M. Musso, D. Salazar
We consider the problem $Δu + λu +u^5 = 0$, $u>0$, in a smooth bounded domain $Ω$ in ${\mathbb R}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at $k$ different points of the domain as $λ$ tends to a special positive value $λ_0$, which we characterize in terms of the Green function of $-Δ- λ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A note on borderline Brezis-Nirenberg type problems
该研究聚焦散度型椭圆算子的Brezis-Nirenberg问题，分析边界上椭圆性点对该问题的影响，为相关边界问题研究提供理论参考。
In this paper, we study the decomposition of Nehari manifold for the Brézis-Nirenberg problem with nonhomogeneous Dirichlet boundary conditions. By using this result, the Lusternik-Schnirelman category and the minimax principle, we establish a multiple result (four solutions) for the Brézis{Nirenberg problem with nonhomogeneous Dirichlet boundary conditions.
In the present paper, we study the effect of an elliptic operator in divergence form L defined in a bounded open set on the existence and nonexistence of positive solutions of the associated nonlocal Brezis-Nirenberg problem involving powers of L.
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.
In this paper, we study the Brezis-Nirenberg problem on bounded smooth domains of R3. Using the algebraic topological argument of Bahri-Coron[2] as implemented in [6] combined with the Brendle[4]- Schoen[8]'s bubble construction, we solve the problem for non-contractible domains.
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 $λ_0>0.$
We consider the classical Brezis-Nirenberg problem in the unit ball of $\mathbb{R}^N$, $N\geq 3$ and analyze the asymptotic behavior of nodal radial solutions in the low dimensions $N=3,4,5,6$ as the parameter converges to some limit value which naturally arises from the study of the associated ordinary differential equation.
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.
We prove some existence and nonexistence results for a class of critical $(p,q)$-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.
On Brezis-Nirenberg problems: open questions and new results in dimension six
本文研究Brezis-Nirenberg问题，回顾其历史，提供六维中的新结果，并列出开放问题。
In this paper, we prove that the Brezis-Nirenberg problem -Δu = |u|^{p-1}u+εu in Ω; u=0 on \partial Ωwhere Ωis a symmetric bounded smooth domain in R^N, N\geq 7 and p = (N+2)/(N-2), has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ε> 0 sufficiently small.
On the $N$-dimensional Schrödinger--Poisson--Slater equation à la Brezis--Nirenberg
本文利用Pohozaev恒等式和Nehari流形，研究含临界指数的N维Schrödinger–Poisson–Slater方程，证明了预定能量解的存在性与多重性。
By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.
We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $λ$ is a real parameter, $N\geq3$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.
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.
In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.
In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.
This work is devoted to the Dirichlet problem for the equation (-Δu = λu + |x|^α|u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $λ$ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided $α$ is small enough. This solution has a variational characterization as a ground state.
In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…
We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space i…
The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-Δ\right)^s u-λu &= α|u|^{p-2}u + β|u|^{2^*-2}u \quad\mbox{in}\quad Ω,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{cases} \end{align*}$$ where $(-Δ)^s$ is the fractional Laplace operator, $s \in (0,1)$, with $N \geq 3s$, $2<p<2^*$, $β>0, λ, α\in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$.
A Brezis-Nirenberg type result for mixed local and nonlocal operators
本文研究拉普拉斯与分数阶拉普拉斯叠加的混合阶算子的临界问题，确定其索伯列夫不等式最优常数永不可达，并建立次临界扰动问题的存在性与不存在性理论。
We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional $p$-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when $p \ne 2$. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained by Brasco, Mosconi and Squassina. The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking t…
Nodal cluster solutions for the Brezis-Nirenberg problem in dimensions $N\geq 7$
该研究针对维数不小于7的光滑有界区域，证明经典Brezis-Nirenberg问题当参数ε趋于0时，存在围绕区域边界某点聚集的结点解，丰富了该问题的解的结构研究。
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.
The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.
Positive solutions for a Kirchhoff problem of Brezis-Nirenberg type in dimension four
本文研究四维有界域上Brezis-Nirenberg型Kirchhoff问题的正解存在性，通过新颖的逼近方法处理高阶项与临界非线性的相互作用，给出参数显式界限并改进已有结果。
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。
Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus
本文研究环形区域中超临界Brezis-Nirenberg问题，建立三维情形正径向解的唯一性结果，证明内半径足够小且外半径在某范围时至少有三个正径向解，指数在临界Sobolev指数与Joseph-Lundgren指数间时对任意正整数$k$至少有$k$个正径向解。
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…
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。
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 3$, $1<α$, $2^{\ast}=\frac{2N}{N-2}$ and $\{u_α\}\subset H_{0}^{1,2α}(Ω)$ be a critical point of the functional \begin{equation*} I_{α,λ}(u)=\frac{1}{2α}\int\limits_Ω [(1+|\nabla u|^2)^α-1 ]dx-\fracλ{2}\int\limits_Ωu^2dx-\frac{1}{2^{\ast}}\int\limits_Ω|u|^{2^{\ast}}dx. \end{equation*} In this paper, we obtain the limit behaviour of $u_α$ ( $α\rightarrow 1$), energy identity, Pohozaev identity, some integral estimates, etc. And using these results, we prove infinitely many solutions for the following Brezis-Nirenberg problem for $…
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…
In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -Δu=|u|^{2^*-2}u+λu&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball of $\R^N$. We compute the asymptotics of the solution $u$ as well as $||u||_\infty$, its first zero and other relevant quantities as $ł$ goes to a critical value $\barł$. Also the sign of $ł-\barł$ is established in all cases. This completes an analogous analysis for $N\ge7$ given in [EGPV].
In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for some nonlinear partial differential systems.
Multiplicity of Solutions to the Brezis-Nirenberg Problem on Hyperbolic Spaces
本文研究超球面上的Brezis-Nirenberg问题解的多重性，通过Dirichlet边界条件和参数λ的谱位置，利用等变Ljusternik-Schnirelmann范畴获得解的下界。
In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \math…
This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-Δ)^2 u=λu+ (-Δ)|u|^{p-1}u,$$ in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-Δu=λ(-Δ)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $λ>0$, up to…
Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group
本文研究海森堡群上含临界 Sobolev 指数项的分数次亚拉普拉斯方程，受 Brezis-Nirenberg 问题启发，证明了在有界连续边界区域中弱解的存在性。
We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ Ω\\ u=0 & \hbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $λ>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $λ$ is close to zero there are no sign-changing solutions of the form $$u_λ=PU_{δ_1,ξ}-PU_{δ_2,ξ}+w_λ, $$ where $PU_{δ_i}$ is the projection on $H_0^1(Ω)$ of the regular positive solution of the critical problem in …
Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian
本文在各向异性 $p$-拉普拉斯算子框架下推广了 Brezis-Nirenberg 经典结果，通过精确估计各向异性 Aubin-Talenti 函数，证明了带 $p$-线性与 $p$-超线性扰动的临界拟线性椭圆问题存在弱正解。
The Brezis-Nirenberg problem in 4D
本文研究四维空间中带势函数的 Brezis-Nirenberg 问题，在回顾相关历史结果的基础上，聚焦于与本文研究问题最接近的已有成果，为后续分析奠定基础。
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…
Sharp quantitative stability estimates for the Brezis-Nirenberg problem
本文研究有界区域临界 Sobolev 嵌入对应的 Brezis-Nirenberg 问题，首次建立其 Sobolev 不等式的定量稳定性估计，并揭示解、线性项、气泡与边界效应导致的特殊稳定性指数。
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…
The quasi-linear Brezis-Nirenberg problem in low dimensions
本文研究低维临界 Sobolev 增长拟线性椭圆方程的正解存在性，指出其判定依赖相应 Green 函数正则部分的符号所体现的全局性质。
Construction of bubbling solutions of the Brezis-Nirenberg problem in general bounded domains (I): the dimensions 4 and 5
本文利用 Ljapunov-Schmidt 约化在4、5维一般有界区域构造精确渐近的鼓泡解，证明 Dirichlet Laplacian 的每个特征值均为集中值，并揭示两维多峰解数量差异。
The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian
本研究探讨了涉及对数拉普拉斯算子的 Brezis-Nirenberg 与 logistic 型椭圆方程，通过推导新的 Sobolev 空间嵌入并利用变分分析证明了最小能量弱解的存在性，同时基于新不等式证明了解的唯一性，并分析了分数阶问题解的渐近收敛性。
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…
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。
For a bounded open set $Ω\subset\mathbb R^3$ we consider the minimization problem $$ S(a+εV) = \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω(|\nabla u|^2+ (a+εV) |u|^2)\,dx}{(\int_Ωu^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+εV)-S$ as $ε\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentrati…
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。
Using a Rellich-Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis-Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of Hn, for 2<n<4.
The Brezis-Nirenberg-type problem for critical cooperative and non-cooperative elliptic systems with general subcritical perturbations
本文研究具有一般次临界扰动的临界合作与非合作椭圆系统的Brezis-Nirenberg型问题，在不同空间维数下证明解的存在性，将经典理论推广至含一般低阶扰动的临界系统。
In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $λ$ is the (positive) parameter in the linear part in $u$, and we consider $λ$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $λ$ such solutions change sign exactly…
Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions
该研究针对四维Brezis-Nirenberg问题构造了变号多泡解，通过Lyapunov-Schmidt约化分析其性质，提出抽象存在性准则并应用于多种对称构型，还证明了双峰解的节点域相关性质。
We consider the positive solutions of the nonlinear eigenvalue problem $-Δ_{\mathbb{H}^n} u = λu + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(Ω),$ where $Ω$ is a geodesic ball of radius $θ_1$ on $\mathbb{H}^n.$ For radial solutions, this equation can be written as an ODE having $n$ as a parameter. In this setting, the problem can be extended to consider real values of $n.$ We show that if $2<n<4$ this problem has a unique positive solution if and only if $λ\in \left(n(n-2)/4 +L^*\,,\, λ_1\right).$ Here $L^*$ is the first positive value of $L = -\ell(\ell+1)$ for which a suitably defined …
Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem
本文研究全空间上含Hardy权函数的分数阶p-Laplace Brézis-Nirenberg问题，通过允许权函数w变号，证明了该问题正解的存在性。
Mountain pass solution to the Brézis-Nirenberg problem with logarithmic perturbation
本文肯定了Hajaiej等人关于带对数扰动的Brézis-Nirenberg问题存在正能量山路解的猜想，通过精细估计证明了ℝ⁴中该问题的解存在性，并推广到三维和五维情形。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。