arXiv subjects

Zongyan Lv

Publications and source records attributed to Zongyan Lv.

7 recordsLinked to original sources

arXiv2025

Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities

In this paper, we consider the existence, multiplicity and the asymptotic behavior of prescribed mass solutions to the following nonlinear Kirchhoff equation with mixed nonlinearities: -(a+b\int|\nabla u|^2\mathrm{d}x)Δu+V(x)u+λu=|u|^{q-2}u+β|u|^{p-2}u \quad&\text{in}\ Ω, \int_Ω|u|^2\mathrm{d}x=α, both on large bounded smooth star-shaped domain Ω\subset\mathbb{R}^{3} and on \mathbb{R}^{3}, where 2<p<\frac{14}{3}<q<6 and V(x) is the potential. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid due to the presence of potential V(x).

math.AP↗

arXiv2025

Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass

In this paper, we consider the following Brézis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -Δu=|u|^{2^*-2} u +λ_ρu\quad \text { in } Ω, u>0, \quad u \in H_0^1(Ω), \quad \int_Ω u^2dx=ρ, \end{cases} \end{equation*} where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $ρ>0$ is a given small constant and $λ_ρ>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $Ω$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Brézis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators.

math.AP↗

arXiv2025

Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrödinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -Δu+V(x)u+λu=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $Ω\subset\mathbb{R}^N$ and on $\mathbb{R}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot)$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a complement of Bartsch-Qi-Zou (Math Ann 390, 4813--4859, 2024), which has addressed an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729--1756, 2021).

math.AP↗

arXiv2025

Normalized solutions on large smooth domains to the Schrödinger equations with potential and combined nonlinearities: The Sobolev critical case

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential. Besides, Our study can be regarded as a Sobolev critical case complement of Bartsch-Qi-Zou (Math Ann 390, 4813-4859, 2024), which has addressed an open problem raised in Bartsch et al.(Commun Partial Differ Equ 46(9):1729--1756,2021).

math.AP↗

arXiv2023

Normalized solutions for a Kirchhoff type equations with potential in $\mathbb{R}^3$

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)Δu+V(x)u+λu=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying the normalized constraint $\displaystyle\int_{\R^3}u^2=c$, where $a,b,c>0$ are prescribed constants, and the nonlinearities $g(u)$ are very general and of mass super-critical. Under some suitable assumptions on $V(x)$ and $g(u)$, we can prove the existence of ground state normalized solutions $(u_c, λ_c)\in H^1(\R^3)\times\mathbb{R}$, for any given $c>0$. Due to the presence of the nonlocal term, the weak limit $u$ of any $(PS)_C$ sequence $\{w_n\}$ may not belong to the corresponding Pohozaev manifold, which is different from the local problem. So we have to overcome some new difficulties to gain the compactness of a $(PS)_C$ sequence.

math.AP↗

arXiv2023

Existence of nontrivial solutions for critical biharmonic equations with logarithmic term

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} Δ^2 u=μΔu+λu+|u|^{2^{**}-2}u+τu\log u^2, \ \ x\inΩ, u|_{\partial Ω}=\frac{\partial u}{\partial n}|_{\partialΩ}=0, \end{cases} \end{equation*} where $μ,λ,τ\in \mathbb{R}$, $|μ|+|τ|\ne 0$, $Δ^2=ΔΔ$ denotes the iterated N-dimensional Laplacian, $Ω\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial Ω$, $2^{**}=\frac{2N}{N-4}(N\ge5)$ is the critical Sobolev exponent for the embedding $H_{0}^{2}(Ω)\hookrightarrow L^{2^{**}}(Ω)$ and $H_0^2 (Ω)$ is the closure of $C_0^ \infty (Ω)$ under the norm $|| u ||:=(\int_Ω|Δu|^2)^\frac{1}{2}$. The uncertainty of the sign of $s\log s^2$ in $(0,+\infty)$ has some interest in itself. To know which of the three terms $μΔu$, $λu$ and $τu \log u^2$ has a greater influence on the existence of nontrivial weak solutions, we prove the existence of nontrivial weak solutions to the above problem for $N\ge5$ under some assumptions of $λ, μ$ and $τ$.

math.AP↗

arXiv2021

Positive normalized solution to the Kirchhoff equation with general nonlinearities of mass super-critical

In present paper, we study the normalized solutions $(λ_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c>0$. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as $c\rightarrow 0^+$ as well as $c\rightarrow+\infty$.

math.AP↗