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

Abstract

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 <p <N, ν is the unit exterior vector on ∂Ω, Δ p is the p-Laplacian operator, that is, Δ p u = div(|∇u| p−2 ∇u), the function M :+ is a continuous function and there is a constant m 0 > 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...

Multiple solutions for a class of p(x)-Kirchhoff type problems with Neumann boundary conditions

“…This problem models several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density, see [6]. Recently, Kirchhoff type problems have been studied in many papers, we refer to [1,4,5,7,9,11,18,19,21,22], in which different methods have been used to get the existence and multiplicity of solutions. Recently, Kirchhoff type problems have been studied in many papers, we refer to [1,4,5,7,9,11,18,19,21,22], in which different methods have been used to get the existence and multiplicity of solutions.…”

“…Moreover, problem (1.1) is related to the stationary version of the Kirchhoff equation which is presented by Kirchhoff in 1883, see [16] for details. One of the important hypotheses in these papers is that the Kirchhoff function M is non-degenerate, i.e., M.t / m 0 > 0 for all t 2 R C 0 WD OE0; C1/: Motivated by the ideas introduced in [8,10,12,22,23], the goal of this paper is to study the multiplicity of solutions for problem (1.1) in the case when the Kirchhoff function M.t/ may be degenerate at zero. One of the important hypotheses in these papers is that the Kirchhoff function M is non-degenerate, i.e., M.t / m 0 > 0 for all t 2 R C 0 WD OE0; C1/: Motivated by the ideas introduced in [8,10,12,22,23], the goal of this paper is to study the multiplicity of solutions for problem (1.1) in the case when the Kirchhoff function M.t/ may be degenerate at zero.…”

“…Due to this reason, [22,Theorem 2.2] is invalid in our framework. We emphasize that the extension from the p-Laplace operator p u to the p.x/-Laplace operator involved in (1.1) is interesting and not trivial, since the new operators have a more complicated structure than the p-Laplace operator, for example they are non-homogeneous.…”