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Yinbin Deng

Publications and source records attributed to Yinbin Deng.

12 recordsLinked to original sources
arXiv2026

Threshold phenomena for least energy solutions of a doubly critical Neumann problem with a critical absorption term

We investigate the existence and nonexistence of least energy solutions for a doubly critical Neumann problem with a critical absorption term. We consider a positive function $u$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^n$ with $n\ge 5$, satisfying $-\Delta u + \lambda u = u^{2^* - 1} -\alpha u^{2^\sharp-1}$ inside $\Omega$, with Neumann boundary condition $\nabla u\cdot \nu = u^{2^\sharp -1}$ on $\partial\Omega$, where $\lambda>0$, $\alpha\ge0$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent and $2^\sharp=\frac{2(n-1)}{n-2}$ denotes the critical trace exponent. The simultaneous presence of the interior and boundary critical exponents together with this critical absorption term creates a new interaction between competing concentration mechanisms. In particular, the absorption term has the same critical trace exponent as the boundary nonlinearity, but with the opposite sign and acting in the interior of $\Omega$. Consequently, it competes with the boundary mean curvature correction at the same asymptotic order in the energy expansion, leading to a sharp threshold phenomenon. Combining several analytic techniques and geometric tools, we prove the existence of a threshold value $\alpha_{0}=\alpha_{0}(\lambda,\Omega)\in(0,+\infty)$ such that the problem admits a least energy solution if $\alpha<\alpha_{0}$, and no least energy solution if $\alpha>\alpha_{0}$. Moreover, the problem also has a least energy solution at $\alpha =\alpha_0$ provided $\alpha_0> C(n)\max_{\partial \Omega} H$, where $C(n)$ is a positive constant depending only on $n$ and $H$ is the mean curvature on $\partial\Omega$.

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arXiv2026

Morse index, Leray-Schauder degree and local uniqueness for multi-peak concentrating solutions of a fractional Schr\"odinger equation

We study positive $k$-peak solutions of the semiclassical fractional Schr\"odinger equation $\varepsilon^{2s}(-\Delta)^s u+V(x)u=u^p$ in $\mathbb{R}^N$, concentrating at different nondegenerate critical points $\xi_1^0,\ldots,\xi_k^0$ of $V$. For every such family satisfying the natural energy quantization, we determine the complete low spectrum of the linearized operator. The first $k$ eigenvalues remain uniformly negative, the next $kN$ eigenvalues are of order $\varepsilon^2$ and are governed by the Hessians $D^2V(\xi_j^0)$, while the remaining spectrum is uniformly separated from zero. Consequently, the Morse index equals $k$ plus the total number of negative eigenvalues of these Hessians, and every such solution is nondegenerate. Combining a unique modulation parametrization with a Leray--Schauder degree computation, we further prove that, for all sufficiently small $\varepsilon$, the prescribed concentrating class contains exactly one positive solution. The result applies to the whole energy-quantized class, not only to a particular solution.

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arXiv2026

On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

We consider ground states of a pseudo-relativistic Fermi system in the $L^2$-critical case. We prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<D_{4/3,2}$, where $D_{4/3,2}\in(0, \infty)$ is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as $a\nearrow D_{4/3,2}$. As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.

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arXiv2026

On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form $(-\Delta)^s-\Delta$, as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical first-order derivative. A major difficulty in this setting stems from the coexistence of operators with different nonlocal structures and incompatible scaling properties, which obstruct the direct use of classical sliding methods. To address this issue, we develop a refined sliding method suited to mixed local-nonlocal operators. As key technical ingredients, we establish new generalized weighted average inequalities, narrow region principles, and maximum principles in bounded and unbounded domains. These tools enable us to derive monotonicity and one-dimensional symmetry results for mixed elliptic equations in bounded domains, half-spaces, and the whole space, and to extend the analysis to parabolic equations with mixed time derivatives. As an application, we resolve the Gibbons' conjecture for a class of mixed fractional equations.

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arXiv2024

On the existence of positive solution for a Neumann problem with double critical exponents in half-space

In this paper, we consider the existence and nonexistence of positive solution for a Neumann problem with double critical exponents and fast increasing weighted in half-space. This problem is closely related to the study of self-similar solutions for nonlinear heat equation. By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for the Mountain Pass level, we obtain the existence of a positive solution under different assumptions. Meanwhile, some nonexistence results for this problem is also obtained by an improved Pohozaev identity and Hardy inequality according to the value of the parameters. Particularly, we give the best lower bound of the parameter for the existence of a positive solution of this problem if dimension N=4.

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arXiv2024

Existence of solutions for critical Neumann problem with superlinear perturbation in the half-space

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu {|u|^{p-2}u}}& \ \ \mbox{in} \ \ \ {{\mathbb{R}}^{N}_{+}}, \frac{{\partial u}}{{\partial n}}&=\sqrt{\lambda}|u|^{{2}_{*}-2}u \ & \mbox{on}\ {{\partial {{\mathbb{R}}^{N}_{+}}}}, \end{aligned} \right. \end{equation} where $ \mathbb{R}^{N}_{+}=\{(x{'}, x_{N}): x{'}\in {\mathbb{R}}^{N-1}, x_{N}>0\}$, $N\geq3$, $\lambda>0$, $\mu\in \mathbb{R}$, $2< p <{2}^{*}$, $n$ is the outward normal vector at the boundary ${{\partial {{\mathbb{R}}^{N}_{+}}}}$, $2^{*}=\frac{2N}{N-2}$ is the usual critical exponent for the Sobolev embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}^{*}}}({\mathbb{R}}^{N}_{+})$ and ${2}_{*}=\frac{2(N-1)}{N-2}$ is the critical exponent for the Sobolev trace embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}_{*}}}(\partial \mathbb{R}^{N}_{+})$. By establishing an improved Pohozaev identity, we show that the problem has no nontrivial solution if $\mu \le 0$; By applying the Mountain Pass Theorem without $(PS)$ condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution for all $\lambda>0$ and the different values of the parameters $p$ and ${\mu}>0$. Particularly, for $\lambda >0$, $N\ge 4$, $2 0$. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all $\mu>0$ and suitable $\lambda$.

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arXiv2023

Semi-classical states for fractional Choquard equations with decaying potentials

This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^su +Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ is the fractional Laplacian, $N>2s$, $s\in(0,1)$, $\alpha\in\big((N-4s)_{+}, N\big)$, $p\in[2, \frac{N+\alpha}{N-2s})$, $I_\alpha$ is a Riesz potential, $V\in C\big(\mathbb{R}^N, [0, +\infty)\big)$ is an electric potential. Under some assumptions on the decay rate of $V$ and the corresponding range of $p$, we prove that the problem has a family of solutions $\{u_\varepsilon\}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. Since the potential $V$ decays at infinity, we need to employ a type of penalized argument and implement delicate analysis on the both nonlocal terms to establish regularity, positivity and asymptotic behaviour of $u_\varepsilon$, which is totally different from the local case. As a contrast, we also develop some nonexistence results, which imply that the assumptions on $V$ and $p$ for the existence of $u_\varepsilon$ are almost optimal. To prove our main results, a general strong maximum principle and comparison function for the weak solutions of fractional Laplacian equations are established. The main methods in this paper are variational methods, penalized technique and some comparison principle developed in this paper.

math.AP↗
arXiv2023

Existence and decays of solutions for fractional Schr\"{o}dinger equations with decaying potentials

We revisit the following fractional Schr\"{o}dinger equation \begin{align}\label{1a} \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes the fractional Laplacian, $s\in(0,1)$, $p\in (2, 2_s^*)$, $2_s^*=\frac {2N}{N-2s}$, $N>2s$, $V\in C\big(\R^N, [0, +\infty)\big)$ is a potential. Under various decay assumptions on $V$, we introduce a uniform penalization argument combined with a comparison principle and iteration process to detect an explicit threshold value $p_*$, such that the above problem admits positive concentration solutions if $p\in (p_*, \,2_s^*)$, while it has no positive weak solutions for $p\in (2,\,p_*)$ if $p_*>2$, where the threshold $p_*\in [2, 2^*_s)$ can be characterized explicitly by \begin{equation*}\label{qdj111} p_*=\left\{\begin{array}{l} 2+\frac {2s}{N-2s} \ \ \ \text { if } \lim\limits_{|x| \to \infty} (1+|x|^{2s})V(x)=0,\vspace{1mm} 2+\frac {\omega}{N+2s-\omega} \text { if } 0<\inf (1+|x|^\omega)V(x)\le \sup (1+|x|^\omega)V(x)< \infty \text { for some } \omega \in [0, 2s],\vspace{1mm} 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { if } \inf V(x)\log(e+|x|^2)>0. \end{array}\right. \end{equation*} Moreover, corresponding to the various decay assumptions of $V(x)$, we obtain the decay properties of the solutions at infinity.

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arXiv2022

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2} &x\in \Omega, \quad \;\:\, u=0& x\in \partial \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\R^N$ is a bounded smooth domain, $\lambda, \mu \in \R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$. The uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $\lambda\in \R, \mu>0$ and $N\geq 4$. While the case of $\mu<0$ is thornier. However, for $N=3,4$ $\lambda\in (-\infty, \lambda_1(\Omega))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $\mu<0$ and $-\frac{(N-2)\mu}{2}+\frac{(N-2)\mu}{2}\log(-\frac{(N-2)\mu}{2})+\lambda-\lambda_1(\Omega)\geq 0$ if $N\geq 3$. Comparing with the results in Br\'ezis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $\mu$ on logarithmic perturbation is not zero.

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arXiv2022

Existence and Concentration Results for the General Kirchhoff Type Equations

We consider the following singularly perturbed Kirchhoff type equations $$-\varepsilon^2 M\left(\varepsilon^{2-N}\int_{\R^N}|\nabla u|^2 dx\right)\Delta u +V(x)u=|u|^{p-2}u~\hbox{in}~\R^N, u\in H^1(\R^N),N\geq 1,$$ where $M\in C([0,\infty))$ and $V\in C(\R^N)$ are given functions. Under very mild assumptions on $M$, we prove the existence of single-peak or multi-peak solution $u_\varepsilon$ for above problem, concentrating around topologically stable critical points of $V$, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. in 2014 [ARMA,213].

math.AP↗
arXiv2021

Sharp interaction estimates and their application: existence of normalized ground states to coupled Schr\"odinger systems with potentials

In this paper, our aim is to prove the existence of normalized ground state for the following Schr\"odinger systems with potentials $$\begin{cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ -\Delta u_2+V_2(x)u_2+\lambda_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ 0 -\infty$, which are allowed to be singular at some points. And the nonlinearities $G(u_1,u_2)$ are considered of the form $$ \begin{cases} G(u_1, u_2):=\sum_{i=1}^{\ell}\frac{\mu_i}{p_i}|u_1|^{p_i}+\sum_{j=1}^{m}\frac{\nu_j}{q_j}|u_2|^{q_j}+\sum_{k=1}^{n}\beta_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}},~~\ell,m,n\in \mathbb{N}^+_0, \mu_i, \nu_j,\beta_k>0, ~2 1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional $J$ on the manifold $S_{a_1,a_2}$. Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.

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