Existence Results for a Nonlocal Problem Involving the p-Laplace Operator

The present paper deals with a nonlocal problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω of R . The problem studied is a stationary version of the original Kirchhoff equation involving the p-Laplace operator. The question of the existence of weak solutions is treated. Using variational approach and applying the Mountain Pass Theorem together with Fountain theorem, the existence and multiplicity of solutions is obtained in the Sobolev space W 1,p (Ω).


Introduction
We study the existence and multiplicity of solutions for a nonlocal elliptic equation with Dirichlet zero-boundary condition, i.e., p-Kirchhoff equation of the following type: where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, M is a continuous function, f satisfies Carathéodory condition and 1 < p < N .
In problem (P), if we consider the case p = 2, and replace the nonlocal term M ( ∫ ) ∆u = f (x, u) in Ω, (1.1) which is related to the stationary analog of the Kirchhoff equation: where M (s) = as + b, a, b > 0. It was proposed by Kirchhoff in [10] as an extension of the classical D'Alambert's wave equations for free vibrations of elastic strings. The Kirchhoff model takes into account the length changes of the spring produced by transverse vibrations. Equation (1.2) received much attention and an abstract framework to the problem was proposed after the work [11]. Some interesting and further results can be found in [4,14] and the references there in. In addition, (1.1) has important physical and biological background. There are many authors who pay more attention to this equation. In particularly, authors concerned with the existence of solutions for (1.1) with zero Dirichlet boundary condition via Galerkin method and built the variational frame in [2,13]. More recently, Perera and Zhang obtained solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods, invariant sets of descent flow, Yang index and critical groups [15,21].
arises in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitational (Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal tunaway in Ohmic Heating, shear bands in metal deformed under high starin rates,among others. Because of importance, in [7,17], the authors similarly studied the existence of solutions for (1.3) with zero Dirichlet boundary condition. On the other hand, elliptic equations with nonlinear boundary conditions have become rather an active area of research; see [3,5,6,16,19,22] and references therein. Those references present necessary and sufficient conditions of solutions of elliptic equations with nonlinear boundary conditions.
In the present paper, we deal with the existence of solutions for an elliptic equation (P) with zero Dirichlet boundary condition based on variational method. The problem (P) involves the p-Laplacian or the p-Laplace with p > 1. The case of p = 2, i.e., linear case of (P), has been investigated in [19], while the general case of p > 1 will be studied in the present paper. When p ̸ = 2, the p-Laplace operator ∆ p is nonlinear. Note also that for p ̸ = 2, the p-Laplace operator is (p − 1)-homogeneous but not additive. For this reason some of the authors, in particular those who work in ODEs, call equations involving the p-Laplacian "half-linear" equations. The word "half-linear" reflects the fact that "one half" of the properties of linearity (i.e. additivity) is lost, while "one half" of the properties is preserved (i.e. homogeneity). The p-Laplace operator is very popular in nonlinear analysis and appears particularly in describing the behavior of compressible fluid in a homogeneous isotropic rigid porous medium (see e.g. [8,12,18]). The energy functional corresponding to problem (P) is defined as I : W 1,p (Ω) → R,

Preliminaries
Thanks to the growth condition (f 1 ), it is not difficult to show that I ∈ C 1 (W 1,p (Ω) , R), and for any u, φ ∈ W 1,p (Ω). Hence, we can infer that critical points of functional I are the weak solutions for problem (P). Proposition 2 [1] Let Ω be an open and bounded subset of The embedding is compact if and only if q ∈ [1, p * ).
Proposition 3 [9] Let X be a Banach space and let define the functional Λ = ∫ Ω (|∇u| p + |u| p ) dx. Then Λ : X → R is convex. The mapping Λ ′ : X → X * is a strictly monotone, bounded homeomorphism, and of (S + ) type, namely u n ⇀ u in X and lim Denition 4 Let X be a Banach space and J : X → R a C 1 -functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short), if any sequence {u n } in X such that {J (u n )} is bounded and J ′ (u n ) → 0 as n → ∞, admits a convergent subsequence.

Main results and proofs
In what follows, the following assumptions related to problem (P) are considered as far as the converse is explicitely stated.
(M 1 ) M : R + → R + is a continuous function and satisfies the growth condition for all t > 0, where A,B and α are positive constants such that B A > α > 1; (f 1 ) f : Ω × R → R satisfies Carathéodory condition and the subcritical growth conditions for all x ∈ Ω and for all t ∈ R, where C is a positive constant and 1 < p < q < p * = (N p) / (N − p); , t → 0 for x ∈ Ω uniformly, where q > αp α ; (AR) Ambrosetti-Rabinovitz's type condition holds, i.e., The first main result of the present paper is: Dividing the above inequality by ∥u n ∥ pα and passing to the limit as n → ∞ we obtain a contradiction. It follows that {∥u n ∥} is bounded in W 1,p (Ω). Thus, there exists u ∈ W 1,p (Ω) such that passing to a subsequence, still denoted by {u n }, it converges weakly to u ∈ W 1,p (Ω).
The second main result of the present paper is: Theorem 8 Suppose (M 1 ), (AR), (f 1 ) and (f 3 ) hold, then I has a sequence of critical points {u n } such that I (u n ) → +∞ and (P) has infinite many pairs of solutions.
PROOF. Proof of Theorem 8 By Lemma 6, I satisfies (PS) condition, and from (f 3 ) it is also an even functional. In the sequel, we will show that if k is large enough, then there exist ρ k > r k > 0 such that (i) b k := inf u∈Z k ,∥u∥=r k I(u) → +∞, as k → +∞, (ii) a k := max u∈Y k ,∥u∥=ρ k I(u) ≤ 0.
Therefore, to obtain the results of Theorem 8, it is enough to apply Fountain theorem (see [20]). (i) For any u ∈ Z k with ∥u∥ big enough, we have Set ∥u∥ = r k := (cqβ q k ) 1 p−q . Then, we have