arXiv subjects

Shengbing Deng

Publications and source records attributed to Shengbing Deng.

At least 19 recordsLinked to original sources

arXiv2026

Concentration, Local Uniqueness, and Morse Index of Multi-bubble Solutions for a Critical Exponential Biharmonic Choquard Equation

Let $\Omega\subset\mathbb R^4$ be a bounded domain of class $C^6$, let $0<\alpha<4$, and let $K\in C^4(\overline\Omega)$ be positive. We study the Navier problem \[ \Delta^2u=\varepsilon^{8-\alpha}K(x)e^{u(x)} \left(\int_\Omega\frac{K(y)e^{u(y)}}{|x-y|^\alpha}\,dy\right), \qquad u=\Delta u=0\quad\text{on }\partial\Omega. \] Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_\alpha=8\pi^2(8-\alpha)$. The concentration points are governed by \[ \mathcal F_m(\boldsymbol\xi) =\sum_{i=1}^m \left[\log K(\xi_i)+\frac{M_\alpha}{2}H(\xi_i,\xi_i)\right] +M_\alpha\sum_{i<j}G(\xi_i,\xi_j). \] Every $C^1$-stable critical point of $\mathcal F_m$ produces a positive $m$-bubble solution whose scales are of order $\varepsilon^{-1}$ and whose nonlinear source converges to $M_\alpha\sum_i\delta_{\xi_i^*}$. If the critical point is nondegenerate, the corresponding $m$-bubble solution is locally unique, modulo permutations, in a fixed scaled modulation neighborhood. The linearized operator is nondegenerate on $H^2(\Omega)\cap H_0^1(\Omega)$, and \[ \operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbol\xi^*)\right). \] A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8\pi^2b_\alpha^2|\log\varepsilon|^{-1}I_m+o(|\log\varepsilon|^{-1})$, where $b_\alpha=(8-\alpha)/2$.

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arXiv2026

Complete Spectrum and Sharp Local Stability for the Critical Exponential Biharmonic Choquard Equation in $\mathbb R^{4}$

We study the conformally invariant exponential biharmonic Choquard equation in $\mathbb R^{4}$. Our principal result is the complete spectral resolution of the linearized operator at its conformal bubbles. After stereographic projection, the operator becomes a bounded zeroth-order perturbation of the Paneitz operator, with a compact Riesz component on $L^{2}(\mathbb S^{4})$. We justify the weak conformal transfer, remove every pole-supported distributional defect, and compute all eigenvalues. The Morse index is one. The kernel is the five-dimensional conformal space. All higher modes satisfy a uniform coercivity estimate. The transverse Hessian of the Adams--Choquard deficit is the same linearized operator. Hence the complete spectrum gives sharp local stability with respect to the Paneitz distance from the conformal extremal manifold. The optimal asymptotic constant is \[ \gamma_{\alpha} =\frac{160-12\alpha-\alpha^{2}}{40(10-\alpha)}, \] and the limiting quotient is minimized precisely by the second spherical harmonics. As a nonlinear preparation, we also prove that every normal distributional finite-mass solution satisfies the hypotheses of Niu's classification theorem and is therefore an explicit translation--dilation bubble.

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arXiv2024

On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} \Delta(|x|^{-\gamma}\Delta u)=|x|^\gamma u^{\frac{N+4+3\gamma}{N-4-\gamma}},\quad u\geq 0 \quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 5$, $-2<\gamma<0$. Then we derive the sharp Stein-Weiss inequality and standard second-order Caffarelli-Kohn-Nirenberg inequality with radially symmetry extremal functions. Moreover, by using standard spherical decomposition, we derive a sharp weighted Rellich-Sobolev inequality. Furthermore, we establish the sharp second-order Caffarelli-Kohn-Nirenberg type inequalities with two variables which have radially symmetry extremal functions. Finally, we derive the weak form Hardy-Rellich inequalities with sharp constants.

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arXiv2024

Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case

Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \int_{\mathbb{R}^N}|x|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N}|x|^{\gamma} |u|^{2^{**}_{\alpha,\beta}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{\alpha,\beta}}}, \quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N\setminus\{0\}), \end{equation} for some $\mathcal{S}=\mathcal{S}(N,\alpha,\beta)>0$, where $N\geq 5$, $\alpha>2-N$, $\frac{N-4}{N-2}\alpha-4 \leq \beta\leq\alpha -2$ and \begin{align*} 2^{**}_{\alpha,\beta}:=\frac{2(N+\gamma)}{N+2\alpha-\beta-4} \quad \mbox{with}\quad (N+\beta)(N+\gamma)=(N+2\alpha-\beta-4)^2. \end{align*} A crucial element is that the functional $\int_{\mathbb{R}^N}|x|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x$ is equivalent to $\int_{\mathbb{R}^N}|x|^{2\alpha-\beta}|\Delta u|^2 \mathrm{d}x$. Firstly, we obtain a symmetry result (with partial translation invariant) when $\alpha=0$ and $\beta=-4$, then existence and non-existence of extremal functions for the best constant $\mathcal{S}$ in \eqref{nsckn} under different conditions are completely given. Moreover, by a result of linearized problem related to radial solution of \eqref{Pwhs0}, we obtain a symmetry breaking conclusion: when $\alpha>0$ and $\frac{N-4}{N-2}\alpha-4<\beta<\beta_{\mathrm{FS}}(\alpha)$ where $\beta_{\mathrm{FS}}(\alpha):= N+2\alpha-4-\sqrt{(N-2+\alpha)^2+4(N-1)}$, the extremal functions for $\mathcal{S}$ are nonradial. Finally, we give a partial symmetry result when $\beta=\frac{N-4}{N-2}\alpha-4$ and $2-N<\alpha<0$, and we also study the stability of extremal functions.

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arXiv2024

Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace

The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p\frac{\mathrm{d}x}{|x|^{\mu}}\geq \mathcal{S} \left(\int_{\mathbb{R}^N}|u|^r\frac{\mathrm{d}x}{|x|^s} \right)^{\frac{p}{r}}, \quad \mbox{for all}\quad u\in C^\infty_c(\mathbb{R}^N), \] where $1<p<p+\mu<N$, $\frac{\mu}{p}\leq \frac{s}{r}<\frac{\mu}{p}+1$, $r=\frac{p(N-s)}{N-p-\mu}$. An important task is investigating the stability of extremals for this inequality. Firstly, we give the classification to the linearized problem related to the extremals which shows the extremals are non-degenerate. Then we investigate the gradient type remainder term of previous inequality by using spectral estimate combined with a compactness argument which partially extends the work of Wei and Wu [Math. Ann., 2022] to a general $p$-Laplace case, and also the work of Figalli and Zhang [Duke Math. J., 2022] to a weighted case.

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arXiv2023

A note on the Lp-Sobolev inequality

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1 0$ independent of $\Omega$ such that \[ \|\nabla u\|^p_{L^p(\Omega)} -\mathcal{S}^p\|u\|^p_{L^{p^*}(\Omega)} \geq \mathcal{C}|\Omega|^{-\frac{\gamma}{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(\Omega)}^{\gamma}\| u\|_{L^{p^*}(\Omega)}^{p-\gamma},\quad \mbox{for all}\ u\in C^\infty_0(\Omega)\setminus\{0\}, \] where $\gamma=\max\{2,p\}$, $\bar{p}=p^*(p-1)/p$, and $\|\cdot\|_{L^{\bar{p}}_w(\Omega)}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in $\mathbb{R}^N$.

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arXiv2023

Multiple blowing-up solutions for a slightly critical Lane-Emden system with non-power nonlinearity

In this paper, we study the following Lane-Emden system with nearly critical non-power nonlinearity \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{lll} -Δu =\frac{|v|^{p-1}v}{[\ln(e+|v|)]^ε}\ \ &{\rm in}\ Ω, \\[2mm] -Δv =\frac{|u|^{q-1}u}{[\ln(e+|u|)]^ε}\ \ &{\rm in}\ Ω, \\[2mm] u= v=0 \ \ & {\rm on}\ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 3$, $ε>0$ is a small parameter, $p$ and $q $ lying on the critical Sobolev hyperbola $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. We construct multiple blowing-up solutions based on the finite dimensional Lyapunov-Schmidt reduction method as $ε$ goes to zero.

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arXiv2023

Normalized solutions for p-Laplacian equations with potential

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{p}u-V(x)\lvert u\rvert^{p-2}u+λ\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{equation*} where $N\geqslant 1$, $p>1$, $p+\frac{p^2}{N} 0$ and $λ\in\mathbb{R}$ is a Lagrange multiplier which appears due to the mass constraint. Firstly, under some smallness assumptions on $V$, but no any assumptions on $a$, we obtain a mountain pass solution with positive energy, while no solution with negative energy. Secondly, assuming that the mass $a$ has an upper bound depending on $V$, we obtain two solutions, one is a local minimizer with negative energy, the other is a mountain pass solution with positive energy.

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arXiv2023

Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities

In this paper we give the first result about the precise symmetry and symmetry breaking regions of extremal functions for weighted second-order inequalities. Firstly, based on the work of C.-S. Lin [Comm. Partial Differential Equations, 1986], a new second-order Caffarelli-Kohn-Nirenberg type inequality will be established, i.e., \begin{equation*} \int_{\mathbb{R}^N}|x|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N} |x|^{\beta}|u|^{p^*_{\alpha,\beta}} \mathrm{d}x\right)^{\frac{2}{p^*_{\alpha,\beta}}},\quad \mbox{for all}\ u\in C^\infty_0(\mathbb{R}^N), \end{equation*} for some constant $\mathcal{S}=\mathcal{S}(N,\alpha,\beta)>0$, where \begin{align*} N\geq 5,\quad \alpha>2-N,\quad \alpha-2<\beta\leq \frac{N}{N-2}\alpha,\quad p^*_{\alpha,\beta}=\frac{2(N+\beta)}{N-4+2\alpha-\beta}. \end{align*} We obtain a symmetry breaking conclusion: when $\alpha>0$ and $\beta_{\mathrm{FS}}(\alpha)<\beta< \frac{N}{N-2}\alpha$ where $\beta_{\mathrm{FS}}(\alpha):= -N+\sqrt{N^2+\alpha^2+2(N-2)\alpha}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. Furthermore, we give a symmetry result when $\beta=\frac{N}{N-2}\alpha$ and $2-N<\alpha<0$...

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arXiv2023

Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} \mathrm{div}(|x|^{\alpha}\nabla(\mathrm{div}(|x|^\alpha\nabla u)))=u^{2^{**}_{\alpha}-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 2$, $\frac{4-N}{2}<\alpha<2$ and $2^{**}_{\alpha}=\frac{2N}{N-4+2\alpha}$. It is obvious that the solutions of above equation are invariant under the scaling $\lambda^{\frac{N-4+2\alpha}{2}}u(\lambda x)$ while they are not invariant under translation when $\alpha\neq 0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $\alpha$ satisfies $(2-\alpha)(2N-2+\alpha)\neq4k(N-2+k)$ for all $k\in\mathbb{N}^+$ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that ``replace'' the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when $N\geq 5$ and $0<\alpha<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{equation*} \int_{\mathbb{R}^N} |\mathrm{div}(|x|^\alpha\nabla u)|^2 \mathrm{d}x \geq C \left(\int_{\mathbb{R}^N}|u|^{2^{**}_{\alpha}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{\alpha}}},\quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N), \end{equation*} and in this case we consider a prescribed perturbation problem by using Lyapunov-Schmidt reduction.

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arXiv2023

On the stability constant of Caffarelli-Kohn-Nirenberg inequality

By using a spectral analysis, we first show that the Caffarelli--Kohn--Nirenberg inequality with gradient remainder term of any order less than $4$ does not hold on the {\em Felli-Schneider} curve $b_{\mathrm{FS}}(a)$. Furthermore, we prove the existence of minimizers of sharp stability constant of Caffarelli--Kohn--Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)(>b_{\mathrm{FS}}(a))$, which extends the work of Wei and Wu [Math. Z., 2024] to a sightly larger region.

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arXiv2023

Normalized solutions for $p$-Laplacian equation with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -Δ_{p}u=λ\lvert u\rvert^{p-2}u+μ\lvert u\rvert^{q-2}u+\lvert u\rvert^{p^*-1}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{align*} where $1 0$, $μ\in\mathbb{R}$ and $λ\in\mathbb{R}$ is a Lagrange multiplier. Using concentration compactness lemma, Schwarz rearrangement, Ekeland variational principle and mini-max theorems, we obtain several existence results under $μ>0$ and other assumptions. We also analyze the asymptotic behavior of there solutions as $μ\rightarrow 0$ and $μ$ goes to its upper bound. Moreover, we show the nonexistence result for $μ<0$ and get that the $p$-Laplacian equation has infinitely solutions by genus theory when $p<q<p+\frac{p^2}{N}$.

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arXiv2023

Sign changing bubble tower solutions to a slightly subcritical elliptic problem with non-power nonlinearity

We study the following elliptic problem involving slightly subcritical non-power nonlinearity $$\left\{\begin{array}{lll} -Δu =\frac{|u|^{2^*-2}u}{[\ln(e+|u|)]^ε}\ \ &{\rm in}\ Ω, \\[2mm] u= 0 \ \ & {\rm on}\ \partialΩ, \end{array} \right.$$ where $Ω$ is a bounded smooth domain in $\mathbb{R}^n$, $n\geq 3$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent, $ε>0$ is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as $ε$ goes to zero.

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arXiv2023

Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian

For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x -C_{\mu,1}\int_{\mathbb{R}^N}\frac{|\nabla u|^2}{|x|^2} \mathrm{d}x +C_{\mu,2}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4} \mathrm{d}x \geq \mathcal{S}_\mu\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}} \mathrm{d}x\right)^\frac{N-4}{N},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \end{align*} where $C_{\mu,1}$, $C_{\mu,2}$ and $\mathcal{S}_\mu$ are constants depending on $N$ and $\mu$, which is a key ingredient in analyzing the blow-up phenomena of solutions to various elliptic equations on bounded or unbounded domains. Then by using spectral analysis combined with a compactness argument, we consider the stability of this inequality. Furthermore, we derive a remainder term inequality in the weak Lebesgue-norm sense in a subdomain with finite Lebesgue measure.

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arXiv2023

Remainder terms of a nonlocal Sobolev inequality1

In this note we study a nonlocal version of the Sobolev inequality \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^2 dx \geq S_{HLS}\left(\int_{\mathbb{R}^N}\big(|x|^{-α} \ast u^{2_α^{\ast}}\big)u^{2_α^{\ast}} dx\right)^{\frac{1}{2_α^{\ast}}}, \quad \forall u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $S_{HLS}$ is the best constant, $\ast$ denotes the standard convolution and $\mathcal{D}^{1,2}(\mathbb{R}^N)$ denotes the classical Sobolev space with respect to the norm $\|u\|_{\mathcal{D}^{1,2}(\mathbb{R}^N)}=\|\nabla u\|_{L^2(\mathbb{R}^N)}$. By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak $L^{\frac{N}{N-2}}$-norm of above inequality for all $0<α<N$.

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arXiv2023

Stability of Hardy-Sobolev inequality involving p-Laplace

This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_β}}{|x|^β} \mathrm{d}x\right)^\frac{p}{p^*_β},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_β>0$, where $1<p<N$, $0\leq β<p$, $p^*_β=\frac{p(N-β)}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $|x|^{-β}$, some new estimates are established. Based on those works, we give the classification to the linearized problem related to the extremals which has its own interest such as in blow-up analysis. Then we investigate the gradient stability of above inequality by using spectral estimate combined with a compactness argument, which extends the work of Figalli and Zhang (Duke Math. J., 2022) to a weighted case.

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arXiv2022

Caffarelli-Kohn-Nirenberg-type inequalities related to weighted $p$-Laplace equations

We use a suitable transform related to Sobolev inequality to investigate the sharp constants and optimizers for some Caffarelli-Kohn-Nirenberg-type inequalities which are related to the weighted $p$-Laplace equations. Moreover, we give the classification to the linearized problem related to the radial extremals. As an application, we investigate the gradient type remainder term of related inequality by using spectral estimate combined with a compactness argument which extends the work of Figalli and Zhang (Duke Math. J. 2022) at least for radial case.

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arXiv2022

Some weighted fourth-order Hardy-Henon equations

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or Hénon (for $α<0$) equation: \begin{equation*} Δ(|x|^αΔu)=|x|^{-α} u^{p^*_α-1},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N. \end{equation*} %Furthermore, we characterize all the solutions to the linearized problem related to above equation at $U_{1,α}$. For $α\neq 0$, it is known the solutions of above equation are invariant for dilations $λ^{\frac{N-4+α}{2}}u(λx)$ but not for translations. However we show that if $α$ is an even integer, there exist new solutions to the linearized problem, which related to above equation at $U_{1,α}$, that "replace" the ones due to the translations invariance. This interesting phenomenon was first shown by Gladiali, Grossi and Neves [Adv. Math. 249, 2013, 1-36] for the second-order Hénon problem. Finally, as applications, we investigate the reminder term of above inequality and also the existence of solutions to some related perturbed equations.

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