Статті в журналах: "Variational critical problems"

1

Ambrosetti, Antonio. "Critical points and nonlinear variational problems." Mémoires de la Société mathématique de France 1 (1992): 1–139. http://dx.doi.org/10.24033/msmf.362.

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2

Prigozhin, Leonid. "Variational inequalities in critical-state problems." Physica D: Nonlinear Phenomena 197, no. 3-4 (October 2004): 197–210. http://dx.doi.org/10.1016/j.physd.2004.07.001.

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3

Ambrosetti, A., J. Garcia Azorero, and I. Peral. "Elliptic Variational Problems in RN with Critical Growth." Journal of Differential Equations 168, no. 1 (November 2000): 10–32. http://dx.doi.org/10.1006/jdeq.2000.3875.

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4

Ghoussoub, Nassif, and Frédéric Robert. "Hardy-singular boundary mass and Sobolev-critical variational problems." Analysis & PDE 10, no. 5 (July 1, 2017): 1017–79. http://dx.doi.org/10.2140/apde.2017.10.1017.

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5

Chow, Shui-Nee, and Reiner Lauterbach. "A bifurcation theorem for critical points of variational problems." Nonlinear Analysis: Theory, Methods & Applications 12, no. 1 (January 1988): 51–61. http://dx.doi.org/10.1016/0362-546x(88)90012-0.

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6

Prigozhin, L. "On the Bean critical-state model in superconductivity." European Journal of Applied Mathematics 7, no. 3 (June 1996): 237–47. http://dx.doi.org/10.1017/s0956792500002333.

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We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

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7

Leonardi, Salvatore, and Nikolaos S. Papageorgiou. "On a class of critical Robin problems." Forum Mathematicum 32, no. 1 (January 1, 2020): 95–109. http://dx.doi.org/10.1515/forum-2019-0160.

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AbstractWe consider a nonlinear parametric Robin problem. In the reaction, there are two terms, one critical and the other locally defined. Using cut-off techniques, together with variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least three nontrivial smooth solutions all with sign information, which converge to zero in {C^{1}(\bar{\Omega})} as the parameter {\lambda\to 0^{+}}.

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8

Alves, C. O., Ana Maria Bertone, and J. V. Goncalves. "A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents." Journal of Mathematical Analysis and Applications 265, no. 1 (January 2002): 103–27. http://dx.doi.org/10.1006/jmaa.2001.7698.

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9

Béhi, Droh Arsène, and Assohoun Adjé. "A Variational Method for Multivalued Boundary Value Problems." Abstract and Applied Analysis 2020 (January 21, 2020): 1–8. http://dx.doi.org/10.1155/2020/8463263.

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In this paper, we investigate the existence of solution for differential systems involving a ϕ−Laplacian operator which incorporates as a special case the well-known p−Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.

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10

Piccione, Paolo, and Daniel V. Tausk. "Lagrangian and Hamiltonian formalism for constrained variational problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 6 (December 2002): 1417–37. http://dx.doi.org/10.1017/s0308210500002183.

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We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

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11

Li, Lin, and Stepan Tersian. "Fractional problems with critical nonlinearities by a sublinear perturbation." Fractional Calculus and Applied Analysis 23, no. 2 (April 28, 2020): 484–503. http://dx.doi.org/10.1515/fca-2020-0023.

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AbstractIn this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.

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12

Wysocki, K. "Multiple critical points for variational problems on partially ordered Hilbert spaces." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 7, no. 4 (July 1990): 287–304. http://dx.doi.org/10.1016/s0294-1449(16)30293-1.

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13

Gazzola, Filippo. "Positive solutions of critical quasilinear elliptic problems in general domains." Abstract and Applied Analysis 3, no. 1-2 (1998): 65–84. http://dx.doi.org/10.1155/s108533759800044x.

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We consider a certain class of quasilinear elliptic equations with a term in the critical growth range. We prove the existence of positive solutions in bounded and unbounded domains. The proofs involve several generalizations of standard variational arguments.

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14

Kang, Dongsheng. "Quasilinear elliptic problems with critical exponents and Hardy terms in ℝN". Proceedings of the Edinburgh Mathematical Society 53, № 1 (12 січня 2010): 175–93. http://dx.doi.org/10.1017/s0013091508000187.

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AbstractWe deal with a singular quasilinear elliptic problem, which involves critical Hardy-Sobolev exponents and multiple Hardy terms. Using variational methods and analytic techniques, the existence of ground state solutions to the problem is obtained.

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15

Stupishin, Leonid U. "Variational Criteria for Critical Levels of Internal Energy of a Deformable Solids." Applied Mechanics and Materials 578-579 (July 2014): 1584–87. http://dx.doi.org/10.4028/www.scientific.net/amm.578-579.1584.

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Variational formulation of the problem of the analysis and synthesis of deformable structures are proposed. It allows studying nonlinear problems of structural mechanics from a single standpoint, from a consideration of the General variational principle.

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16

Plotnikov, P. I., and J. F. Toland. "Variational problems in the theory of hydroelastic waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2129 (August 20, 2018): 20170343. http://dx.doi.org/10.1098/rsta.2017.0343.

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This paper outlines a mathematical approach to steady periodic waves which propagate with constant velocity and without change of form on the surface of a three-dimensional expanse of fluid which is at rest at infinite depth and moving irrotationally under gravity, bounded above by a frictionless elastic sheet. The elastic sheet is supposed to have gravitational potential energy, bending energy proportional to the square integral of its mean curvature (its Willmore functional), and stretching energy determined by the position of its particles relative to a reference configuration. The equations and boundary conditions governing the wave shape are derived by formulating the problem, in the language of geometry of surfaces, as one for critical points of a natural Lagrangian, and a proof of the existence of solutions is sketched. This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

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17

Liu, Shibo. "Multiple solutions for elliptic resonant problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 138, no. 6 (November 12, 2008): 1281–89. http://dx.doi.org/10.1017/s0308210507000443.

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Two non-trivial solutions for semilinear elliptic resonant problems are obtained via the Lyapunov—Schmidt reduction and the three-critical-points theorem. The difficulty that the variational functional does not satisfy the Palais—Smale condition is overcome by taking advantage of the reduction and a careful analysis of the reduced functional.

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18

Kyritsi, Sophia Th, and Nikolaos S. Papageorgiou. "Multiple Solutions for Nonlinear Periodic Problems." Canadian Mathematical Bulletin 56, no. 2 (June 1, 2013): 366–77. http://dx.doi.org/10.4153/cmb-2011-154-5.

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Abstract We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term f (t; x) that exhibits a (p – 1)-superlinear growth in x 2 R near 1 and near zero. A special case of the differential operator is the scalar p-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

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19

Ribeiro, Bruno. "Critical elliptic problems in ℝ2 involving resonance in high-order eigenvalues". Communications in Contemporary Mathematics 17, № 01 (16 грудня 2014): 1450008. http://dx.doi.org/10.1142/s0219199714500084.

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In this paper we deal with the following class of problems [Formula: see text] where Ω ⊂ ℝ2 is bounded with smooth boundary, g has a unilateral critical behavior of Trudinger–Moser type and λk denotes the kth eigenvalue of [Formula: see text], k ≥ 2. We prove existence of nontrivial solution for this problem using variational methods.

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20

Wei, Yongfang, and Zhanbing Bai. "Superlinear damped vibration problems on time scales with nonlocal boundary conditions." Nonlinear Analysis: Modelling and Control 27 (July 19, 2022): 1–21. http://dx.doi.org/10.15388/namc.2022.27.28343.

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This paper studies a class of superlinear damped vibration equations with nonlocal boundary conditions on time scales by using the calculus of variations. We consider the Cerami condition, while the nonlinear term does not satisfy Ambrosetti–Rabinowitz condition such that the critical point theory could be applied. Then we establish the variational structure in an appropriate Sobolev’s space, obtain the existence of infinitely many large energy solutions. Finally, two examples are given to prove our results.

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21

Lian, Chun-Bo, Bei-Lei Zhang, and Bin Ge. "Multiple Solutions for Double Phase Problems with Hardy Type Potential." Mathematics 9, no. 4 (February 13, 2021): 376. http://dx.doi.org/10.3390/math9040376.

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In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.

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22

Khodabakhshi, Mehdi, Abdolmohammad Aminpour, and Mohamad Tavani. "Infinitely many weak solutions for some elliptic problems in RN." Publications de l'Institut Math?matique (Belgrade) 100, no. 114 (2016): 271–78. http://dx.doi.org/10.2298/pim1614271k.

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23

Katzourakis, Nikos, and Tristan Pryer. "Second-order L∞ variational problems and the ∞-polylaplacian." Advances in Calculus of Variations 13, no. 2 (April 1, 2020): 115–40. http://dx.doi.org/10.1515/acv-2016-0052.

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AbstractIn this paper we initiate the study of second-order variational problems in {L^{\infty}}, seeking to minimise the {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})}, for the functional\mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{}the associated equation is the fully nonlinear third-order PDE\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{}Special cases arise when {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the {\infty}-polylaplacian and the {\infty}-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

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24

Yang, Dianwu. "A Variational Principle for Three-Point Boundary Value Problems with Impulse." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/840408.

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We construct a variational functional of a class of three-point boundary value problems with impulse. Using the critical points theory, we study the existence of solutions to second-order three-point boundary value problems with impulse.

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25

Chen, Yu. "G-α-preinvex functions and non-smooth vector optimization problems". Yugoslav Journal of Operations Research, № 00 (2021): 8. http://dx.doi.org/10.2298/yjor200527008c.

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In this paper, we proposed the non-smooth G-?-preinvexity by generalizing ?-invexity and G-preinvexity, and discussed some solution properties about non-smooth vector optimization problems and vector variational-like inequality problems under the condition of non-smooth G-?-preinvexity. Moreover, we also proved that the vector critical points, the weakly efficient points and the solutions of the non-smooth weak vector variational-like inequality problem are equivalent under non-smooth pseudo-G-?-preinvexity assumptions.

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26

Alves, Claudianor O., and Geilson F. Germano. "Ground state solution for a class of indefinite variational problems with critical growth." Journal of Differential Equations 265, no. 1 (July 2018): 444–77. http://dx.doi.org/10.1016/j.jde.2018.02.039.

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27

Shibata, Tetsutaro. "Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems." Abstract and Applied Analysis 2003, no. 11 (2003): 671–84. http://dx.doi.org/10.1155/s1085337503212045.

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We study the nonlinear two-parameter problem−u″(x)+λu(x)q=μu(x)p,u(x)>0,x∈(0,1),u(0)=u(1)=0. Here,1<q<pare constants andλ,μ>0are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurveμ(λ)asλ→∞. We emphasize that the critical case concerning the decaying rate of the second term isp=(3q−1)/2and this kind of criticality is new for two-parameter problems.

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28

Mishra, S. K., and Vivek Laha. "On V-r-invexity and vector variational-like inequalities." Filomat 26, no. 5 (2012): 1065–73. http://dx.doi.org/10.2298/fil1205065m.

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In this paper, we consider the multiobjective optimization problems involving the differentiable V-r-invex vector valued functions. Under the assumption of V-r-invexity, we use the Stampacchia type vector variational-like inequalities as tool to solve the vector optimization problems. We establish equivalence among the vector critical points, the weak efficient solutions and the solutions of the Stampacchia type weak vector variational-like inequality problems using Gordan?s separation theorem under the V-r-invexity assumptions. These conditions are more general than those appearing in the literature.

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29

Shen, Yansheng. "Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities." Advanced Nonlinear Studies 21, no. 1 (September 13, 2019): 77–93. http://dx.doi.org/10.1515/ans-2019-2056.

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Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

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30

Galewski, Marek, and Donal O'Regan. "ON WELL POSED IMPULSIVE BOUNDARY VALUE PROBLEMS FOR P(T)-LAPLACIAN'S." Mathematical Modelling and Analysis 18, no. 2 (April 1, 2013): 161–75. http://dx.doi.org/10.3846/13926292.2013.779600.

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In this paper we investigate via variational methods and critical point theory the existence of solutions, uniqueness and continuous dependence on parameters to impulsive problems with a p(t)-Laplacian and Dirichlet boundary value conditions.

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31

Badiale, Marino, and Alessio Pomponio. "Bifurcation results for semilinear elliptic problems in RN." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 1 (February 2004): 11–32. http://dx.doi.org/10.1017/s0308210500003048.

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In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

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32

Wei, Yongfang, Suiming Shang, and Zhanbing Bai. "Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses." Nonlinear Analysis: Modelling and Control 27 (February 4, 2022): 1–13. http://dx.doi.org/10.15388/namc.2022.27.26253.

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In this paper, we study the multiple solutions for some second-order p-Laplace differential equations with three-point boundary conditions and instantaneous and noninstantaneous impulses. By applying the variational method and critical point theory the multiple solutions are obtained in a Sobolev space. Compared with other local boundary value problems, the three-point boundary value problem is less studied by variational method due to its variational structure. Finally, two examples are given to illustrate the results of multiplicity.

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33

Chen, Ching-yu, and Tsung-fang Wu. "Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 4 (July 24, 2014): 691–709. http://dx.doi.org/10.1017/s0308210512000133.

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In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

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34

Alves, Claudianor O., Daniel C. de Morais Filho та Marco A. S. Souto. "Multiplicity of positive solutions for a class of problems with critical growth in ℝN". Proceedings of the Edinburgh Mathematical Society 52, № 1 (лютий 2009): 1–21. http://dx.doi.org/10.1017/s0013091507000028.

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AbstractUsing variational methods, we establish the existence and multiplicity of positive solutions for the following class of problems:where λ,β∈(0,∞), q∈(1,2*−1), 2*=2N/(N−2), N≥3, V,Z:ℝN→ℝ are continuous functions verifying some hypotheses.

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35

Liu, Xia, Tao Zhou, and Haiping Shi. "Boundary value problems of a discrete generalized beam equation via variational methods." Open Mathematics 16, no. 1 (December 26, 2018): 1412–22. http://dx.doi.org/10.1515/math-2018-0121.

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AbstractThe authors explore the boundary value problems of a discrete generalized beam equation. Using the critical point theory, some sufficient conditions for the existence of the solutions are obtained. Several consequences of the main results are also presented. Examples are given to illustrate the theorems.

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36

Khodadadian, Amirreza, Nima Noii, Maryam Parvizi, Mostafa Abbaszadeh, Thomas Wick, and Clemens Heitzinger. "A Bayesian estimation method for variational phase-field fracture problems." Computational Mechanics 66, no. 4 (July 14, 2020): 827–49. http://dx.doi.org/10.1007/s00466-020-01876-4.

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Abstract In this work, we propose a parameter estimation framework for fracture propagation problems. The fracture problem is described by a phase-field method. Parameter estimation is realized with a Bayesian approach. Here, the focus is on uncertainties arising in the solid material parameters and the critical energy release rate. A reference value (obtained on a sufficiently refined mesh) as the replacement of measurement data will be chosen, and their posterior distribution is obtained. Due to time- and mesh dependencies of the problem, the computational costs can be high. Using Bayesian inversion, we solve the problem on a relatively coarse mesh and fit the parameters. In several numerical examples our proposed framework is substantiated and the obtained load-displacement curves, that are usually the target functions, are matched with the reference values.

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37

Alves, Claudianor O., and Geilson F. Germano. "Existence and Concentration Phenomena for a Class of Indefinite Variational Problems with Critical Growth." Potential Analysis 52, no. 1 (September 24, 2018): 135–59. http://dx.doi.org/10.1007/s11118-018-9734-2.

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38

Papageorgiou, Nikolaos S., and Vicenţiu D. Rădulescu. "Semilinear Robin problems resonant at both zero and infinity." Forum Mathematicum 30, no. 1 (January 1, 2018): 237–51. http://dx.doi.org/10.1515/forum-2016-0264.

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Abstract We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at {\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at {\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

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39

Li, Hai Chun, and Yu Long Zhang. "Variational Method to Nonlinear Fourth-Order Impulsive Partial Differential Equations." Advanced Materials Research 261-263 (May 2011): 878–82. http://dx.doi.org/10.4028/www.scientific.net/amr.261-263.878.

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Анотація:

By using the critical point theory of variational method and Lax-Milgram theorem, we obtain the new results for the solution existence of the fourth-order impulsive partial differential equations with periodic boundary value problems.

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40

Hoffman, Kathleen A. "Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2057 (April 20, 2005): 1357–81. http://dx.doi.org/10.1098/rspa.2004.1435.

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Problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history. Constrained variational problems have been much less studied, particularly the theory of stability, which determines which solutions are physically realizable. In this paper, we develop stability exchange results appropriate for parameter-dependent calculus of variations problems with two particular features: either the parameter appears in the boundary conditions, or there are isoperimetric constraints. In particular, we identify an associated distinguished bifurcation diagram, which encodes the direction of stability exchange at folds. We apply the theory to a twisted elastic loop, which can naturally be formulated as a calculus of variations problem with both isoperimetric constraints and parameter-dependent boundary conditions. In combination with a perturbation expansion that classifies certain pitchfork bifurcations as sub- or super-critical, the distinguished diagram for the twisted loop provides a classification of the stability properties of all equilibria. In particular, an unanticipated sensitive dependence of stability properties on the ratio of twisting to bending stiffness is revealed.

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41

AIZICOVICI, SERGIU, NIKOLAOS S. PAPAGEORGIOU, and VASILE STAICU. "NODAL AND MULTIPLE SOLUTIONS FOR NONLINEAR PERIODIC PROBLEMS WITH COMPETING NONLINEARITIES." Communications in Contemporary Mathematics 15, no. 03 (May 19, 2013): 1350001. http://dx.doi.org/10.1142/s0219199713500016.

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Анотація:

We consider a nonlinear periodic problem drive driven by a nonhomogeneous differential operator which incorporates as a special case the scalar p-Laplacian, and a reaction which exhibits the competition of concave and convex terms. Using variational methods based on critical point theory, together with suitable truncation techniques and Morse theory (critical groups), we establish the existence of five nontrivial solutions, two positive, two negative and the fifth nodal (sign-changing). In the process, we also prove some auxiliary results of independent interest.

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42

Candela, A. M., G. Palmieri, and A. Salvatore. "Multiple solutions for some symmetric supercritical problems." Communications in Contemporary Mathematics 22, no. 08 (November 28, 2019): 1950075. http://dx.doi.org/10.1142/s0219199719500755.

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The aim of this paper is to investigate the existence of one or more critical points of a family of functionals which generalizes the model problem [Formula: see text] in the Banach space [Formula: see text], where [Formula: see text] is an open bounded domain, [Formula: see text] and the real terms [Formula: see text] and [Formula: see text] are [Formula: see text] Carathéodory functions on [Formula: see text]. We prove that, even if the coefficient [Formula: see text] makes the variational approach more difficult, if it satisfies “good” growth assumptions then at least one critical point exists also when the nonlinear term [Formula: see text] has a suitable supercritical growth. Moreover, if the functional is even, it has infinitely many critical levels. The proof, which exploits the interaction between two different norms on [Formula: see text], is based on a weak version of the Cerami–Palais–Smale condition and a suitable intersection lemma which allow us to use a Mountain Pass Theorem.

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43

Yang, Lianwu. "Existence and Multiple Solutions for Higher Order Difference Dirichlet Boundary Value Problems." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 5 (July 26, 2018): 539–44. http://dx.doi.org/10.1515/ijnsns-2017-0176.

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AbstractIn this paper, a higher order nonlinear difference equation is considered. By using the critical point theory, we obtain the existence and multiplicity for solutions of difference Dirichlet boundary value problems and give some new results. The proof is based on the variational methods and linking theorem.

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44

Belaouidel, Hassan, Mustapha Haddaoui, and Najib Tsouli. "Results of singular Direchelet problem involving the $p(x)$-laplacian with critical growth." Boletim da Sociedade Paranaense de Matemática 41 (December 23, 2022): 1–18. http://dx.doi.org/10.5269/bspm.52984.

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In this paper, we study the existence and multiplicity of solutions for Dirichlet singular elliptic problems involving the $p(x)$-Laplace equation with critical growth. The technical approach is mainly based on the variational method combined with the genus theory.

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45

Liang, Sihua, Giovanni Molica Bisci, and Binlin Zhang. "Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponents." Nonlinear Analysis: Modelling and Control 27 (March 28, 2022): 1–20. http://dx.doi.org/10.15388/namc.2022.27.26575.

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In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problem with critical variable exponent. By using constraint variational method and quantitative deformation lemma we show the existence of one least energy solution, which is strictly larger than twice of that of any ground state solution.

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46

Hadjian, Armin, and Juan J. Nieto. "Existence of solutions of Dirichlet problems for one dimensional fractional equations." AIMS Mathematics 7, no. 4 (2022): 6034–49. http://dx.doi.org/10.3934/math.2022336.

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<abstract><p>We establish the existence of infinitely many solutions for some nonlinear fractional differential equations under suitable oscillating behaviour of the nonlinear term. These problems have a variational structure and we prove our main results by using a critical point theorem due to Ricceri.</p></abstract>

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47

Matallah, Atika, Safia Benmansour, and Hayat Benchira. "Existence and nonexistence of nontrivial solutions for a class of p-Kirchhoff type problems with critical Sobolev exponent." Filomat 36, no. 9 (2022): 2971–79. http://dx.doi.org/10.2298/fil2209971m.

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48

Shokooh, Saeid, Ghasem A. Afrouzi, and John R. Graef. "Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces." Mathematica Slovaca 68, no. 4 (August 28, 2018): 867–80. http://dx.doi.org/10.1515/ms-2017-0151.

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Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

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49

Afrouzi, Ghasem A., and Armin Hadjian. "A variational approach for boundary value problems for impulsive fractional differential equations." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1565–84. http://dx.doi.org/10.1515/fca-2018-0082.

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Abstract By using an abstract critical point result for differentiable and parametric functionals due to B. Ricceri, we establish the existence of infinitely many classical solutions for fractional differential equations subject to boundary value conditions and impulses. More precisely, we determine some intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable oscillatory behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. No symmetric condition on the nonlinear term is assumed. Two examples are then given.

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50

Gao, David Yang. "Dual Extremum Principles in Finite Deformation Theory With Applications to Post-Buckling Analysis of Extended Nonlinear Beam Model." Applied Mechanics Reviews 50, no. 11S (November 1, 1997): S64—S71. http://dx.doi.org/10.1115/1.3101852.

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The critical points of the generalized complementary energy variation principles are clarified. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoff type stress only) is constructed, and a complete duality theory in geometric nonlinear system is established. We prove that the well-known generalized Hellenger-Reissner’s energy L(u,s) is a saddle point functional if and only if the Gao-Strang gap function is positive. In this case, the system is stable and the minimum potential energy principle is equivalent to a unique maximum dual variational principle. However, if this gap function is negative, then L(u,s) is a so-called ∂+-critical point functional. In this case, the system has two extremum complementary principles. An interesting trinity theorem for nonconvex variational problem is discovered, which can be used to study nonlinear bifurcation problems, phase transitions, variational inequality, and other things. In order to study the shear effects in frictional post-buckling problems, a new second order 2-D nonlinear beam model is developed. Its total potential is a double-well energy. A stability criterion for post-buckling analysis is proposed, which shows that the minimax complementary principle controls a stable buckling state. The unilaterial buckling state is controled by a minimum complementary principle. However, the maximum complementary principle controls the phase transitions.

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