Singular Liouville Equations Research Articles

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{−Δun=Wn(x)unpn,un>0,inΩ,un=0,on∂Ω,∫ΩpnWn(x)unpndx≤C, where Ω is a smooth bounded domain in R2, Wn(x)≥0 are bounded functions with zeros in Ω, and pn→∞ as n→∞. A typical example is Wn(x)=|x|2α with 0∈Ω, i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for α=0 has been well studied in the literature. While for α>0, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case α>0 and prove a quantization property (suppose 0 is a concentration point)pn|x|2αun(x)pn−1+t→8πet2∑i=1kδai+8π(1+α)et2ctδ0,t=0,1,2, for some k≥0, ai∈Ω∖{0} and some c≥1. Moreover, for α∉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 Hénon equation.

Laplacian vanishing theorem for a quantized singular Liouville equation

In this article we establish a vanishing theorem for a singular Liouville equation with a quantized singular source. If a blowup sequence tends to infinity near the quantized singular source and the blowup solutions violate the spherical Harnack inequality around the singular source (non-simple blow-ups), the Laplacian of the coefficient function must tend to zero. This seems to be the first second order estimates for a Liouville equation with a quantized source and non-simple blowups. This result as well as the key ideas of the proof will be useful for various applications.

On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

Non-simple blow-up for singular Liouville equations in unit ball

Non-simple blow-up for singular Liouville equations in unit ball

Simple blow-up solutions of singular Liouville equations

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.

A singular Liouville equation on two-dimensional domains

A singular Liouville equation on two-dimensional domains

Classification and a priori estimates for the singular prescribing Q-curvature equation on 4-manifold

On (M,g) a compact riemannian 4-manifold we consider the prescribed Q-curvature equation defined on M with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on R4 and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are isolated simple blow up points.

Existence of nontopological solutions for the generalized Maxwell-Chern-Simons-Higgs model

In this paper, we study nontopological solutions of the generalized self-dual Maxwell-Chern-Simons-Higgs model in the whole plane. By introducing a scaling parameter ε>0 and regarding the main equations as perturbation of a singular Liouville equation for small ε, we verify the existence of one parametric family of nontopological solutions.

Vanishing estimates for Liouville equation with quantized singularities

In this article we continue with the research initiated in our previous work on singular Liouville equations with quantized singularity. The main goal of this article is to prove that as long as the bubbling solutions violate the spherical Harnack inequality near a singular source, the first derivatives of coefficient functions must tend to zero.

Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

Abstract We study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$\begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}$$under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

Uniform bounds for solutions to elliptic problems on simply connected planar domains

We consider the following elliptic problems on simply connected planar domains: { − Δ u = λ | x | 2 α K ( x ) e u a m p ; in Ω u = 0 a m p ; on ∂ Ω ; { − Δ u = | x | 2 α K ( x ) u p a m p ; in Ω u > 0 a m p ; in Ω u = 0 a m p ; on ∂ Ω \begin{equation*} \left \{\begin {array}{ll}-\Delta u=\lambda |x|^{2\alpha }K(x)e^u&\text {in }\Omega \\u=0&\text {on }\partial \Omega \end{array}\right . ;\quad \quad \left \{\begin {array}{ll}-\Delta u=|x|^{2\alpha }K(x)u^p&\text {in }\Omega \\u>0&\text {in }\Omega \\u=0&\text {on }\partial \Omega \end{array}\right . \end{equation*} with α > − 1 , λ > 0 , p > 1 , 0 > K ( x ) ∈ C 1 ( Ω ¯ ) \alpha >-1,\lambda >0,p>1,0>K(x)\in C^1\left (\overline \Omega \right ) . We show that any solution to each problem must satisfy a uniform bound on the mass, which is given, respectively, by λ ∫ Ω | x | 2 α K ( x ) e u d x \lambda \int _\Omega |x|^{2\alpha }K(x)e^u\mathrm dx and p ∫ Ω | x | 2 α K ( x ) u p + 1 d x p\int _\Omega |x|^{2\alpha }K(x)u^{p+1}\mathrm dx . The same results apply to some systems and more general nonlinearities. The proofs are based on the Riemann mapping theorem and a Pohožaev-type identity.

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $$\Sigma $$ admitting conical singularities of orders $$\alpha _i$$ ’s at points $$p_i$$ ’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $$\chi (\Sigma )+\sum _i \alpha _i$$ approaches a positive even integer, where $$\chi (\Sigma )$$ is the Euler characteristic of the surface $$\Sigma $$ .

On Singular Liouville Equations and Systems

Abstract We consider some singular Liouville equations and systems motivated by uniformization problems in a non-smooth setting, as well as from models in mathematical physics. We will study the existence of solutions from a variational point of view, using suitable improvements of the Moser–Trudinger inequality. These reduce the problem to a topological one by studying the concentration property of conformal volume, which will be constrained by the functional inequalities of geometric flavour. We will mainly describe some common strategies from the papers [11, 12, 20] in simple situations to give an idea to the non-expert reader about the general methods we use.

REMARKS ON NONTOPOLOGICAL SOLUTIONS IN THE SELF-DUAL CHERN-SIMONS GAUGED O(3) SIGMA MODELS

In this paper, we prove the existence of nontopological solutions to the self-dual equations arising from the Chern-Simons gauged O(3) sigma models. The property of solutions depends on a parameter <TEX>${\tau}{\in}[-1,1</TEX><TEX>]</TEX><TEX>$</TEX> appearing in the nonlinear term. The case <TEX>${\tau}=1$</TEX> lies on the borderline for the existence of solutions in the previous results [4, 5, 7]. We prove the existence of solutions in this case when there are only vortex points. Moreover, if <TEX>$-1{\leq}{\tau}$</TEX><1, we establish solutions which are perturbed from the solutions of singular Liouville equations.

Lane Emden problems with large exponents and singular Liouville equations

We consider the Lane–Emden Dirichlet problem{−Δu=|u|p−1u,in B,u=0,on ∂B, where p>1 and B denotes the unit ball in R2. We study the asymptotic behavior of the least energy nodal radial solution up, as p→+∞. Assuming w.l.o.g. that up(0)<0, we prove that a suitable rescaling of the negative part up− converges to the unique regular solution of the Liouville equation in R2, while a suitable rescaling of the positive part up+ converges to a (singular) solution of a singular Liouville equation in R2. We also get exact asymptotic values for the L∞-norms of up− and up+, as well as an asymptotic estimate of the energy. Finally, we have that the nodal line Np:={x∈B:|x|=rp} shrinks to a point and we compute the rate of convergence of rp.

On solution sequences of the singular Liouville equations with an integral constraint

In this paper, we analyze the blow-up behavior of sequences {uk} satisfying the following conditions $$- \Delta u_k = \left| x \right|^{2\alpha _k } V_k e^{u_k } in \Omega ,$$ (0.1) where Ω ⊂⊂ ℝ2, Vk → V in C1, |∇Vk| ≤ A, 0 < a ≤ Vk ≤ b, 0 ≤ αk → α ∈ (0, ∞), and $$\int_\Omega {\left| x \right|^{2\alpha _k } e^{u_k } dx \leqslant \Lambda _1 .}$$ (0.2) Furthermore, we assume that there exists some q ∈ (1, 2) such that $$r^{q - 2} \int_{{\rm B}r(p)} {\left| {\nabla u_k } \right|^q dx \leqslant \Lambda _2 }$$ (0.3) for any Br(p) ⊆ Ω. As a result, we give a new proof of the concentration-compactness theorem for the mean field equation.

On the solvability of singular Liouville equations on compact surfaces of arbitrary genus

On the solvability of singular Liouville equations on compact surfaces of arbitrary genus

An Improved Geometric Inequality via Vanishing Moments, with Applications to Singular Liouville Equations

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.

A sup × inf-type inequality for conformal metrics on Riemann surfaces with conical singularities

A sup × inf-type inequality is proved for the regular part of conformal factors for Riemannian metrics on surfaces with conical singularities of positive order α>0. The proof is based on the explicit representation formula for solutions of the singular Liouville equation and generalizes to the singular case an argument by I. Shafrir.

Weighted barycentric sets and singular Liouville equations on compact surfaces

Given a closed surface, we prove a general existence result for some elliptic PDE with exponential nonlinearities and negative Dirac deltas, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional might be unbounded from below, we define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of Chen and Li (1991) [11] and then employ a min–max scheme based on conical construction, jointly with the blow-up analysis in Bartolucci and Montefusco (2007) [4] (after Bartolucci and Tarantello, 2002; Brezis and Merle, 1991 [5,7]). This study is motivated by abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature with conical singularities (generalizing a problem raised in Kazdan and Warner, 1974 [24]).