Existence of two local minima for functionals on reflexive Banach spaces

In this paper, we establish an existence theorem of two local minima for functionals on reflexive Banach space. As a consequence, we obtain an existence theorem of three critical points which improves the conclusion of a result in a recent paper of G.Bonanno concerning some remarks on a three-critical-points theorem established by B. Ricceri.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

On a class of nonlinear variational–hemivariational inequalities

A three critical points theorem for nondifferentiable functions is pointed out and an existence result of multiple solutions for a Neumann elliptic variational–hemivariational inequality involving the p-laplacian is established. As an application, a Neumann problem for elliptic equations with discontinuous nonlinearities is studied.     

On a variational problem with non-differentiable constraints

We study the regularity of minimizers and critical points of the Dirichlet energy under an integral constraint given by a non-differentiable function. We obtain existence of a Lipschitz continuous minimizer for a relaxed problem. In two dimensions, some regularity can also be proved for critical points.     

似梯度型系统的连结轨线

本文讨论Ｒｎ中似梯度型系统奇点的连结轨线．给出了存在连结轨线的一个判定准则;证明了当系统仅有的两个奇点分别为吸引子和排斥子时,连结轨线集是无界的开集．     

Bahri-Coron type theorem for the scalar curvature problem on high dimensional spheres

We consider the existence and multiplicity results for the prescribed scalar curvature problem on the standard spheres of high dimension n ?? 7. Given a C ² positive function K, using the theory of critical points at infinity, we prove an existence result as Bahri-Coron theorem. Our case is a generalization of Li (J Differ Equ 120:319?C410, 1995). Indeed, here the function K is flat near some critical points as in Li (J Differ Equ 120:319?C410, 1995) and it can have some nondegenerate critical points with ?? K ?? 0. Furthermore, using some topological arguments, we prove another kind of result.     

Three weak solutions for elliptic Dirichlet problems

An existence result of three non-zero solutions for non-autonomous elliptic Dirichlet problems, under suitable assumptions on the nonlinear term, is presented. The approach is based on a recent three critical points theorem for differentiable functionals.     

A Critical Points Theorem and Nonlinear Differential Problems

The existence of two intervals of positive real parameters λ for which the functional Φ + λΨ has three critical points, whose norms are uniformly bounded in respect to λ belonging to one of the two intervals, is established. As an example of an application to nonlinear differential problems, a two point boundary value problem is considered and multiplicity results are obtained.     

Applications of two critical point results for non-differentiable indefinite functionals

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained.     

Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter

We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the \(p\) -Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow.     

Nonsymmetric bifurcations of solutions of the 2D Navier–Stokes system

We consider the 2D Navier–Stokes system written for the stream function with periodic boundary conditions and construct a set of initial data such that initial critical points bifurcate from 1 to 2 and then to 3 critical points in finite time. The bifurcation takes place in a small neighborhood of the origin. Our construction does not require any symmetry assumptions or the existence of special fixed points. For another set of initial data we show that 3 critical points merge into 1 critical point in finite time. We also construct a set of initial data so that bifurcation can be generated by the Navier–Stokes flow and do not require the existence of an initial critical point.     

On a nonlinear elliptic problem with critical potential in R~2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in R2. By establishing a weighted inequality with the best constant, determine the critical potential in R2, and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexis tence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

Three Solutions for a Partial Differential Inclusion Via Nonsmooth Critical Point Theory

The existence of three solutions for a partial differential inclusion involving a perturbed nonlinearity and two real parameters is proved. Moreover, an estimate of the norms of solutions, independent of both the parameters and the perturbation, is achieved. The main theoretical tool is an extension to nonsmooth functionals of a very recent three critical points theorem of Ricceri.     

Circle action with prescribed number of fixed points

Given any two positive integers k and n, this paper is concerned with the existence of a circle action on a closed, smooth orientable n-dimensional manifold with precisely k isolated fixed points. We first show that this existence problem can be reduced to that of an n-dimensional manifold with exactly three fixed points. Then by using a rigidity result, we determine possible weights on these three fixed points when n = 4.     

Existence and multiplicity of normal geodesics in Lorentzian manifolds

In this paper we shall prove some results pertaining to the existence and multiplicity of normal geodesics joining two given submanifolds of an orthogonal splitting Lorentzian manifold. To this aim, we look for critical points of an unbounded suitable functional by using a Saddle-Point Theorem and the relative category theory.     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Multiplicity of solutions for a NLS equations with magnetic fields in \mathbb {R}^{N}

We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Schödinger equation. The main result is established by using minimax methods and Lusternik–Schnirelman theory of critical points.     

具有临界两点异宿环的二次系统

本文给出二次系统存在临界两点异宿环的充要条件,并证明二次系统的临界两点异宿环必由双曲线的一支和直线或由椭圆和直线构成,其内部的奇点必是中心。推广所研究的这种系统,本文对[1]中提出的一个公开问题也给出了解答。