arXiv subjects

Huxiao Luo

Publications and source records attributed to Huxiao Luo.

8 recordsLinked to original sources

arXiv2025

Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials

We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$ \sup_{u\in W^{\frac{1}{2},2}_0(I),\|u\|_{W_0^{\frac{1}{2},2}}\leq1}\int_{I} \int_{I} \log \frac{1}{|x-y|} G(u(x))G(u(y)) \, dx \, dy<+\infty,$$ where $G(s)\leq C\frac{e^{πs^{2}}}{(1 + |s|)^γ}~ \forall s\in\mathbb{R}$ with some constant $C>0,γ\geq1$, the domain $I\subset\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\mathbb{R}$ is also considered. Moreover, we study the existence of corresponding extremal functions. In addition, by the moving plane method, we obtain the radial symmetry and radial decreasing property of positive solutions to the corresponding Euler-Lagrange equation.

math.AP↗

arXiv2025

On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$

We prove the following Limiting Bliss inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{β\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,β), \ \hbox{ for } β\le 1 \end{equation} The inequalities are optimal with respect to $β\le 1$; there is compactness for $β<1$, and along the infinitesimal Moser sequence for $β= 1$. Moreover, we show that the improved inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+γ\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,γ) \end{equation} hold for $γ\leq1$, and for $γ=1$ the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from $N=2$ to general dimensions $N\geq2$.

math.AP↗

arXiv2023

Uniqueness and nondegeneracy of ground states for the Schrödinger-Newton equation with power nonlinearity

In this article, we study the Schrödinger-Newton equation \begin{equation} -Δu+λu=\frac{1}{4π}\left(\frac{1}{|x|}\star u^{2}\right)u+|u|^{q-2}u \quad \text{in}~\mathbb{R}^3, \end{equation} where $λ\in\mathbb{R}_+$, $q\in (2,3)\cup(3, 6)$. By investigating limit profiles of ground states as $λ\to0^+$ or $λ\to+\infty$, we prove the uniqueness of ground states. By the action of the linearized eqaution with respect to decomposition into spherical harmonics, we obtain the nondegeneracy of ground states.

math.AP↗

arXiv2023

Maximum principles for nonlinear integro-differential equations and symmetry of solutions

In this paper, we study the semilinear integro-differential equations \begin{equation*} \mathcal{L}_{K}u(x)\equiv C_n\text{P.V.}\int_{\R^n}\left(u(x)-u(y)\right)K(x-y)dy=f(x,u), \end{equation*} and the full nonlinear integro-differential equations \begin{equation*} F_{G,K}u(x)\equiv C_n\text{P.V.}\int_{\R^n}G(u(x)-u(y))K(x-y)dy=f(x,u), \end{equation*} where $K(\cdot)$ is a symmetric jumping kernel and $K(\cdot)\geq C|\cdot|^{-n-α}$, $G(\cdot)$ is some nonlinear function without non-degenerate condition. We adopt the direct method of moving planes to study the symmetry and monotonicity of solutions for the integro-differential equations, and investigate the limit of some non-local operators $\mathcal{L}_{K}$ as $α\to2.$ Our results extended some results obtained in \cite{CL} and \cite{CLLG}.

math.AP↗

arXiv2023

Normalized solutions for nonautonomous Schrödinger-Poisson equations

In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger-Poisson equations \begin{equation}\nonumber -Δu+λu +\left(\vert x \vert ^{-1} * \vert u \vert ^{2} \right) u=A(x)|u|^{p-2}u,\quad \text{in}~\R^3, \end{equation} where $λ\in\R$, $A \in L^\infty(\R^3)$ satisfies some mild conditions. Due to the nonconstant potential $A$, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For $p\in (2,3)$, $p\in(3,\frac{10}{3})$ and $p\in(\frac{10}{3}, 6)$, we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently, respectively.

math.AP↗

arXiv2023

Uniqueness and nondegeneracy of ground states for $(-Δ)^su+u=2(I_2\star u^2)u$ in $\mathbb{R}^N$ when $s$ is close to 1

In this article, we study the uniqueness and nondegeneracy of ground states to a fractional Choquard equation of the form: $(-Δ)^su+u=2(I_2\star u^2)u$ where $s\in(0,1)$ is sufficiently close to $1$. Our method is to make a continuation argument with respect to the power $s\in(0,1)$ appearing in $(-Δ)^s$. This approach is based on [M. M. Fall and E. Valdinoci, Comm. Math. Phys., 329 (2014) 383-404].

math.AP↗

arXiv2023

Normalized ground states solutions for nonautonomous Choquard equations

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-Δu-λu=\left(\frac{1}{|x|^μ}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$ where $c>0$, $0<μ<N$, $λ\in\mathbb{R}$, $A\in C^1(\mathbb{R}^N,\mathbb{R})$. For $p\in(2_{*,μ}, \bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2^*_μ)$, we find a normalized solution, which is not a global minimizer. $2^*_μ$ and $2_{*,μ}$ are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. $\bar{p}$ is $L^2-$critical exponent. Our results generalize and extend some related results.

math.AP↗

arXiv2022

Bifurcation into spectral gaps for strongly indefinite Choquard equations

We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-Δu+V(x)u=\left(I_α\ast |u|^p\right)|u|^{p-2}u+λu\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right. $$ where $I_α$ is a Riesz potential, $p\in(\frac{N+α}N,\frac{N+α}{N-2})$, $N\geq3$, and $V $ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-Δ+ V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $λ\in(a, b)$, which bifurcate from $b$ if $\frac{N+α}N< p < 1 +\frac{2+α}{N}$. Moreover, $b$ is the unique gap-bifurcation point (from zero) in $[a,b]$. When $λ=a$, we find infinitely many geometrically distinct solutions in $H^2_{loc}(\mathbb R^N)$. Final remarks are given about the eventual occurrence of a bifurcation from infinity in $λ=a$.

math.AP↗