Kyril Tintarev
In the paper we show that critical sequences associated with the mountain pass level for semilinear elliptic problems on $\R^N$ converge when the non-linearity is subcritical, superlinear and satisfies the penalty condition $F_\infty(s)<F(x,s)$. The proof suggests a concentration compactness framework for minimax problems, similar to that of P.-L.Lions for constrained minima.
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The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous crit…
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…
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$.
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 $x^3=y$?
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.
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 …
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.
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 give a linking theorem that strengthens and unifies some many minimax theorems including Ambrosetti-Rabinowitz ``mountain pass theorem'', Rabinowitz ``multidimensional mountain pass theorem'', Rabinowitz ``saddle point theorem'' and Silva's variants of these results. We focus our attention especially on ``the limiting case'', known to be true for the mountain pass principle, where some information on the location of the critical points is given. Two forms of this theorem are given: the first one is established via a deformation lemma and we use Ekeland's variational principle to get the sec…
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…
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 revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for all most all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bo…
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…
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。
We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1…
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 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.
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。
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.
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…
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。
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…
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。
We consider the following nonlinear singular elliptic equation $$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$ where $g$ belongs to an appropriate weighted Sobolev space, and $p$ denotes the Caffarelli-Kohn-Nirenberg critical exponent associated to $a$, $b$, and $N$. Under some natural assumptions on the positive potential $K(x)$ we establish the existence of some $\la\_0>0$ such that the above problem has at least two distinct solutions provided that $\la\in(0,\la\_0)$. The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem withou…
In this paper we consider the nonlinear Choquard equation $$ -Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<μ<N$, $N\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \cite{BC1…
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.
On a class of critical Schrödinger-Poisson systems involving the (p,q)-Laplacian
本文研究了一类在R^3中涉及(p,q)-拉普拉斯算子的临界薛定谔-泊松系统，通过直接变分方法、山路定理和集中紧性原理，建立了正解的存在性。
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。
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
本文通过对称山路引理和非局部集中紧性方法，证明了含临界增长的分数阶Choquard型方程存在无穷多解。
We study the existence of nontrivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in $\mathbb{R}^N$. By combining variational methods and the concentration-compactness principle we obtain a nontrivial solution for asymptotically periodic problem and a ground state solution for the periodic problem. In the proofs we apply the Mountain Pass Theorem and its local version.
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。
Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. The corresponding study of higher critical points was recently initiated in Jerison and Perera [30, 31]. In particular, the existence and nondegeneracy of a mountain pass point in a superlinear and subcritical free boundary problem related to plasma confinement was proved in [30]. In this paper we study ground states of a critical free boundary problem related to the Brezis-Nirenberg problem [5]. We extend the results of [30] to this problem by combining the method introdu…
Critical quasilinear Schroedinger equations with electromagnetic fields
研究N维欧氏空间中受外部电磁势影响的p-Laplace算子，利用变分方法克服双重紧致性缺失与复杂拟线性框架的困难，证明了1<p<N情形下加权临界方程存在至少一个山路型解。
Weighted biharmonic equations involving continous potentiel under exponential nonlinear growth
本文研究四维单位球中含对数权重与连续正势的临界指数增长双调和方程，利用集中紧性与渐近条件克服紧性缺失，并通过山路定理证明非平凡正弱解的存在性。
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,…
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。
In this paper we study the existence problem for the $p(x)-$Laplacian operator with a nonlinear critical source. We find a local condition on the exponents ensuring the existence of a nontrivial solution that shows that the Pohozaev obstruction does not holds in general in the variable exponent setting. The proof relies on the Concentration--Compactness Principle for variable exponents and the Mountain Pass Theorem.
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。
Existence solutions for a weighted equation of p-biharmonic type in the unit ball of $\mathbb{R}^{N}$ with critical exponential growth
本文在单位球中研究带正连续位势的加权N/2双调和方程，在临界指数增长条件下利用山路定理、集中紧性结果和渐近条件证明了非平凡弱解的存在性。
In this paper we establish the existence of mountain pass and negative energy weak solutions for a Kirchhoff-Schrödinger type problem in $\mathbb R^4$ involving a critical nonlinearity and a suitable small perturbation. The arisen competition between the terms due to the nonlocal coefficient and critical nonlinearity turns out to be rather interesting. The main tools used in the present work are variational methods and the Lions' Concentration Compactness Principle.
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…
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.…
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。
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.
We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-Δ)^s u = ε\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $ε>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and~$h\in L^1(\R^n)\cap L^\infty(\R^n)$. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficult…
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。
The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian
本文证明了与$p$-Laplacian相关的正则化两相自由边界问题在边界数据较小时存在山隘点解，从而产生三个弱解，并通过径向对称情形下的常微分方程揭示了自由边界Bernoulli条件导致的分岔现象。
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…
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