On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions

On a fractional Nirenberg problem, part II: existence of solutions

This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were given on a fractional Nirenberg problem. We prove two existence results stated there. We also obtain a fractional Aubin inequality.

Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five

We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.

三维空间中Brezis-Nirenberg问题的多气泡爆炸分析

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

本文研究三维空间中Brezis-Nirenberg问题的多气泡爆炸现象，推导了精确的爆炸速率并刻画了集中点位置。

布雷齐斯 - 尼伦伯格方程的精细爆破分析及其应用：\(N\geq4\)时的单泡情形

A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$

研究\(N\geq4\)时布雷齐斯 - 尼伦伯格方程，基于反约化论证发展精细爆破分析，对单泡情形下随参数\(\lambda\)变化的施特鲁韦分解进行分类，并证明\(4d\)方程在一般有界区域对特定\(\lambda\)有非平凡解，完善相关存在性理论。

Sign changing solutions of the Brezis-Nirenberg problem in the Hyperbolic space

In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.

Grushin型的加权Gagliardo--Nirenberg不等式：对称性、紧性与最优解

Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers

本文研究ℝⁿ×ℝᵐ上Grushin型算子相关的加权Gagliardo--Nirenberg插值不等式，确定参数条件与最优缩放指数，利用径向对称性恢复紧嵌入，证明β=0时存在最优解。

The Nirenberg problem on half spheres: A bubbling off analysis

In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical sol…

高维球面上的 Nirenberg 问题：带有残余质量现象的爆破

Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon

本文研究高维球面上Nirenberg问题次临界逼近中带残余质量的爆破现象，刻画爆破点与速率，计算拓扑贡献，并在弱极限非退化时构造爆破解。

On a Fractional Nirenberg problem involving the square root of the Laplacian on $\mathbb{S}^{3}$

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $σ=1/2,$ when the prescribing $σ$-curvature function satisfies the $(n-2σ)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM…

Convergence for the fractional $p$-Laplacian and its corresponding extended Nirenberg problem

The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of nonnegative sequence of functions with $p>2$. Further, we show the blow-up phenomena for solutions to the extended Nirenberg problem modeled by fractional $p$-Laplacian with the prescribed negative functions.

Nodal solutions of the Brezis-Nirenberg problem in dimension 6

We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$

三维临界Hartree型布雷齐斯-尼伦伯格问题的爆破渐近性

Blow-up asymptotics for a critical Hartree-type Brézis--Nirenberg problem in dimension three

研究三维临界Hartree问题，在自然非退化假设下，推导\(u_{\varepsilon}\)精确渐近展开，确定爆破率与浓度点，获浓度点附近及远离时的逐点爆破行为。

双调和Brézis-Nirenberg问题的能量渐进行为与爆炸现象

Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem

本文研究了n≥8维下双调和Brézis-Nirenberg问题在Navier边界条件下的商泛函，通过匹配上界和下界估计，建立了能量差S(0)-S(εV)的精确渐进行为，并描述了近极小序列的爆炸剖面及集中点的性质。

The number of positive solutions to the Brezis-Nirenberg problem

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~Ω},\\ u>0, &{\text{in}~Ω},\\ u=0, &{\text{on}~\partial Ω}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem fo…

Towers of bubbles for Yamabe-type equations and for the Brézis-Nirenberg problem in dimensions $n \ge 7$

Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $ξ_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at $ξ_0$. The method of proof combines a finite-dimensional reduction with the sharp…

三维空间中Choquard型Brezis-Nirenberg问题的爆破解

Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three

本文研究三维光滑有界区域上带临界指数的Choquard型Brezis-Nirenberg问题，通过约化论证刻画了临界参数$λ_0$，证明当$λ$略大于$λ_0$时存在在Robin函数临界点处爆破集中的解，并拓展至零Neumann边界条件情形。

低维分数阶 Brezis--Nirenberg 问题的临界函数与爆破渐近性

Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

本文研究了低维情形下分数阶Brezis--Nirenberg极小化问题，证明了临界函数下Robin函数的下确界为零，并精确刻画了特定维数下能量泛函的爆破渐近行为、集中速度与集中点位置。

涉及混合局部非局部算子的Brézis-Nirenberg型问题的全局紧性结果

Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

本文针对混合局部非局部算子驱动的Brézis-Nirenberg型问题，建立了Palais-Smale序列的全局紧性结果，并据此证明了λ=0时Coron型问题正解的存在性。

Non simple blow ups for the Nirenberg problem on half spheres

In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \e…

An integral type Brezis-Nirenberg problem on the Heisenberg group

This paper is devoted to study a class of integral type Brezis-Nirenbreg problem on the Heisenberg group. It is a class of new nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on the CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities, the nonexistence and existence results are obtained by Pohozaev type identity, variational method and blow-up analysis, respectively.

Compactness与退化临界椭圆型方程解的存在性

Compactness and existence results for degenerate critical elliptic equations

本文研究Caffarelli-Kohn-Nirenberg型退化临界椭圆型方程。利用Blow-up分析技巧，证明了一个连续函数加权空间中的先验估计。利用勒雷-绍德尔度的同伦不变性，证明了该问题解的存在性.

奇异拟线性椭圆型方程的Brezis-Nirenberg型问题的可解性

The solvability of Brezis-Nirenberg type problems of singular quasilinear elliptic equation

本文考虑一类带奇异权的Brezis-Nirenberg型问题非平凡解的存在性和非存在性。首先，我们得到了一个紧嵌入定理，它是经典的Rellich-Kondrachov紧嵌入定理的推广，并考虑了相应的本征值问题。其次，我们推导了一个Pohozaev型恒等式，得到了一个不存在的结果。第三，基于广义集中紧性原理，给出了泛函满足（PS）$_c$条件的一些抽象条件。最后，基于极值函数的显式形式，我们将得到问题的一些存在性结果。

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatial growing nonlinearity

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

We establish theorems on the existence and compactness of solutions to the $σ_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order der…

CR Nirenberg问题，第二部分：存在性与拓扑度

The CR Nirenberg Problem, Part II: Existence and Topological Degree

本文延续CR Nirenberg问题第一部分的工作，在标准球面上，利用对称性、Leray-Schauder度公式及Aubin型不等式，得到了该问题的存在性与拓扑度相关结果。

带有超线性扰动的混合局部-非局部算子的Brezis-Nirenberg问题：紧致性及其应用

Brezis-Nirenberg problems for mixed local-nonlocal operators with superlinear perturbations: compactness and applications

针对混合局部-非局部算子的Brezis-Nirenberg问题，在特定参数范围内证明了次临界/临界问题解序列的紧致性，并应用该结果得到无穷多变号解的存在性。

On the $σ_{k}$-Nirenberg problem

We consider the problem of prescribing the $σ_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for $n = 4$ and $k = 2$.

高维Brezis-Nirenberg问题中的细多气泡分析

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

本文研究了高维Brezis-Nirenberg问题中正解的吹胀行为，分析了多个集中点的孤立性及集中速度，并与N=3情况的论文共同完善了该问题的理论框架。

Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case

For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |u|^2 \, dx}{\left(\int_Ω|u|^q \, dx \right)^{2/q}}, \] where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $Ω\subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(εV)$ to leading order as $ε\to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient in…

Nirenberg问题的双凸起轴对称解

Two-bump axisymmetric solutions of the Nirenberg problem

针对Nirenberg问题，在临界超曲面Σ⁰附近一侧构造了n≥4维的双凸起轴对称解，揭示了从该侧发生双凸起爆破而另一侧解集保持紧致的一侧精细化现象。

拟线性椭圆型问题的局部compactness

On local compactness in quasilinear elliptic problems

Palais-Smale序列的紧性是非线性椭圆问题中的一个主要困难。在他们著名的工作\cite{BN}中，Brézis和Nirenberg在拉普拉斯齐次本征值问题的临界扰动的情况下引入了这些序列的临界水平的概念。本文给出了一类非线性椭圆型临界问题的临界水平的一个自然的一般公式。我们的公式的清晰度是建立在适当的Palais-Smale序列的建设是不是相对紧凑。

四维 Brezis-Nirenberg 问题的多泡解

Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

本文解决了四维有界区域内 Brezis-Nirenberg 问题多点爆破解的存在性这一长期开放问题，展示了存在大量多泡解的区域实例，并与 Konig 和 Laurin 的渐近分析工作形成对应。

二维空间中的一个加权Moser-Trudinger不等式及其与Caffarelli-Kohn-Nirenberg不等式的关系

A weighted Moser-Trudinger inequality and its relation to the Caffarelli-Kohn-Nirenberg inequalities in two space dimensions

首先证明了二维欧几里得空间中一个依赖于参数的Moser-Trudinger型加权不等式。如果参数大于-1，则不等式对径向函数成立。在没有对称性假设的情况下，当且仅当参数在区间（-1，0]内时，它成立。不等式给了我们一些洞察力的对称性破缺现象的极值功能的Hardy-Sobolev不等式，建立了由Caffarelli-Kohn-Nirenberg，在两个空间维度。事实上，对于合适的参数集（渐近尖锐），我们证明对称性或对称性破缺的爆破方法。这样，加权Moser-Trudinger不等式就成为Hardy-Sobolev不等式的一个极限情形。

Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations

Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.

具有超耗散的Navier-斯托克斯方程的一个廉价的Caffarelli-Kohn-Nirenberg不等式

A cheap Caffarelli-Kohn-Nirenberg inequality for Navier-Stokes equations with hyper-dissipation

证明了对于具有耗散项$(-Δ)^α$的斯托克斯方程，其中$1<α<{5/4}$，初始数据光滑，第一次爆破时奇异集的豪斯多夫维数至多为$5-4α$ .这统一了两个方向，人们可以从这两个方向来处理克莱奖问题。

A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case

We consider in this paper some class of perturbation for the semilinear wave equation with critical (in the conformal transform sense) power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using a two-step argument based on interpolation and a critical Gagliardo-Nirenberg inequality, we show that the blow-up rate of any sigular solution is given by the solution of the non perturbed associated ODE, namely $u" = u^p$.

三维CR Yamabe问题的紧性结果

A compactness result for the CR Yamabe problem in three dimensions

本文通过blow-up分析证明了三维紧致严格伪凸CR流形上CR Yamabe问题解集的紧性，并得出不CR等价于$S^3$的CR-可嵌入流形紧性成立的推论。

Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-Δ)^m u=\varepsilon u+u^{p} & \text { in }\, Ω, \\ u>0 & \text { in }\, Ω, \\ u=-Δu=\cdots=(-Δ)^{m-1} u=0 & \text { on }\, \partial Ω, \end{cases} \ee where $Ω$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,λ_{1})$, $λ_{1}$ is the first Navier eigenvalue for $(-Δ)^{m}$ in $Ω$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $λ…

Compactness of solutions to higher order elliptic equations

We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.

Existence and non existence results for the singular Nirenberg problem

In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function $K$ changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, by a min-max approach and a new compactness argument we give sufficient conditions on $K$, concerning…

Brezis-Nirenberg问题多峰解的渐近行为：次临界扰动情形

Asymptotic behavior of multi-peak solutions to the Brezis-Nirenberg problem. The sub-critical perturbation case

本文研究次临界扰动下Brezis-Nirenberg问题的多峰解，刻画了ε→0时解的渐近轮廓、爆破速率、集中特性，证明了解的唯一性与非退化性，指出指数q对解的性质有显著影响，完善了相关猜想框架下的多峰爆破理论。

Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2σ$ on $n$-dimensional standard sphere when $ n-2σ=2$, $σ=1+m/2,$ $m\in \mathbb{N}_{+}.$ The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case…

A compactness result for an equation with Holderian condition

We give blow-up analysis for a Brezis and Merle's problem with Dirichlet and Holderian condition. Also, we derive a compactness criterion.

About Brezis-Merle Problem with holderian condition: the case of one or two blow-up points

We give a compactness result for Brezis-Merle Problem with holderian condition. We look to the case of one or two blow-up points.

缺少compactness的奇异椭圆问题

Singular elliptic problems with lack of compactness

考虑下面的非线性奇异椭圆方程$$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$，其中$g$属于适当的加权Sobolev空间，$p$表示与$a$，$b$和$N$相关的Caffarelli-Kohn-Nirenberg临界指数.在对正势$K(x)$的一些自然假设下，我们建立了某个$\la\_0>0$的存在性，使得上述问题至少有两个不同的解，只要$\la\in(0,\la\_0)$ .证明依赖于Ekeland变分原理和山路定理，而不需要Palais-Smale条件，结合Brezis-Lieb引理的加权变体。

A compactness result for an elliptic equation in dimension 2

We give blow-up analysis for the solutions of an elliptic equation under some conditions. Also, we derive a compactness result for this equation.

子椭圆方程在海森堡群上解的紧致性

Compactness of Solutions to Sub-Elliptic Equations with Potential on the Heisenberg Group

本文研究了海森堡群上带非负势函数的临界子椭圆方程非负解的紧致性，证明在势函数满足一定非退化条件时解集紧致，并探讨了解序列发散时势函数及其子拉普拉斯算子在发散点处消失的性质。

序列爆破解到椭圆方程的量化

Quantization for sequences of blow-up solutions to an elliptic equation having nonlocal exponential nonlinearity

本文研究了具有非局部指数非线性的椭圆方程解序列的渐进行为，通过集中-紧致性替代方案和能量量化结果揭示了解的爆破现象。

The Nirenberg problem and its generalizations: A unified approach

Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining operator of order $2σ$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $σ=1$, it is the prescribing scalar curvature problem or the Nirenberg problem,…

Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a la…

Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem

We study the asymptotic behavior, as $λ\rightarrow 0$, of least energy radial sign-changing solutions $u_λ$, of the Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ B_1\\ u=0 & \hbox{on}\ \partial B_1, \end{cases} \end{equation*} where $λ>0$, $2^*=\frac{2n}{n-2}$ and $B_1$ is the unit ball of $\R^n$, $n\geq 7$. We prove that both the positive and negative part $u_λ^+$ and $u_λ^-$ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of $u_λ^+$ and $u_λ^-$ conve…

A compactness result for solutions to an equation with boundary singularity

We give a blow-up behavior for solutions to a problem with singularity and with Dirichlet condition. An application, we have a compactness of the solutions to this Problem with singularity and Lipschitz conditions.

$L^2$ concentration of blow-up solutions for the mass-critical NLS with inverse-square potential

In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.

New multiplicity results for critical $p$-Laplacian problems

We prove new multiplicity results for the Brezis-Nirenberg problem for the $p$-Laplacian. Our proofs are based on a new abstract critical point theorem involving the ${\mathbb Z}_2$-cohomological index that requires less compactness than the (PS) condition.