arXiv subjects

Houwang Li

Publications and source records attributed to Houwang Li.

8 recordsLinked to original sources

arXiv2026

On the Classification of blow-up solutions of a singular Liouville equation on the disk

We study the blow-up behavior of solutions to the singular Liouville equation \[ \Delta \tilde u+\lambda e^{\tilde u}=4\pi\alpha\delta_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $\alpha>0$, $\lambda>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $\alpha>0$ and $\lambda\in(0,\lambda_\alpha)$, the singular Liouville equation has exactly $\lceil \alpha\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil \alpha\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

math.AP↗

arXiv2025

Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness

Let $(\Sigma, g_1)$ be a compact Riemann surface with conical singularites of angles in $(0, 2\pi)$, and $f: \Sigma\to\mathbb R$ be a positive smooth function. In this paper, by establishing a sharp quantization result, we prove the compactness of the set of positive critical points for the Moser-Trudinger functional \[F_1(u)=\int_{\Sigma}(e^{u^2}-1)f dv_{g_1}\] constrained to $u\in\mathcal E_\beta:=\{u\in H^1(\Sigma,g_1) : \|u\|_{H^1(\Sigma,g_1)}^2=\beta\}$ for any $\beta>0$. This result is a generalization of the compactness result for the Moser-Trudinger functional on regular compact surfaces, proved by De Marchis-Malchiodi-Martinazzi-Thizy (Inventiones Mathematicae, 2022, 230: 1165-1248). The presence of conical singularities brings many additional difficulties and we need to develop different ideas and techniques. The compactness lays the foundation for proving the existence of critical points of the Moser-Trudinger functional on conical singular surfaces in a sequel work.

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arXiv2024

Normalized solutions for a class of Sobolev critical Schrodinger systems

This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the multiplicity of normalized solutions for Schrodinger systems with Sobolev critical growth, with potential for future development and broader applicability.

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arXiv2023

Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials \[\begin{cases} -\Delta u_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega, u_n=0,\quad\text{on}~\partial\Omega, \int_\Omega p_n W_n(x)u_n^{p_n}dx\le C, \end{cases}\] where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $W_n(x)\geq 0$ are bounded functions with zeros in $\Omega$, and $p_n\to\infty$ as $n\to\infty$. A typical example is $W_n(x)=|x|^{2\alpha}$ with $0\in\Omega$, i.e. the equation turns to be the well-known H\'enon equation. The asymptotic behavior for $\alpha=0$ has been well studied in the literature. While for $\alpha>0$, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case $\alpha>0$ and prove a quantization property (suppose $0$ is a concentration point) \[p_n|x|^{2\alpha}u_n(x)^{p_n-1+t}\to 8\pi e^{\frac{t}{2}}\sum_{i=1}^k\delta_{a_i}+8\pi(1+\alpha)e^{\frac{t}{2}}c^t\delta_0, \quad t=0,1,2,\] for some $k\ge0$, $a_i\in\Omega\setminus\{0\}$ and some $c\ge1$. Moreover, for $\alpha\not\in\mathbb{N}$, we show that the blow up must be simple, i.e. $c=1$. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the H\'enon equation.

math.AP↗

arXiv2022

Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains

We study the Lane-Emden system $$\begin{cases} -Δu=v^p,\quad u>0,\quad\text{in}~Ω, -Δv=u^q,\quad v>0,\quad\text{in}~Ω, u=v=0,\quad\text{on}~\partialΩ, \end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq Λ$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $Ω$ is convex, then the solution of this system is unique and nondegenerate for large $p, q$.

math.AP↗

arXiv2022

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $α,β>1$, $2<α+β<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $ν<0$. When $ν>0$ and $α+β\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<ν<ν_0$, the constant $ν_0$ will be explicitly given. In the case $α+β>2+\frac{4}{N}$ we prove the existence of a threshold $ν_1\ge 0$ such that a normalized ground state solution exists for $ν>ν_1$, and does not exist for $ν<ν_1$. We also give conditions for $ν_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $ν\to0^+$ or $ν\to+\infty$.

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arXiv2021

Quasilinear Schr\"odinger equations: ground state and infinitely many normalized solutions

In the present paper, we study the normalized solutions for the following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$ We first consider the mass-supercritical case $p>4+\frac{4}{N}$, which has not been studied before. By using a perturbation method, we succeed to prove the existence of ground state normalized solutions, and by applying the index theory, we obtain the existence of infinitely many normalized solutions. Then we turn to study the mass-critical case, i.e., $p=4+\frac{4}{N}$, and obtain some new existence results. Moreover, we also observe a concentration behavior of the ground state solutions.

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arXiv2021

Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -Δu+λ_1u=μ_1 |u|^{p-2}u+βr_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -Δv+λ_2v=μ_2 |v|^{q-2}v+βr_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $β$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.

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