On weak solutions for fourth‐order problems involving the Leray–Lions type operators

Existence of Positive Solutions for a Pertubed Fourth-Order Equation

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2104.623h ◽

2021 ◽

Vol 45 (4) ◽

pp. 623-633

Author(s):

MOHAMMAD REZA HEIDARI TAVANI ◽

◽

ABDOLLAH NAZARI ◽

Keyword(s):

Differential Equation ◽

Fourth Order ◽

Weak Solutions ◽

Nonlinear Term ◽

Point Theory ◽

Order Equation ◽

Fourth Order Equation ◽

Equation Problem ◽

Existence Of Positive Solutions ◽

Fourth Order Problems

In this paper, a special type of fourth-order diﬀerential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the diﬀerential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

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Infinitely many weak solutions for fourth-order equations depending on two parameters

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i4.31997 ◽

2018 ◽

Vol 36 (4) ◽

pp. 131-147 ◽

Cited By ~ 1

Author(s):

Saeid Shokooh ◽

Ghasem A. Afrouzi ◽

Hossain Zahmatkesh

Keyword(s):

Variational Principle ◽

Fourth Order ◽

Weak Solutions ◽

Two Parameters ◽

Fourth Order Equations ◽

Fourth Order Problems

In this paper, by employing Ricceri variational principle, we prove the existence of infinitely many weak solutions for fourth-order problems depending on two real parameters. We also provide some particular cases and a concrete example in order to illustrate the main abstract results of this paper.

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Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drab003 ◽

2021 ◽

Author(s):

Andreas Dedner ◽

Alice Hodson

Keyword(s):

Fourth Order ◽

Perturbation Parameter ◽

Two Dimensions ◽

Optimal Error Estimates ◽

Projection Operators ◽

Varying Coefficients ◽

Virtual Element Methods ◽

Virtual Element ◽

Perturbation Problems ◽

Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

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