On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition

Corrêa, Francisco Júlio S. A.; Nascimento, Rúbia G.

doi:10.1016/j.mcm.2008.03.013

Cited by 58 publications

(24 citation statements)

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“…In [8], [3], [4], [6], [10] and references therein, variational methods have been applied to prove existence and multiplicity of positive solutions for systems as (1.1). In [1] and [2] the sub-supersolution method has been used to prove the existence of solution with M i increasing and bounded from above and below for positive constants, that is, there exist positive constants 0 < m i ≤ m ∞ i < ∞ such that 0 < m i ≤ M i (t) ≤ m ∞ i < ∞ i = 1, 2, ∀t ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

The sub-supersolution method for Kirchhoff systems: applications

Figueiredo

Suárez

2015

Progress in Nonlinear Differential Equations and Their Applications

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Section: Introductionmentioning

confidence: 99%

The sub-supersolution method for Kirchhoff systems: applications

Figueiredo

Suárez

2015

Progress in Nonlinear Differential Equations and Their Applications

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“…The generalization of Kirchhoff equations to p(x)-Kirchhoff equation is a quite new topic, so there exists only a few paper [10]. In [10], Dai and Hao study 9) and show the existence and multiplicity of solutions of (1.9), where f : Ω ×R → R satisfies Carathéodory condition, M (t) satisfies (1.4) and…”

Section: Introductionmentioning

confidence: 99%

On an Elliptic System of P(x)-Kirchhoff-Type Under Neumann Boundary Condition

Yucedag

Avcı

Mashiyev

2012

Mathematical Modelling and Analysis

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“…The reader may consult [2][3][4][5][6][7][8] and the references therein for similar problem in several cases.…”

Section: Introductionmentioning

confidence: 99%

Existence results for a class of nonlocal problems involving p-Laplacian

Yang

Zhang

2011

Bound Value Probl

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This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.Keywords: Nonlocal problems, Neumann problem, p-Kirchhoff's equation IntroductionIn this paper, we deal with the nonlocal p-Kirchhoff type of problem given by:where Ω is a smooth bounded domain in R N , 1

0, such thatf (x, t) : × R → R is a continuous function and satisfies the subcritical condition:where C denotes a generic positive constant. Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes some mathematical difficulties which makes the study of such a problem particulary interesting. This problem has a physical motivation when p = 2. In this case, the operator M (∫ Ω |∇u| 2 dx)Δu appears in the Kirchhoff equation which arises in nonlinear vibrations, P-Kirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult [2][3][4][5][6][7][8] and the references therein for similar problem in several cases.This work is organized as follows, in Section 2, we present some preliminary results and in Section 3 we prove the main results. PreliminariesBy a weak solution of (1.1), then we say that a function u ε W 1,p (Ω) such that , which is called the space of functions of W 1,p (Ω) with null mean in Ω. ThusAs it is well known the Poincaré's inequality does not hold in the space W 1,p (Ω).However, it is true in W 0 .Let us also recall the following useful notion from nonlinear operator theory. If X is a Banach space and A : X X* is an operator, we say that A is of type (S + ), if for every sequence {x n } n≥1 ⊆ X such that x n ⇀ x weakly in X, and lim sup n→∞ A(x n ), x n − x ≤ 0. we have that x n x in X. Let us consider the map A :We have the following result: Lemma 2.2 [9,10]The map A :and of type (S + ).In the next section, we need the following definition and the lemmas. 1 (E, R), we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence {u n } ⊂ E for which J(u n ) is bounded and J'(u n ) 0 as n ∞ possesses a convergent subsequence.Lemma 2.3 [11]Let X be a Banach space with a direct sum decomposition X = X 1 ⊕ X 2 , with k = dimX 2 < ∞, let J be a C 1 function on X, satisfying (PS) condition. Assume that, for some r > 0,Assume also that J is bounded below and inf X J < 0. Then J has at least two nonzero critical points.Lemma 2.4 [12]Let X = X 1 ⊕ X 2 , where X is a real Banach space and X 2 ≠ {0}, and is finite dimensional. Suppose J C 1 (X, R) satisfies (PS) and (i) there is a constant a and a bounded neighbo...

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On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition

Cited by 58 publications

References 4 publications

The sub-supersolution method for Kirchhoff systems: applications

The sub-supersolution method for Kirchhoff systems: applications

On an Elliptic System of P(x)-Kirchhoff-Type Under Neumann Boundary Condition

Existence results for a class of nonlocal problems involving p-Laplacian

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