Searcharxiv⌕ Search

arXiv subjects

Jianqing Chen

Publications and source records attributed to Jianqing Chen.

7 recordsLinked to original sources
arXiv2023

Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in $\mathbb R^N$: $$\left\{\begin{align}&-Δu+A(x)u=\frac{2p}{p+q} \bigl(I_α\ast |v|^q\bigr)|u|^{p-2}u,\\ &-Δv+B(x)v=\frac{2q}{p+q}\bigl(I_α\ast|u|^p\bigr)|v|^{q-2}v,\\ &\ u(x)\to0\ \ \hbox{and}\ \ v(x)\to0\ \ \hbox{as}\ |x|\to\infty,\end{align}\right.$$ where $α\in(0,N)$ and $\frac{N+α}{N}<p,\ q<2_*^α$, in which $2_*^α$ denotes $\frac{N+α}{N-2}$ if $N\geq 3$ and $2_*^α:= \infty$ if $N=1,\ 2$. The function $I_α$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of positive ground state solution in the case of bounded potential and periodic potential respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\neq q$ and $u(x)\neq v(x)$. Moreover it seems to be the first existence result for the case of $p\neq q$.

math.AP↗
arXiv2023

Multiple nonsymmetric nodal solutions for quasilinear Schrödinger system

In this paper, we consider the quasilinear Schrödinger system in $\mathbb R^{N}$($N\geq3$): $$\left\{\begin{align} &-Δu+ A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2α}{α+β}|u|^{α-2}u|v|^β,\\ &-Δv+ Bv-\frac{1}{2}\triangle(v^{2})v=\frac{2β}{α+β}|u|^α|v|^{β-2}v,\end{align}\right. $$ where $α,β>1$, $2<α+β<\frac{4N}{N-2}$,$B>0$ is a constant. By using a constrained minimization on Nehari-Pohožaev set, for any given integer $s\geq2$, we construct a non-radially symmetrical nodal solution with its $2s$ nodal domains.

math.AP↗
arXiv2023

Ground states solution of Nehari-Pohožaev type for periodic quasilinear Schrödinger system

This paper is concerned with a quasilinear Schrödinger system in $\mathbb R^{N}$ $$\left\{\aligned &-Δu+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2α}{α+β}|u|^{α-2}u|v|^β,\\ &-Δv+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2β}{α+β}|u|^α|v|^{β-2}v,\\ & u(x)\to 0\ \hbox{and}\quad v(x)\to 0\ \hbox{as}\ |x|\to \infty,\endaligned\right. $$ where $α,β>1$ and $2<α+β<\frac{4N}{N-2}$ ($N \geq 3$). $A(x)$ and $B(x)$ are two periodic functions. By minimization under a convenient constraint and concentration-compactness lemma, we prove the existence of ground states solution. Our result covers the case of $α+β\in(2,4)$ which seems to be the first result for coupled quasilinear Schrödinger system in the periodic situation.

math.AP↗
arXiv2023

Existence of ground state solution of Nehari-Pohožaev type for a quasilinear Schrödinger system

This paper is concerned with the following quasilinear Schrödinger system in the entire space $\mathbb R^{N}$($N\geq3$): $$\left\{\begin{align} &-Δu+A(x)u-\frac{1}{2}\triangle(u^{2})u = \frac{2α}{α+β}|u|^{α-2}u|v|^β,\\ &-Δv+Bv-\frac{1}{2}\triangle(v^{2})v=\frac{2β}{α+β}|u|^α|v|^{β-2}v.\end{align}\right. $$ By establishing a suitable constraint set and studying related minimization problem, we prove the existence of ground state solution for $α,β>1$, $2<α+β<\frac{4N}{N-2}$. Our results can be looked on as a generalization to results by Guo and Tang (Ground state solutions for quasilinear Schrödinger systems, J. Math. Anal. Appl. 389 (2012) 322).

math.AP↗
arXiv2022

A local pointwise inequality for a biharmonic equation with negative exponents

In this paper, we are inspired by Ngô, Nguyen and Phan's [15] study of the pointwise inequality for positive $C^{4}$-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality $$-\frac{Δu}{u}+α\frac{|\nabla u|^{2}}{u^{2}}+βu^{-\frac{q+1}{2}}\leq\frac{C}{R^{2}}$$ for positive $C^{4}$-solutions of the biharmonic equations with negative exponent $$-Δ^{2}u=u^{-q} \ in \ B_{R}$$ where $B_{R}$ denotes the ball centered at $x_{0}$ with radius $R$, $n\geq3$, $q>1$, and some constants $α\geq0$, $β>0$, $C>0$.

math.AP↗
arXiv2010

Instability of Standing Waves to the Inhomogeneous Nonlinear Schrödinger Equation with Harmonic Potential

We study the instability of standing-wave solutions $e^{iωt}ϕ_ω(x)$ to the inhomogeneous nonlinear Schrödinger equation $$iϕ_t=-\triangleϕ+|x|^2ϕ-|x|^b|ϕ|^{p-1}ϕ, \qquad \in\mathbb{R}^N, $$ where $ b > 0 $ and $ ϕ_ω $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $ω$ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $

math.AP↗