On positive weak solutions for new Kirchhoff type systems with Dirichlet boundary condition

Subject Areas : Statistics

Mohammad Bagher  Ghaemi ¹ (Department of Mathematics, Iran University of Science and Technology, Tehran, Iran)
Mehdi  Choubin ² (Department of Mathematics, Velayat University, Iranshahr, Iran,)

Received: 2020-09-19 Accepted : 2021-09-07 Published : 2022-03-21

Keywords: شرط مرزی دیریکله, جواب‌ ضعیف مثبت, معادله کیرشهف,

Abstract :

The Kirchhoff equation(*) [rho frac{{{partial ^2}u}}{{partial {t^2}}} - left( {frac{{{P_0}}}{h} + frac{E}{{2L}}int_0^L {left| {frac{{partial u}}{{partial x}}} right|} dx} right)frac{{{partial ^2}u}}{{partial {x^2}}} = 0]extends the classical d'Alembert's wave equation by considering the effects of the changes in the length of the strings during the vibrations. The parameters in equation (*) have the following meanings: L is the length of the string, h is the area of cross-section, E is the Young modulus of the material, rho is the mass density and P_0 is the initial tension. In recent years, some applicable generalization of Kirchhoff equation have been proposed and studied in many papers. In this paper, we study the existence of positive weak solutions for new Kirchhoff type systems with multiple parameters. We will show under what conditions these systems have a positive weak solution for any positive parameters. Our approach in this paper is based on the sub- and supersolution method. approach in this paper is based on the sub- and supersolution method.

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