Mountain pass solutions for a mean field equation from two-dimensional turbulence
Tonia Ricciardi
利用Struwe的“单调技巧”和最近的爆破分析Ohtsuka和铃木，我们证明了山口解决方案的二维湍流中产生的平均场方程的存在性。
Using Struwe's "monotonicity trick" and the recent blow-up analysis of Ohtsuka and Suzuki, we prove the existence of mountain pass solutions to a mean field equation arising in two-dimensional turbulence.
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We obtain the existence of mountain pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity trick proved by the second author. The main results are new also for the p-Laplacian operator.
This paper addresses the Mountain Pass Theorem for locally Lipschitz functions on finite-dimensional vector spaces in terms of tangencies. Namely, let $f \colon \mathbb R^n \to \mathbb R$ be a locally Lipschitz function with a mountain pass geometry. Let $$c := \inf_{γ\in \mathcal A}\max_{t\in[0,1]}f(γ(t)),$$ where $\mathcal{A}$ is the set of all continuous paths joining $x^*$ to $y^*.$ We show that either $c$ is a critical value of $f$ or $c$ is a tangency value at infinity of $f.$ This reduces to the Mountain Pass Theorem of Ambrosetti and Rabinowitz in the case where the function $f$ is def…
In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding variational techniques. Furthermore, we establish the sharpness of our central assumptions. As far as we know, our approach is new, even for the Orlicz case.
We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ${\mathbb R}^N$ ($N\geq 2$): $$ (*)_m \left\{ \eqalign{ -&Δu = g(u) -μu \quad \hbox{in}\ {\mathbb R}^N, \cr &\| u\|_{L^2({\mathbb R}^N)} = m, \cr &u \in H^1({\mathbb R}^N), \cr} \right. $$ where $g(ξ)\in C({\mathbb R},{\mathbb R})$, $m>0$ is a given constant and $μ\in {\mathbb R}$ is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of the problem $(*)_m$. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general …
An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems
本文建立了一个无限维的临界点定理，扩展了Rabinowitz的经典广义山峰定理，并应用于证明具有不定权重的二维非线性椭圆系统至少存在一个解。
We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation (-Δ)^{s} u = g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian and $g: \mathbb{R} \rightarrow \mathbb{R}$ is an odd $\mathcal{C}^{1, α}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence o…
In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian $(-Δ)^{α/2}$ for $α\in\left( 1,2\right) $ and some superlinear and subcritical nonlinearity $G_{z}$ provided with a nonhomogeneous Dirichlet exterior boundary condition. Some sufficient conditions under which the set of weak solutions to the boundary value problem is nonempty and depends continuously in the Painleve-Kuratowski sense on distributed parameters and exterior boundary data are stated. The proofs of the existence results rely on the Mountain Pass Theorem. The application…
Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments
我们引入山路型参数的背景下，轨道不稳定性的克莱因-戈登方程。我们的目的是通过两个例子来说明这些论点如何有助于简化证明并得出轨道稳定性/不稳定性的新结果。对于一个功率型非线性，我们证明了相关的固定方程的基态是极小的功能“行动”上的各种约束。对于一般的非线性方程，我们将Shatah在1985年证明的非线性Klein-Gordon方程定常解的经典不稳定性结果推广到维数N=2，其中维数N>2.
In this paper we study the existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ where the nonlinearity has a critical growth for $N \geq 2$. By combining a penalization scheme found in del Pino and Felmer [18] with a penalization method due to Bensoussan and Lions [9], we improve a recent result by Alves, Barros and Torres [1].
We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. Heteroclinic mountain pass solution in the second laminations is new to the generalized Frenkel-Kontorova model. Our proof follows that of Bolotin and Rabinowitz for an Allen-Cahn equation, which is different with heat flow method for finding critical point of Frenkel-Kontorova model in the literature. The proofs depend on suitable choices of functionals and working spaces. We also study the multiplicity of these mountain pass solutions.
Mountain pass for the Ginzburg-Landau energy in a strip: solitons and solitonic vortices
研究在无限条带中应用相印刻的吉本斯-兰道能量临界点，发现临界宽度下孤子解为山峰解，临界宽度以上形成孤子涡，并探讨其在不同宽度下的极小值解行为。
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。
Concentration-compactness principle for mountain pass problems
本文证明了当非线性项是次临界、超线性且满足罚条件$F_\infty(s)<F(x,s)$时，$\R^N$上半线性椭圆问题的临界序列收敛于山路水平.证明表明了极大极小问题的集中紧性框架，类似于P。L.Lions for constrained minima.
In this paper, we obtain a new quantitative deformation Lemma so that we can obtain more critical points, especially for supinf critical value $c_1$, $x=φ^{-1}(c_1)$ is a new critical point. For $infmax$ critical value $c_2$, we can obtain two new critical points $x = 0$ (valley point) and $x = e$(peak point) ,comparing with Willem's variant of the mountain pass theorem of Ambrosetti-Rabinowitz,in which $φ(e)\leqφ(0)<c_2$, but in our new mountain pass theorem, $ φ(e)=c_2$.
Concentration-compactness at the mountain pass level in semilinear elliptic problems
本文给出了无紧性的半线性椭圆型方程的集中紧性框架，该框架最初由P. - Lions在齐次非线性情况下求约束极小值的方法，在这里推广到Ambrosetti-Rabinowitz的标准山路条件下的一般非线性情况。在这些设置中，在山口水平的解决方案的存在性进行了验证下一个单一的惩罚条件类似的狮子的情况下。考虑了全空间问题和临界非线性问题。特别注意的是非齐次临界非线性振荡的“关键干”。
Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler-Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part $K-W$ and a forcing term. We consider two situations: $G$ satisfying $Δ_2\cap\nabla_2$ in infinity and globally. We give conditions on the growth of the potential near zero for both situations.
We formulate some global invertibility and implicit function theorems. We extend the result of Idczak, Skowron and Walczak on the invertibility of the operators to the case of the operators with critical points. The proof relies on the Mountain Pass Theorem combined with the Palais-Smale condition guaranteeing the claim by the invertibility of the first or the third derivative. I. e. how to solve $x^3=y$?
Solutions of the stationary semilinear Cahn--Hilliard equation -Δ^2 u - u -Δ(|u|^{p-1}u)=0 in R^N, with p>1, which are exponentially decaying at infinity, are studied. Using the Mounting Pass Lemma allows us the determination of two different solutions. On the other hand, the application of Lusternik--Schnirel'man (L--S) Category Theory shows the existence of, at least, a countable family of solutions. However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable fami…
The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schrödinger equation $(-Δ)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the…
Cylinder Buckling: The Mountain Pass as an Organizing Center
我们重新讨论了轴向压缩圆柱薄长壳的屈曲问题。通过检查能量景观的完美的圆柱体，我们推导出的估计的敏感性壳的缺陷。获得这一点的关键是该系统存在一个山口点。我们证明存在于有界域上的所有最所有的负载，然后数值计算示例山口解决方案，这样的解决方案。在数值上，能量最低的山口解具有单凹窝的形式。我们解释了这些结果，并验证了下界对文献中的一些实验结果。
We develop a min-max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two.
On the Mountain-pass algorithm for the quasi-linear Schrodinger equation
针对拟线性薛定谔方程对应的非光滑泛函，提出一种改进的山口算法并证明其收敛性。
A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities
针对具有超二次非线性的非局部方程，利用经典山形路径定理证明非平凡解的存在性，并开发基于梯度下降的数值算法。
In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $γ$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $γ$. In order to do so, we develop min-max methods similar to those of De Lellis and Ramic, references in the paper, adapted to the discrete setting of Almgren and Pitts.
For a degenerate autonomous Kirchhoff equation which is set on $\mathbb{R}^N$ and involves the Berestycki-Lions type nonlinearity, we cope with the cases $N=2,3$ and $N\geq5$ by using mountain pass and symmetric mountain pass approaches and by using Clark theorem respectively.
Mountain-Pass Solutions in 2D Turbulence with Generalized Sobolev Operators
本文通过引入广义加权Sobolev算子，结合山路定理与爆破分析，在更广泛的边界条件下证明了二维湍流平均场方程非平凡解的存在性，克服了经典方法因缺乏紧性而失效的问题，并增强了模型对复杂几何区域湍流现象的适用性。
Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator
针对 Grushin 次椭圆算子驱动的半正问题，在参数 a 的不同范围内，证明了 D_γ(R^N) 中正山路解的存在性。
Mountain Pass Critical Points of the Volume Constrained Area Functional
本文在固定体积约束下构造周长泛函的山口临界点，得到非零常平均曲率的光滑几乎嵌入超曲面，并利用新连通性结果证明任意两个光滑Cacciopoli集合可由F-连续映射连接。
Level set methods for finding critical points of mountain pass type
提出一种数值方法，用于寻找非光滑情况下的临界点，在光滑有限维情形下具有局部超线性收敛性，并应用于Wilkinson问题中计算矩阵到最近重特征值矩阵的距离。
Convergence of a mountain pass type algorithm for strongly indefinite problems and systems
针对强不定泛函，研究山脊型算法生成序列的子列收敛到临界点，并通过步长选择建立全序列收敛性，应用于不定薛定谔方程和超线性薛定谔系统。
On the Linking Principle
我们给出了一个连接定理，它加强和统一了许多极小极大定理，包括Ambrosetti-Rabinowitz的“山路定理”，Rabinowitz的“多维山路定理”，Rabinowitz的"鞍点定理"和Silva的这些结果的变体.我们把我们的注意力，特别是在“极限情况”，已知是真实的山路原理，其中一些信息的位置上的临界点。给出了该定理的两种形式：第一种是通过变形引理建立的，第二种是利用Ekeland的变分原理得到的。
We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean--Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value $β=β_c$. On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transitio…
For a sinh-Poisson type problem with asymmetric exponents of interest in hydrodynamic turbulence, we establish the optimal lower bounds for the blow-up masses. We apply this result to construct solutions of mountain pass type on two-dimensional tori.
Existence of positive solutions to anisotropic elliptic equations in the entire Euclidean space
本文针对全欧氏空间中非线性项具指数或临界增长的各向异性椭圆方程，运用山路引理及Ekeland变分原理，证明其存在非平凡正弱解，且扰动后有两个不同正弱解。
The One-Phase Bifurcation For The p-Laplacian
本文研究与 p-Laplacian 相关的奇异摄动相变自由边界问题，利用山路引理揭示边界数据低于阈值时第三解出现的分岔，并证明演化收敛于稳定解而山路解不稳定。
Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case
本文研究带弱吸引位势的Sobolev临界非线性薛定谔方程在指定L²范数下的正解，证明局部极小解与山路解的存在性，并应用于遍历平均场博弈系统。
Bifurcation phenomena in one-phase quasilinear equations with nonstandard growth
本文研究由g-拉普拉斯算子控制的单相自由边界问题，证明边界数据低于阈值时出现分岔，存在第三个解，并分析演化问题的渐近稳定性。
Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms
研究一类含双非局部项的$p(x)$-基尔霍夫问题，利用变分方法证明两个非平凡解及无穷多解的存在性，推广了已有结果。
Three bound states with prescribed angular momentum to the cubic-quintic NLS equations in $\mathbb{R}^{3}$
本文研究三维三次-五次NLS方程具有给定角动量和质量的束缚态，证明了双约束问题存在局部极小值、山隘型和全局极小值三个解，并构造新山隘路径揭示其几何联系与能级比较。
\ In this paper, the following biharmonic elliptic problem \begin{eqnarray*} \begin{cases} Δ^2u-λ\frac{|u|^{q-2}u}{|x|^s}=|u|^{2^{**}-2}u+ f(x,u), &x\inΩ,\\ u=\dfrac{\partial u}{\partial n}=0, &x\in\partialΩ\end{cases} \end{eqnarray*} is considered. The main feature of the equation is that it involves a Hardy term and a nonlinearity with critical Sobolev exponent. By combining a careful analysis of the fibering maps of the energy functional associated with the problem with the Mountain Pass Lemma, it is shown, for some positive parameter $λ$ depending on $s$ and $q$, that the problem admits at…
For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $γ_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq γ_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $γ_p(u,v)<\infty$ for $p\in [1,k)$, and using their construction, one can obtain an estimate of the form $γ_p(u,v)\leq \frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $γ_p(u,…
The present work is concerned with the following version of Choquard Logarithmic equations $ -Δ_p u -Δ_N u + a|u|^{p-2}u + b|u|^{N-2}u + λ(\ln|\cdot|\ast G(u))g(u) = f(u) \textrm{ in } \mathbb{R}^N $ , where $ a, b, λ>0 $, $ \max\{\frac{N}{2}, 2 \} < p< N $, $f, g: \mathbb{R} \rightarrow \mathbb{R} $ are continuous functions that behave like $ \exp(α|u|^{\frac{N}{N-1}}) $ at infinity, for $ α>0 $, and that has polynomial growth, respectively, and $ G(s)=\int\limits_{0}^{s}g(τ)dτ$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution.…
We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.
By using variational methods, we study the existence of mountain pass solution to the following doubly critical Schrödinger system: $$ \begin{cases} -Δu-μ_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u &=h(x)α|u|^{α-2}|v|^βu\quad \rm{in}\; \R^N, -Δv-μ_2\frac{v}{|x|^2}-|v|^{2^{*}-2}v &= h(x)β|u|^α|v|^{β-2}v\quad \rm{in}\; \R^N, \end{cases} $$ where $α\geq 2, β\geq 2, α+β\leq 2^*$;\; $ μ_1, μ_2\in [0, \frac{(N-2)^2}{4})$. The weight function $h(x)$ is allowed to be sign-changing so that the nonlinearities include a large class of indefinite weights. We show that the $PS$ condition is satisfied at higher energy…
In this paper, we study Mountain Pass Characterization of the second eigenvalue of the operator $-\De_p u -\De_{J,p}u$ and study shape optimization problems related to these eigenvalues.
We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimension.
Normalized ground states for the mass supercritical Schrödinger-Bopp-Podolsky system: existence, uniqueness, limit behavior, strong instability
本文研究质量超临界 Schrödinger-Bopp-Podolsky 系统归一化基态的存在性、唯一性、极限行为与强不稳定性，通过 $L^2$-球面上的山路论证和等价极小化问题，揭示了山路能量水平作为全局存在性阈值的作用。
In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and als…
On the existence of positive solution for a Neumann problem with double critical exponents in half-space
本文利用无 (PS) 条件的山路定理、精细水平估计、改进 Pohozaev 恒等式与 Hardy 不等式，研究半空间中带双临界指数及快速增长权的 Neumann 问题正解存在与不存在性，并在 N=4 时给出参数最佳下界。
We study the existence of positive solutions with prescribed $L^2$-norm for the Schrödinger equation \[ -Δu-V(x)u+λu=|u|^{p-2}u\qquadλ\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in\left(2+\frac 4 N,2^*\right)$, $2^*:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^*:=+\infty$ if $N=1,2$. We treat two cases. Firstly, under an explicit smallness assumption on $V$ and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is small…
The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic o…
Existence and qualitative properties of ground state solutions for the Schrödinger-Bopp-Podolsky system
本文研究 Schrödinger-Bopp-Podolsky 系统，利用山路引理和基态能量估计证明基态解存在，并分析其正性、对称性、衰减性及径向情形下关于参数 $a$ 的渐近行为。
On the existence of nontrivial solutions for a nonlinear equation relative to a measure-valued Lagrangian on homogeneous spaces
证明了与测度值拉格朗日量有关的非线性方程非平凡解的存在性。结果是基于一个紧凑的嵌入定理的拉格朗日域和应用程序的山路定理加入了一个Palais-Smale条件。
Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities
本文研究了非均匀非线性情况下，$L^2$超临界非线性薛定谔方程在外部领域中归一化山峰解的存在性，通过新型极小-极大论证揭示了非自主非线性在非紧致领域中的新现象。
Existence and asymptotical behavior of normalized solutions to focusing biharmonic HLS upper critical Hartree equation with a local perturbation
本文研究带局部扰动的聚焦双调和HLS上临界Hartree方程正规化解，通过山路水平估计和验证(PS)条件证明解的存在性，将已有次临界扰动结果推广到更广的指数范围，并分析能量渐近行为。
Existence results for a superlinear singular equation of Caffarelli-Kohn-Nirenberg type
本文利用山路引理和链接论证，证明了当参数$λ\in (0, λ_2)$、$λ_2$是Caffarelli-Kohn-Nirenberg型拟线性椭圆型方程的第二正特征值时，Caffarelli-Kohn-Nirenberg型超线性方程狄利克雷问题非平凡弱解的存在性。
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_λ) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscr{P}_λ) and we show that the problem tr…
In this paper we prove the existence of two solutions having a prescribed $L^2$-norm for a quasi-linear Schrödinger equation. One of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global. To overcome the lack of differentiability of the associated functional, we rely on a perturbation method developed in [27].
Level set methods for finding saddle points of general Morse index
本文基于多维山脊定理，提出并分析了几种水平集方法以寻找一般Morse指标的鞍点，证明了非光滑情形下的收敛性和光滑有限维情形下的局部超线性收敛。
The Obukhov--Corrsin spectrum of passive scalar turbulence through anomalous regularization
该研究通过异常正则化证明了被动标量湍流的Obukhov-Corrsin谱在特定条件下的成立，利用了傅里叶空间的能量等式和加权晶格Poincaré不等式。
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