The Experts below are selected from a list of 213 Experts worldwide ranked by ideXlab platform
Shaowei Chen - One of the best experts on this subject based on the ideXlab platform.
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a variant of clark s theorem and its applications for nonsmooth functionals without the palais Smale Condition
Siam Journal on Mathematical Analysis, 2017Co-Authors: Shaowei Chen, Zhaoli Liu, Zhiqiang WangAbstract:By introducing a new notion of the genus with respect to the weak topology in Banach spaces, we prove a variant of Clark's theorem for nonsmooth functionals without the Palais--Smale Condition. In this new theorem, the Palais--Smale Condition is replaced by a weaker assumption, and a sequence of critical points converging weakly to zero with nonpositive energy is obtained. As applications, we obtain infinitely many solutions for a quasi-linear elliptic equation which is very degenerate and lacks strict convexity, and we also prove the existence of infinitely many homoclinic orbits for a second-order Hamiltonian system for which the functional is not in $C^1$ and does not satisfy the Palais--Smale Condition. These solutions cannot be obtained via existing abstract theory.
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existence of nontrivial solution for a 4 sublinear schrodinger poisson system
Applied Mathematics Letters, 2014Co-Authors: Shaowei Chen, Dawei ZhangAbstract:Abstract We consider a Schrodinger–Poisson system in R 3 with potential indefinite in sign and a general 4-sublinear nonlinearity. Its variational functional does not satisfy the Palais–Smale Condition in general. We use a finite-dimensional approximation method to obtain a sequence of approximate solutions. Then we prove that these approximate solutions converge weakly to a nontrivial solution of this system.
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existence of a nontrivial solution for a strongly indefinite periodic schrodinger poisson system
arXiv: Analysis of PDEs, 2014Co-Authors: Shaowei Chen, Liqian XiaoAbstract:We consider the Schr\"odinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -\Delta u+V(x) u+|u|^{p-2}u=\lambda \phi u, & \mbox{in}\mathbb{R}^{3},\\ -\Delta\phi= u^{2}, & \mbox{in}\mathbb{R}^{3}. \end{array} \right.\nonumber \end{eqnarray} where $\lambda>0$ is a parameter, $3< p<6$, $V\in C(\mathbb{R}^{3}) $ is $1$-periodic in $x_j$ for $j = 1,2,3$ and 0 is in a spectral gap of the operator $-\Delta+V$. This system is strongly indefinite, i.e., the operator $-\Delta+V$ has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities $|u|^{p-2}u$ and $\lambda \phi u$. Moreover, the functional corresponding to this system does not satisfy the Palai-Smale Condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small $\lambda>0,$ this system has a nontrivial solution.
Andrea Ratto - One of the best experts on this subject based on the ideXlab platform.
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higher order energy functionals
Advances in Mathematics, 2020Co-Authors: Volker Branding, Stefano Montaldo, Cezar Oniciuc, Andrea RattoAbstract:Abstract The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called E S − r -energy functionals E r E S ( φ ) = ( 1 / 2 ) ∫ M | ( d ⁎ + d ) r ( φ ) | 2 d V , where φ : M → N is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an E S − r -harmonic map, i.e., a critical point of E r E S ( φ ) . That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of E r E S ( φ ) when N = S m ( r ≥ 4 , m ≥ 3 ) , and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for E r E S ( φ ) for r = 4 . We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4 , we shall also show that, even if 2 r > dim M , the functionals E r E S ( φ ) may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.
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higher order energy functionals
arXiv: Differential Geometry, 2019Co-Authors: Volker Branding, Stefano Montaldo, Cezar Oniciuc, Andrea RattoAbstract:The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r (\varphi)|^2\,dV$, where $ \varphi:M \to N$ is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an $ES-r$-harmonic map, i.e, a critical point of $ E_r^{ES}(\varphi)$. That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of $E_r^{ES}(\varphi)$ when $N={\mathbb S}^m$ $(r \geq4,\, m\geq3)$, and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for $E_r^{ES}(\varphi)$ for $r=4$. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if $2 r > \dim M$, the functionals $ E_r^{ES}(\varphi)$ may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.
Michel Willem - One of the best experts on this subject based on the ideXlab platform.
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origin and evolution of the palais Smale Condition in critical point theory
Journal of Fixed Point Theory and Applications, 2010Co-Authors: Jean Mawhin, Michel WillemAbstract:In 1963–64, Palais and Smale have introduced a compactness Condition, namely Condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This Condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type Conditions. The paper describes their evolution.
Abdellaziz Harrabi - One of the best experts on this subject based on the ideXlab platform.
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on the palais Smale Condition
Journal of Functional Analysis, 2014Co-Authors: Abdellaziz HarrabiAbstract:Abstract We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz Condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth Condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes Condition [8] to obtain Cerami Condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth Condition, while the strong superlinear Condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.
Pietro Majer - One of the best experts on this subject based on the ideXlab platform.
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on the palais Smale Condition for action integrals
Journal of Functional Analysis, 1992Co-Authors: Pietro MajerAbstract:Abstract We study the Palais-Smale Condition for the action integral f(u) = ∝ 0 1 { 1 2 ¦ u ¦ 2 − V(u)} dt, u ϵ H 1 ( R Z , R N ) in terms of the potential V . In particular, we prove that for any V in a ∥ · ∥ ∞ -dense subset of C 1 b ( R N ) this Condition never holds at large levels. The same result is proven in the time-dependent case.
