Existence Of Multiple Critical Points Research Articles

Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to L^{infty }, which implies C^{1,alpha } smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

Multiple Critical Points for Symmetric Functionals without upper Growth Condition on the Principal Part

This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For this purpose, nonsmooth variational methods are applied.

Analysis of Dual Nature of Heat Flux Predicted by Monin‐Obukhov Similarity Theory: An Impact of Empirical Forms of Stability Correction Functions

AbstractThe behavior of sensible heat flux H with stability parameter ζ in the stable atmospheric surface layer is analyzed within the framework of Monin‐Obukhov similarity theory (MOST). Using MOST equations, H is expressed as a function of ζ and wind shear ∂U/∂z. A systematic mathematical analysis is carried out to analyze the behavior of H with ζ utilizing the linear as well as various nonlinear empirical functional forms of the similarity functions. In case of linear form, for a given value of H, two different values of ζ along with the “downward heat flux maximum” with a unique critical point are found to occur. However, for the other selected empirical similarity functions, more than two values of ζ having the same magnitude of H may occur suggesting the existence of multiple critical points. Further, the observational variation of H with ζ is analyzed using data from three different sites. Data suggest the existence of two different stability regimes for the same magnitude of heat flux. However, the data do not support the existence of multiple values of ζ for the same magnitude of H, as predicted by MOST with the other selected empirical forms of similarity functions. This suggests that the phenomenon of existence of downward heat flux maximum, as predicted by MOST equations, depends on the functional form of the similarity functions, and it appears to break down for some selected empirical forms of similarity functions under very stable conditions.

A generalized mountain-pass theorem: the existence of multiple critical points

We analyse the existence of multiple critical points for an even functional $$\begin{aligned} J:H\rightarrow {\mathbb {R}}\end{aligned}$$ in the following context: the Hilbert space H can be split into an orthogonal sum \(H = Y \oplus Z\) in such a way that $$\begin{aligned} \inf \{ J(u) : u \in Z \text { and } \Vert u\Vert = \rho \} \ge \alpha > J(0) \end{aligned}$$ and that here exits a point \(b \in H\) with \(\Vert b\Vert > \rho \) and with \(J(b) \le J(0)\). We develop a new variational characterization of multiple critical levels without an assumption on the dimension of Y. Our characterization is simple and natural: we can for example avoid the notion of pseudo-index and the definition of the activated levels does not substantially differs from the one used for the lowest critical level, giving us in this way a unified view of critical levels. We apply our results to a semi-linear Schrodinger equation of the form $$\begin{aligned} \left\{ \begin{array}{ll} &{} -\Delta u + V (x) u - q (x) |u|^ \sigma u = \lambda u, \qquad x\in {\mathbb {R}}^ N\\ &{} u \in {H^1({\mathbb {R}}^N)}\setminus \{ 0 \} \end{array} \right. \end{aligned}$$ where \(\lambda \) is inside a spectral gap bounded on both sides by parts of the essential spectrum.

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by Geba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant \({\omega(\nabla \Phi)}\), expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of \({\Phi}\) exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here \({f : V \rightarrow \mathbb{R}}\) is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant \({\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}\) in the case nk ≠ 0 (for maximal twisted orbit types (Hk)) guarantees the existence of at least |nk| different G-orbits of periodic solutions with symmetries at least (Hk). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of \({\omega(\,\nabla \Phi)}\) evaluated.

Kinetic effects of trapped energetic particles on stability of external kink modes with a resistive wall

Kinetic effects of trapped energetic particles (EPs) on stability of the external kink mode with a resistive wall are investigated in detail, on the basis of the theory model developed in a previous paper [G. Z. Hao, A. K. Wang, Y. Q. Liu, and X. M. Qiu, Phys. Rev. Lett. 107, 015001 (2011)]. The results indicate that, when the perpendicular beta β* of the trapped EPs exceeds a critical value βc*, a bursting fishbone-like mode (FLM) instability, with external kink eigenstructure ,can be triggered, which rapidly grows with increasing β*(>βc*), and eventually becomes a dominant instability. Detailed physical analyses are carried out, comparing radial profiles of the EPs kinetic energy for both the FLM and the conventional resistive wall mode (RWM). On the other hand, a mode conversion between the FLM and RWM can directly occur. This work also presents a systematic investigation of effects of various physical parameters on the FLM instability. An interesting new finding is the existence of multiple critical points in β*, for the FLM triggering. The number of critical points depends sensitively on the trapped EPs pitch angle. In addition, it is found that there can be a critical value of the pitch angle, beyond which the critical βc*, for triggering the FLM, jumps from a large value to a small one. The FLM instability, with the m/n=3/1 mode structure, can also be triggered by the trapped EPs.

Nabla theorems and multiple solutions for some noncooperative elliptic systems

We study some variational principles which imply the existence of multiple critical points for a functional $f$, using the properties of both $f$ and $\nabla f$ on some suitable sets. We derive some multiplicity theorems for a certain class of strongly indefinite functionals and we apply these results for finding multiple solutions of an elliptic system of reaction-diffusion type.

Existence of multiple critical points for an asymptoticallyquadratic functional with applications

Morse theory for isolated critical points at infinity is used for the existence of multiple critical points for an asymptotically quadratic functional. Applications are also given for the existence of multiple nontrivial periodic solutions of asymptotically Hamiltonian systems.

Multiple critical points for variational problems on partially ordered Hilbert spaces

We are studying the existence of multiple critical points for functionals whose potential operators preserve an order structure. By using Morse type arguments we prove that the existence of local minima of a functional Φ which are ordered in a special way « forces » Φ to have many additional critical points. We also show how these abstract results apply to a concrete situation.

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.

A Borsuk-Ulam theorem for orthogonal Tk, and Zpr actions and applications

A version if the Borsuk-Ulam theorem is proved for the torus T k and p-torus Z p r . We deal with the case of a nonempty fixed point set of groups on the sphere of representation. As an application, a geometrical index theory is introduced for these groups. The index is used to obtain the existence of multiple critical points for functions invariant under T k , or Z p r action.

MULTIPLE CRITICAL POINTS OF PERTURBED SYMMETRIC FUNCTIONALS IN [formula omitted

This paper shudies the existence of multiple critical points of perturbed symmetric functionals in W01,p (Ω) (p⩾2). For p>2, this kind of problems have never been studied befor and the main result of this paper generlizes the main results of [10].

On Critical Point Theory for Indefinite Functionals in The Presence of Symmetries

We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.

On critical point theory for indefinite functionals in the presence of symmetries

We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.