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Shijie Qi

Publications and source records attributed to Shijie Qi.

4 recordsLinked to original sources
arXiv2023

Normalized solutions to Schödinger equations with potential and inhomogeneous nonlinearities on large convex domains

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem \[ -Δu+V(x)u+λu = |u|^{q-2}u+β|u|^{p-2}u, \quad \|u\|^2_2=\int|u|^2dx = α, \] both on $\mathbb{R}^N$ as well as on domains $rΩ$ where $Ω\subset\mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponents satisfy $2<p<2+\frac4N<q<2^*=\frac{2N}{N-2}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schrödinger equations with potential and find conditions on $V$ so that normalized solutions exist. Our results are new even in the case $β=0$.

math.AP↗
arXiv2018

Semiclassical states of a linearly coupled critical fractional Schrödinger system

This paper focuses on the linearly coupled critical fractional Schrödinger system \begin{equation*} \begin{cases} ε^{2s}(-\triangle)^s u +a(x)u=u^p+λv\quad &\text{in}\ \mathbb{R}^N,\\ ε^{2s}(-\triangle)^s v +b(x)v=v^{2_s^*-1}+λu\quad &\text{in}\ \mathbb{R}^N, \end{cases} \end{equation*} where $N>2s,$ $s\in(0,1),$ $p\in(1,2_s^*),$ $ε$ and $λ$ are positive parameters, $a,b\in C{(\mathbb{R}^N)}$ are positive potentials, and $(-\triangle)^s$ is the fractional Laplacian operator. Under certain assumptions on $a$ and $λ,$ we obtain the existence, decay estimates and concentration property of positive vector ground states for small $ε.$ Furthermore, under an additional assumption on potentials $a$ and $b$, we consider the multiplicity of positive vector solutions for small $ε$, which turn out to have similar decay estimate and concentration property to those of the ground state for small $ε$.

math.AP↗
arXiv2018

A Hopf lemma and regularity for fractional $p-$Laplacians

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.

math.AP↗