arXiv subjects

Shengbing Deng

Publications and source records attributed to Shengbing Deng.

31 records · Page 2Linked to original sources

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.

math.AP↗

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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arXiv2022

Normalized solutions for a Choquard equation with exponential growth in $\mathbb{R}^{2}$

In this paper, we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} &-Δu+λu=(I_α\ast F(u))f(u), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$ is prescribed, $λ\in \mathbb{R}$, $α\in(0,2)$, $I_α$ denotes the Riesz potential, $\ast$ indicates the convolution operator, the function $f(t)$ has exponential growth in $\mathbb{R}^{2}$ and $F(t)=\int^{t}_{0}f(τ)dτ$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.

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arXiv2022

Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following system \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\big(a_1+b_1\|u\|^{2(θ_1-1)}\big)Δu= λH_u(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] -\big(a_2+b_2\|v\|^{2(θ_2-1)}\big)Δv= λH_v(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary,\ $\|u\|=\big(\int_Ω|\nabla u|^2dx\big)^{1/2}$, $H_u$ and $H_v$ behave like $e^{β|s|^2}$ when $|s|\rightarrow \infty$ for some $β>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $θ_1,\ θ_2> 1$ and $λ$ is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.

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arXiv2022

Normalized solutions for Schrödinger systems in dimension two

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-Δu+λ_{1}u=H_{u}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &-Δv+λ_{2} v=H_{v}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2},\quad \int_{\mathbb{R}^{2}}|v|^{2}dx=b^{2}, \end{aligned} \right. \end{align*} where $a,b>0$ are prescribed, $λ_{1},λ_{2}\in \mathbb{R}$ and the functions $H_{u},H_{v}$ are partial derivatives of a Carathéodory function $H$ with $H_{u},H_{v}$ have exponential growth in $\mathbb{R}^{2}$. Our main results are totally new for Schrödinger systems in $\mathbb{R}^{2}$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.

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arXiv2022

Existence of solution for a class of fractional Hamiltonian-type elliptic systems with exponential critical growth in R

In this paper, we study the following class of fractional Hamiltonian systems: \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} (-Δ)^{\frac{1}{2}} u + u = \Big(I_{μ_{1}}\ast G(v)\Big)g(v) \ \ \ & \mbox{in} \ \mathbb{R},\\[2mm] (-Δ)^{\frac{1}{2}} v + v = \Big(I_{μ_{2}}\ast F(u)\Big)f(u) \ \ \ & \mbox{in} \ \mathbb{R}, \end{array} \right. \end{aligned} \end{eqnarray*} where $(-Δ)^{\frac{1}{2}}$ is the square root Laplacian operator, $μ_{1},μ_{2}\in(0,1)$, $I_{μ_{1}},I_{μ_{2}}$ denote the Riesz potential, $\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{R}$. Using the linking theorem and variational methods, we establish the existence of at least one positive solution to the above problem.

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arXiv2022

On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth

In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} -Δu + V(x)u = \Big(I_{μ_{1}}\ast \frac{G(v)}{|x|^α}\Big)\frac{g(v)}{|x|^α} \ \ \ & \mbox{in} \ \mathbb{R}^{2},\\[2mm] -Δv + V(x)v = \Big(I_{μ_{2}}\ast \frac{F(u)}{|x|^β}\Big)\frac{f(u)}{|x|^β} \ \ \ & \mbox{in} \ \mathbb{R}^{2}, \end{array} \right. \end{aligned} \end{eqnarray*} where $μ_{1},μ_{2}\in(0,2)$, $0<α\leq \frac{μ_{1}}{2}$, $0<β\leq \frac{μ_{2}}{2}$, $V(x)$ is a continuous positive potential, $I_{μ_{1}}$ and $I_{μ_{2}}$ denote the Riesz potential, $\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{R}^{2}$. Using the linking theorem and variational methods, we establish the existence of solutions to the above problem.

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arXiv2022

Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system: \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} \varepsilon^{2s}(-Δ)^{s}u+u+ϕu=u^p,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] (-Δ)^{s}ϕ=u^2,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] u=ϕ=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus Ω, \end{array} \right. \end{eqnarray*} where $s\in(0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, and $(-Δ)^{s}$ is the fractional Laplacian operator, $\varepsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_\varepsilon,ϕ_\varepsilon)$ such that the peak of $u_\varepsilon$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance $\varepsilon^{2/3}$ away from the boundary $\partialΩ$ as $\varepsilon$ tends to 0, some new estimates and analytic technique are used.

math.AP↗

arXiv2022

On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -m\left(\sum^k_{j=1}\|u_j\|^2\right)Δu_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^β}+\varepsilon h_i(x),\ \ & \mbox{in}\ \ Ω, \ \ i=1,\ldots,k ,\\[2mm] u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin with smooth boundary, $β\in [0,2)$, $m$ is a Kirchhoff type function, $\|u_j\|^2=\int_Ω|\nabla u_j|^2dx$, $f_i$ behaves like $e^{βs^2}$ when $|s|\rightarrow \infty$ for some $β>0$, and there is $C^1$ function $F: Ω\times\mathbb{R}^k\to \mathbb{R}$ such that $\left(\frac{\partial F}{\partial u_1},\ldots,\frac{\partial F}{\partial u_k}\right)=\left(f_1,\ldots,f_k\right)$, $h_i\in \left(\big(H^1_0(Ω)\big)^*,\|\cdot\|_*\right)$. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when $\varepsilon>0$ is small.

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arXiv2017

Concentration at submanifolds for an elliptic Dirichlet problem near high critical exponents

Let $Ω$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partialΩ$. We consider the equation $ Δu + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,Ω$, under zero Dirichlet boundary condition, where $\varepsilon$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partialΩ$, which is non-degenerate, and along which a certain weighted average of sectional curvatures of $\partialΩ$ is negative. Under these assumptions, we prove existence of a sequence $\varepsilon=\varepsilon_j$ and a solution $u_{\varepsilon}$ which concentrate along $K$, as $\varepsilon \to 0^+$, in the sense that $$ |\nabla u_{\varepsilon} |^2\,\rightharpoonup \, S_{n-k}^{\frac{n-k}{2}} \,δ_K \quad \mbox{as} \ \ \varepsilon \to 0 $$ where $δ_K $ stands for the Dirac measure supported on $K$ and $S_{n-k}$ is an explicit positive constant. This result generalizes the one obtained by del Pino-Musso-Pacard, where the case $k=1$ is considered.

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arXiv2017

High energy sign-changing solutions for Coron's problem

We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} Δu+|u|^{p-1}u=0 \quad & {\rm in} \quad Ω_ε; u=0 \quad & {\rm on} \quad\partial Ω_ε, \end{array} \right. $$ where $p=\frac{n+2}{n-2}$ is the critical Sobolev exponent and $Ω_ε$ is a bounded smooth domain in ${\mathcal R}^n$, $n\geq 3$, with the form $Ω_ε=Ω\backslash B(0,ε)$ with $Ω$ a smooth bounded domain containing the origin $0$ and $B(0,ε)$ the ball centered at the origin with radius $ε>0$. We construct a new type of sign-changing solutions with high energy to problem $(P)$, when the parameter $ε$ is small enough.

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arXiv2014

Concentration on minimal submanifolds for a Yamabe type problem

We construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.

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