arXiv subjects

Yueqiang Song

Publications and source records attributed to Yueqiang Song.

6 recordsLinked to original sources

arXiv2026

Concentration Phenomena of Normalized Solutions of Critical Biharmonic Equations with Combined Nonlinearities in $\mathbb{R}^{N}$

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} Δ^{2}u+V(\varepsilon x)u=λu+μ|u|^{q-2}u+|u|^{2^{**}-2}u \mbox{ in }\ \mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2}, \end{equation*} where $Δ^{2}$ is the biharmonic operator, $N\geq5$, $μ,c>0$, $\varepsilon>0,$ $λ\in\mathbb{R}$, $q\in(2,2+\frac{8}{N}),$ and $2^{**}=\frac{2N}{N-4}$ is the Sobolev critical exponent. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\varepsilon$ is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.

math.AP↗

arXiv2025

Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s} 0$ small enough. In a certain sense, we generalize some previously known results.

math.AP↗

arXiv2024

On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity

In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-Δ)_{p, A}^{s} u+V(x)|u|^{p-2} u=λ\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{μ, s}^{*}}}{|x-y|^μ} \mathrm{d}y\right)|u|^{p_{μ, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$$ where $0<s<1<p$, $ps < N$, $p<q<2p^{*}_{s,μ}$, $0<μ<N$, $λ$ and $k$ are some positive parameters, $p^{*}_{s,μ}=\frac{pN-p\fracμ{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.

math.AP↗

arXiv2023

High and low perturbations of the critical Choquard equation on the Heisenberg group

We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-Δ_H u }=μ |u|^{q-2}u+\int_Ω \frac{|u(η)|^{Q_λ^{\ast}}} {|η^{-1}ξ|^λ} dη|u|^{Q_λ^{\ast}-2}u &\mbox{in }\ Ω, u=0 &\mbox{on }\ \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{H}^N$ is a smooth bounded domain, $Δ_H$ is the Kohn-Laplacian on the Heisenberg group $\mathbb{H}^N$, $1 0$, $0<λ<Q=2N+2$, and $Q_λ^{\ast}=\frac{2Q-λ}{Q-2}$ is the critical exponent. Using the concentration compactness principle and the critical point theory, we prove that the above problem has the least two positive solutions for $1<q<2$ in the case of low perturbations (small values of $μ$), and has a nontrivial solution for $2<q<Q_λ^\ast$ in the case of high perturbations (large values of $μ$). Moreover, for $1<q<2$, we also show that there is a positive ground state solution, and for $2<q<Q_λ^\ast$, there are at least $n$ pairs of nontrivial weak solutions.

math.AP↗

arXiv2023

On $p$-Laplacian Kirchhoff-Schrödinger-Poisson type systems with critical growth on the Heisenberg group

In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form: \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_Ω|\nabla_{H} u|^{p}dξ)Δ_{H,p}u-μϕ|u|^{p-2}u}=λ|u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ Ω, \\ -Δ_{H}ϕ=|u|^{p} &\mbox{in}\ Ω, \\ u=ϕ=0 &\mbox{on}\ \partialΩ, \end{array} \right. \end{equation*} where $a,b$ are positive real numbers, $Ω\subset \mathbb{H}^N$ is a bounded region with smooth boundary, $1<p<Q$, $Q = 2N + 2$ is the homogeneous dimension of the Heisenberg group $\mathbb{H}^N$, $Q^{\ast}=\frac{pQ}{Q-p}$, $q\in(2p, Q^{\ast})$, and $Δ_{H,p}u=\mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u)$ is the $p$-horizontal Laplacian. Under some appropriate conditions for the parameters $μ$ and $λ$, we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).

math.AP↗

arXiv2023

On the Schrödinger-Poisson system with $(p,q)$-Laplacian

We study a class of Schrödinger-Poisson systems with $(p,q)$-Laplacian. Using fixed point theory, we obtain a new existence result for nontrivial solutions. The main novelty of the paper is the combination of a double phase operator and the nonlocal term. Our results generalize some known results.

math.AP↗