Abstract
This paper is aiming at obtaining weak solution for a bi-nonlocal fourth order elliptic problem with Navier boundary condition. Our approach is based on variational methods and critical point theory.
Highlights
Introduction and main resultsIn recent years, a great deal of attention has been paid to the study of problems involving nonlocaloperators, both in the pure mathematical research and in the concrete real-world applications, such as, optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, and game theory, see [7, 8, 6] and references therein.In this paper, we are interested in the existence of weak solutions for the following fourth order elliptic equations of Kirchhoff type, with an additional nonlocal term, ⎧ ⎨ ⎩ M μZ Ω¶ 1 |∆u|p(x)dx p(x) ∆2p(x)u = λ|u|q(x)−2u ∙ZΩr |u|q(x)dx q(x) in
A great deal of attention has been paid to the study of problems involving nonlocaloperators, both in the pure mathematical research and in the concrete real-world applications, such as, optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, and game theory, see [7, 8, 6] and references therein
We are interested in the existence of weak solutions for the following fourth order elliptic equations of Kirchhoff type, with an additional nonlocal term
Summary
A great deal of attention has been paid to the study of problems involving nonlocaloperators, both in the pure mathematical research and in the concrete real-world applications, such as, optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, and game theory, see [7, 8, 6] and references therein. We are interested in the existence of weak solutions for the following fourth order elliptic equations of Kirchhoff type, with an additional nonlocal term,. Since the two equation in (1.1) contains an integral over Ω, it is no longer a pointwise identity; it is often called bi-nonlocal problem. There has been increasing attention to study a problem like (1.1) because such equations can describe some phenomena appeared in physics and engineering, such as describing the theorem of beam vibration, image processing, and so on. In the [7], the authors study the existence and multiplicity of solutions via Krasnoselskii’s Genus, to the following bi-nonlocal p(x)-Kirchhoff equation,.
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