Abstract
This paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zero Dirichlet boundary condition in bounded domainΩ⊂ℝNby using the subsuper solutions method.
Highlights
In this paper, we consider the following system of differential equations: ⎧⎪⎪⎪⎪⎪⎨ − A |∇u|2dxΔu Ω λ1ua + μ1vb in Ω, ⎪⎪⎪⎪⎪⎩B |∇u|2dxΔv λ2uc μ2vd in (1)u v 0 on zΩ, where Ω ⊂ RN (N ≥ 3) is a bounded smooth domain with C2 boundary zΩ, A, B: R+ ⟶ R+ are continuous functions, and λ1, λ2, μ1, and μ2 are positive parameters, where a + c < 1 and b + d < 1. e peculiarity of this type of problem, and by far the most important, is that it is not local
We study the existence of weak positive solutions for a sublinear Kirchhoff elliptic systems in bounded domains by using the subsuper solutions method (SSM) combined with comparison principle which have been widely applied in many work.Validity of the comparison principle and of the SSM for local and nonlocal problems as the stationary Kirchhoff Equation was an important subject in the last few years
It is worth to notice that in [4], Alves and Correa developed a new SSM for problem (1) to deal with the increasing M case. e result is obtained by using a kind of Minty–Browder theorem for a suitable pseudomonotone operator, but instead of constructing a subsolution, the authors assumed the existence of a whole family of functions which satisfy a stronger condition than just being subsolutions; the inconvenience is that these stronger conditions restrict the possible right hand sides in (1). Another SSM for nonlocal problems is obtained in [4] for a problem involving a nonlocal term with a Lebesgue norm, instead of the Sobolev norm appearing in (1)
Summary
We consider the following system of differential equations:. u v 0 on zΩ , where Ω ⊂ RN (N ≥ 3) is a bounded smooth domain with C2 boundary zΩ , A, B: R+ ⟶ R+ are continuous functions, and λ1, λ2, μ1, and μ2 are positive parameters, where a + c < 1 and b + d < 1. e peculiarity of this type of problem, and by far the most important, is that it is not local. Where T is a positive constant and u0 and u1 are given functions In such problems, u expresses the displacement, h(x, u) the extreme force, M(r) a1r + b1, b1 the initial tension, and a1 relates to the intrinsic properties of the wire material (such as Young’s modulus). Kirchhoff took into account the changes caused by transverse oscillations along the length of the wire With their implications in other disciplines, and given the breadth of their fields of application, nonlocal problems will be used to model several physical phenomena, and they intervene in biological systems or describe a process dependent on its average, such as particle density population. Problems relating to Kirchhoff operators have been studied in several papers (we refer to [6]), where the authors used different methods to obtain solutions (1) in the case of single equation (see [6]).
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