scholarly journals Multiple Critical Points for Symmetric Functionals without Upper Growth Condition on the Principal Part

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 898
Author(s):  
Marco Degiovanni ◽  
Marco Marzocchi
Keyword(s):  
Variational Methods ◽  
Growth Condition ◽  
Calculus Of Variations ◽  
Critical Points ◽  
Principal Part ◽  
Dimensional Case

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.

Critical points approaches to a nonlocal elliptic problem driven by 𝑝(𝑥)-biharmonic operator

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shapour Heidarkhani ◽  
Shahin Moradi ◽  
Mustafa Avci
Keyword(s):  
Variational Methods ◽  
Elasticity Theory ◽  
Critical Points ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Biharmonic Operator ◽  
Technical Approach ◽  
Nonlinear Elasticity Theory

Abstract Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by p ⁢ ( x ) p(x) -biharmonic operator. Our technical approach is based on variational methods. Some applications illustrate the obtained results. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

scholarly journals On a class of non-local discrete boundary value problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Anass Ourraoui ◽  
Abdesslem Ayoujil
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Critical Points ◽  
Boundary Value ◽  
Discrete Problem ◽  
Three Solutions ◽  
Content Type ◽  
Three Critical Points Theorem ◽  
Discrete Boundary Value Problem ◽  
Non Local

PurposeIn this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.Design/methodology/approachThe approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.FindingsThe authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.Originality/valueAn other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.

scholarly journals Solutions for Impulsive Fractional Differential Equations via Variational Methods

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Hui Wang ◽  
Zheqing Li
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Critical Points ◽  
Existence Results ◽  
Fractional Order Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

Linking sandwich pairs

2009 ◽  
Vol 147 (3) ◽  
pp. 679-700 ◽  
Author(s):  
MARTIN SCHECHTER
Keyword(s):  
Calculus Of Variations ◽  
Critical Points ◽  
The Other ◽  
Distinct Advantage ◽  
Cerami Sequence ◽  
Image Position ◽  
Clear Method

AbstractSince the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice, the solutions are critical points. In searching for critical points, there is a distinct advantage if the functional G is semibounded. In this case one can find a Palais–Smale (PS) sequence or even a Cerami sequence These sequences produce critical points if they have convergent subsequences. However, there is no clear method of finding critical points of functionals which are not semibounded. Linking subsets do provide such a method. They can produce a PS sequence provided they separate the functional. In the present paper we show that there are pairs of subsets that can produce Cerami-like sequences even though they do not separate the functional. All that is required is that the functional be bounded from above on one of the sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. We apply the method to several situations.

scholarly journals MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATION VIA VARIATIONAL METHODS

2010 ◽  
Vol 82 (3) ◽  
pp. 446-458 ◽  
Author(s):  
JUNTAO SUN ◽  
HAIBO CHEN ◽  
TIEJUN ZHOU
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Fourth Order ◽  
Critical Points ◽  
Impulsive Differential Equation ◽  
Classical Solutions ◽  
Multiplicity Of Solutions ◽  
Three Critical Points Theorem ◽  
New Criteria

AbstractIn this paper, we deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

scholarly journals Multipleweak solutions for a kind of time-dependent equation involving singularity

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4567-4574
Author(s):  
F. Abdolrazaghi ◽  
A. Razani ◽  
R. Mirzaei
Keyword(s):  
Variational Methods ◽  
Critical Points ◽  
Weak Solutions ◽  
Source Function ◽  
Time Dependent

The existence of at least three weak solutions for a kind of nonlinear time-dependent equation is studied. In fact, we consider the case that the source function has singularity at origin. To this aim, the variational methods and the well-known critical points theorem are main tools.

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