Multiplicity Results for a Perturbed Elliptic Neumann Problem

and Applied Analysis 3 2. Preliminaries and Basic Notations Our main tools are three critical point theorems that we recall here in a convenient form. The first has been obtained in 6 , and it is a more precise version of Theorem 3.2 of 7 . The second has been established in 7 . Theorem 2.1 see 6, Theorem 2.6 . LetX be a reflexive real Banach space,Φ : X → R a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X∗, Ψ : X → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that


Introduction
Here and in the sequel, Ω ⊂ R n is a bounded open set, with a boundary of class C 1 , q ∈ L ∞ Ω with ess inf Ω q > 0, p > n; f : Ω × R → R and g : Ω × R → R are L 1 -Carathéodory functions.
The aim of this paper is to study the following perturbed boundary value problem with Neumann conditions: −Δ p u q x |u| p−2 u λf x, u μg x, u , in Ω, ∂u ∂ν 0, on ∂Ω, where Δ p div |∇u| p−2 ∇u is the p-Laplacian, ν is the outer unit normal to ∂Ω, λ and μ are positive real parameters.
Nonlinear boundary value problems involving the p-Laplacian operator Δ p with p / 2 arise from a variety of physical problems. They are used in non-Newtonian fluids, reaction-diffusion problems, flow through porous media, and petroleum extraction see, e.g., 1, 2 . In the last years, several researchers have studied nonlinear problems of this type through different approaches. In 1 , the authors have obtained results on the existence of a solution for the problem by using the perturbation result on sums of ranges of nonlinear accretive operators. Subsequently, Wei and Agarwal, in 2 , have studied the same problem by developing some new techniques in the wake of 1 . Problem P λ,μ , when q 0, λ 1 and g does not depend on u, has been studied in 3 . In this paper, the authors have obtained the existence of at least three solutions for small μ, by using Implicit Function Theorem and Morse Theory. By using variational methods and in particular critical point results given by Ricceri in 4 , Faraci, in her nice paper 5 , has dealt with a Neumann Problem involving the p-Laplacian for any p of type ∂u ∂ν 0, on ∂Ω.

1.2
In particular, 5, Theorems 8, 9 assure the existence of three solutions for the problem given above.
In the present paper, we establish some results Theorems 3.1, 3.2 , which assure the existence of at least three weak solutions for the problem P λ,μ . In particular the following result is a consequence of Theorem 3.2. has at least three nonzero classical solutions.
With respect to 3, 5 , we stress that our results hold under different assumptions see Remarks 3.4 and 3.5 . In particular, in Theorem 1.1, no asymptotic condition at infinity is required on the nonlinear term. We also point out that in Theorems 3.1 and 3.2, precise estimates of parameters λ and μ are given.
Abstract and Applied Analysis 3

Preliminaries and Basic Notations
Our main tools are three critical point theorems that we recall here in a convenient form. The first has been obtained in 6 , and it is a more precise version of Theorem 3.2 of 7 . The second has been established in 7 .
Theorem 2.1 see 6, Theorem 2.6 . Let X be a reflexive real Banach space, Φ : X → R a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist r > 0 and x ∈ X, with r < Φ x , such that: Then, for each λ ∈ Λ r , the functional Φ − λΨ has at least three distinct critical points in X.
Theorem 2.2 see 7, Corollary 3.1 . Let X be a reflexive real Banach space, Φ : X → R a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist two positive constants r 1 , r 2 , and x ∈ X, with 2r 1 < Φ x < r 2 /2, such that x } and for every x 1 , x 2 ∈ X, which are local minima for the functional Φ − λΨ, and such that Ψ x 1 ≥ 0 and Ψ x 2 ≥ 0 one has inf t∈ 0,1 Ψ tx 1 1 − t x 2 ≥ 0.

Now we recall some basic definitions and notations. A function
We also recall that a weak solution of the problem P λ,μ is any u ∈ W 1,p Ω , such that

4 Abstract and Applied Analysis
Put k sup If Ω is convex, an explicit upper bound for the constant k is Moreover, set G c : Ω max |ξ|≤c G x, ξ dx for all c > 0 and G d : inf Ω× 0,d G for all d > 0. Clearly, G c ≥ 0 and G d ≤ 0.

Main Results
In this section, we present our main results on the existence of at least three weak solutions for the problem P λ,μ .

Abstract and Applied Analysis 5
Then, for every λ ∈ Λ : q 1 d p /p Ω F x, d dx, c p /pk p Ω max |ξ|<c F x, ξ dx and for every L 1 Ω -Carathédory function g : there exists δ > 0 given by 3.2 such that, for each μ ∈ 0, δ , Problem P λ,μ has at least three weak solutions.
Proof. Fix λ, g, and μ as in the conclusion. Take X W 1,p Ω endowed with the norm On the space C 0 Ω , we consider the norm u ∞ : sup x∈Ω |u x |. Since p > n, X is compactly embedded in C 0 Ω , we have Put, for each u ∈ X,

3.5
Since the critical points of the functional Φ−λΨ on X are weak solutions of problem P λ,μ , our aim is to apply Theorem 2.1 to Φ and Ψ. To this end, taking into account that the regularity assumptions of Theorem 2.1 on Φ and Ψ are satisfied, we will verify a 1 and a 2 .
Put r 1/p c/k p taking into account 3.4 , one has Now, fix u d. Clearly u ∈ X and moreover Φ u > r. One has

3.13
This leads to the coercivity of Φ − λΨ, and condition a 2 of Theorem 2.1 is verified. Since, from 3.8 and 3.10 , λ ∈ Φ u Ψ u , r sup Φ u ≤r Ψ u , 3.14 Theorem 2.1 assures the existence of three critical points for the functional Φ − λΨ, and the proof is complete.