Existential results for a class of fourth-order Elliptic problems with Robin boundary conditions

Subject Areas : Statistics

Atieh Ramzannia jalali ¹ ( Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. )
Ghasem Alizadeh afrouzi ² ( Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. )

Received: 2020-01-26 Accepted : 2020-09-19 Published : 2021-03-01

Keywords: جادهی فشرده, فضای سوبولف با توان متغیر, عملگر مرتبه چهارم, نظریه گذرگاه کوهی,

Abstract :

In recent years, fourth-order differential equations in mathematical physics have been considered by many researchers. These applications include Micro Electro Mechanical systems, thin film theory, surface diffusion on solids, flow in Hele-Shaw cells and phase field models of multiphase systems.[ 9, 20] The importance of studying such equations is due to the justification of many physical examples using mathematical modeling, which can be seen mostly in the field of Newtonian fluids and elastic mechanics, in particular, electrological fluids (smart liquids). See [11, 21] for more details.In this paper, using variational methods, sufficient conditions for the existence of at least two weak non-trivial solutions of a fourth-order elliptic boundary value problem with the Rubin boundary conditions are investigated. Our analysis mainly relies on the variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue–Sobolev spaces. Our work starting point is the paper "Continuous spectrum of a fourth-order nonhomogenous elliptic equation with variable exponent" by A. Ayoujil, A.R. El Amrouss of [3] where the authors considered the problem (1) with the Navier boundary conditions. This paper's guarantee the exsitence of at least two nontrivial weak solutions for the problem (1) with Robin boundary conditions.More precisely, by applying Ambrosetti and Rabinowitz’s mountain pass theorem and under appropriate conditions, we show that there exists a positive number λ_*such that the problem (1) has at least two nontrivial weak solutions.

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