approach is applicable to a wide range of operators and domains provided that our hypotheses are verified. Additionally, we prove new results about the non-degeneracy and uniqueness of positive GSS in a general setting. Our findings are applicable to fractional nonlinear Schrodinger equations with non-autonomous nonlinearities. In particular, we were able to extend the main results of [13, 14] to general non-autonomous and mixed nonlinearities. This does not seem possible by using the approach developed by the authors of the above breakthrough papers. The orbital stability/instability of the standing waves will be addressed thanks to the non-degeneracy.

math.AP↗

arXiv2023

On the Eigenvalues of the $p\&q-$ Fractional Laplacian

We consider the eigenvalue problem for the fractional $p \& q-$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|^{p-2}u\quad & \text{in } Ω\\ u=0\quad& \text{in}\quad\R^N\backslashΩ, \end{aligned}\right. \end{equation} where $Ω$ is an open bounded, and possibly disconnected domain, $λ\in\R$, $1 0$ with a weight function in $L^\infty(Ω)$ that is allowed no change sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter $μ\to 0^+.$ In addition, a stability property of eigenvalues as $s\to 1^-$ is established.

math.AP↗

arXiv2022

On existence and stability results for normalized ground states of mass-subcritical biharmonic NLS on $\mathbb{R}^d\times\mathbb{T}^n$

We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from \cite{TTVproduct2014} for determining the sharp $y$-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that $y$-independence of ground states with small mass still holds in the case $β>0$ and $α\in(0,4/(d+n))$. Additionally, we also prove that ground states with sufficiently large mass must possess non-trivial $y$-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.

math.AP↗

arXiv2022

A New Method to prove the Existence, Non-existence, Multiplicity, Uniqueness, and Orbital Stability/Instability of standing waves for NLS with partial confinement

We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction appears for some parameters. We also find different bifurcation phenomena from the cases with full confinement.

math.AP↗

arXiv2022

Threshold for Existence, Non-existence and Multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in RN . In this paper, we will show that in the exterior domain they are strongly stable.

math.AP↗