Some Multiplicity Results to the Existence of Three Solutions for a Dirichlet Boundary Value Problem Involving the p-Laplacian

Abstract In this paper we prove the existence of two intervals of positive real parameters λ for a Dirichlet boundary value problem involving the p-Laplacian which admit three weak solutions, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. Our main tool is a three critical points theorem due to G. Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim., 28:249–258, 2004].


Introduction
The purpose of this paper is to establish the existence of two intervals of positive real parameters λ for which the problem where Δ p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator, Ω ⊂ R N (N 1) is a non-empty bounded open set with smooth boundary ∂Ω, p > N, λ is a positive parameter and f : Ω × R → R is an L 1 -Carathéodory function, admits three weak solutions, whose norms are uniformly bounded in respect to λ belonging to one of the two intervals.
We recall that a function f : Ω × R → R is said to be L 1 -Carathéodory if (δ 1 ) x → f (x, t) is measurable for every t ∈ R; (δ 2 ) t → f (x, t) is continuous for almost every x ∈ Ω; (δ 3 ) for every > 0 there exists a function l ∈ L 1 (Ω) such that sup |t| |f (x, t)| l (x) for almost every x ∈ Ω.
We say that u is a weak solution to the problem (1.1) if u ∈ W 1,p 0 (Ω) and for every v ∈ W 1,p 0 (Ω). In recent years, many publications [1,7,8,9,10,11,12,14] have appeared about elliptic problems with Dirichlet boundary conditions which have been used in a great variety of application. For example, Ramaswamy and Shivaji in [14] established the existence of three positive solutions for classes of nondecreasing, p-sublinear functions f belonging to C 1 ([0, ∞)) for a p-Laplacian version of [3], i.e., the problem where p > 1, λ > 0 is a parameter and Ω is a bounded domain in R N ; N ≥ 2 with ∂Ω of class C 2 and connected. Uniqueness of positive solutions to the problem (1.2) when p > 1 and f (u)/u p−1 is decreasing on (0, +∞) was obtained in Guo and Webb [11] and Drabek and Hernandez [9]. A natural question is that, whether uniqueness holds under the weaker condition than f (u)/u p−1 is decreasing for large u. When Ω is a ball, Hai and Shivaji [12] showed that the answer is affirmative. However, the approach used in [12] depends on ordinary differential equations techniques and cannot be applied to the case of a general domain. In [7], Ricceri's three critical points theorem [15] has been successfully used to obtain existence of at least three weak solutions to the problem (1.1) in W 1,p 0 (Ω). In [1], based on Ricceri's three critical points theorem [15] we obtained the existence of an interval Λ ⊆ [0, +∞[ and a positive real number q such that for each λ ∈ Λ problem where Ω ⊂ R N (N 2) is non-empty bounded open set with smooth boundary ∂Ω, p > N, λ > 0, f : Ω × R → R is a continuous function and positive weight function a(x) ∈ C(Ω), admits at least three weak solutions whose norms in W 1,p 0 (Ω) are less than q that we extended the main result of [4] by using of the results of [7] to the general case. In [8], the authors employing Ricceri's three critical points theorem [16] obtained multiple weak solutions for the following BVP where Ω ⊂ R N is a non-empty bounded open set with smooth boundary ∂Ω, p > N, f, g : Ω × R → R are two Carathéodory functions and λ, μ are two positive parameters. Bonanno in [6] established the existence of two intervals of positive real parameters λ for which the functional Φ + λΨ has three critical points, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. He illustrated the result for a two point boundary value problem, and here we are interested to illustrate this result to the problem (1.1). Our main result is Theorem 1 that ensures the existence of two intervals Λ 1 and Λ 2 such that, for each λ ∈ Λ 1 ∪ Λ 2 , the problem (1.1) admits at least three weak solutions whose norms are uniformly bounded with respect to λ ∈ Λ 2 . The technique used in our proof has been introduced in [7].
As an immediate consequences of Theorem 1, we obtain Corollary 1, in which the function f has separated variables. The applicability of the result is illustrated by Example 1. Finally, we present the application of Theorem 1 in the ordinary case with p = 2, that Example 2 illustrates the result.

Main Results
First we recall for the reader's convenience Theorem 3.1 of [6] (see also [2,5,13,15,16] for related results) to transfer the existence of three solutions of the problem (1.1) into the existence of critical points of the Euler functional: Theorem A ([6, Theorem 3.1]) Let X be a separable and reflexive real Banach space; Φ : X −→ R a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X * ; J : X −→ R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there Further, assume that there are r > 0, x 1 ∈ X such that : Then, for each the equation Φ (u)−λJ (u) = 0 has at least three solutions in X and, moreover, for each h > 1, there exist an open interval and a positive real number σ such that, for each λ ∈ Λ 2 , the equation given above has at least three solutions in X whose norms are less than σ.
Here and in the sequel, X will denote the Sobolev space W 1,p 0 (Ω) with the norm where Γ denotes the Gamma function and m(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball. Now, fix x 0 ∈ Ω and pick r 1 , r 2 with 0 < r 1 < r 2 such that where S(x 0 , r i ) denotes the ball with center at x 0 and radius of r i for i = 1, 2. Put We formulate our main result as follows: Assume that there exist three positive constants θ, τ and γ with k 1 τ > θ, γ < p and a function μ ∈ L 1 (Ω) + such that where k 1 is given in (2.1). Then, for each , the problem (1.1) admits at least three weak solutions in X and, moreover, for each h > 1, there exist an open interval and a positive real number σ such that, for each λ ∈ Λ 2 , the problem (1.1) admits at least three weak solutions in X whose norms are less than σ.
Proof. In order to apply Theorem A, we begin by setting for each u ∈ X. It is well known that J is a continuously Gâteaux differentiable functional whose Gâteaux derivative at the point u ∈ X is the functional J (u) ∈ X * , given by for every v ∈ X. We claim that J : X → X * is a compact operator. To this end, it is enough to show that J is strongly continuous on X. For this, for fixed u ∈ X let u n → u weakly in X as n → +∞, then we have u n converges uniformly to u on Ω as n → +∞ (see [17]). Since F (x, ·) is C 1 in R for every x ∈ Ω, so it is continuous in R for every x ∈ Ω, and we get that F (x, u n ) → F (x, u) strongly as n → +∞ which follows J (u n ) → J (u) strongly as n → +∞. Thus we proved that J is strongly continuous on X, which implies that J is a compact operator by Proposition 26.2 of [19]. Hence the claim is true. Moreover, the functional Φ is a continuously Gâteaux differentiable whose Gâteaux derivative at the point u ∈ X is the functional Φ (u) ∈ X * , given by Φ admits a continuous inverse on X * . Indeed, owing to (2.2) of [17], for every u, v ∈ X there exists a positive constant c p such that where ·,· denotes the usual inner product in R. So, we have for every u, v ∈ X, namely Φ is an uniformly monotone operator in X, and since Φ is coercive and hemicontinuous in X, by applying Theorem 26.A. [19], we have that Φ admits a continuous inverse on X * . Using again that Φ is monotone, we obtain that Φ is sequentially weakly lower semi continuous (see [19,Proposition 25.20]). Thanks to (α 3 ), for each λ > 0 one has that x ∈ S(x 0 , r 1 ) and r = 1 p ( θ c ) p . It is easy to see that u * ∈ X and, in particular, one has F (x, t) dx, and since 0 u * (x) τ for each x ∈ Ω, the condition (α 1 ) ensures that Therefore, owing to our assumptions, we have Now, we can apply Theorem A. Taking into account that and with x 0 = 0, x 1 = u * , and see Λ 1 ⊆ Λ 1 , Λ 2 ⊆ Λ 2 , and also taking into account that the weak solutions of the problem (1.1) are exactly the solutions of the equation from Theorem A it follows that, for each λ ∈ Λ 1 , the problem (1.1) admits at least three weak solutions, and there exist an open interval Λ 2 ⊆ [0, ρ] and a real positive number σ such that, for each λ ∈ Λ 2 , the problem (1.1) admits at least three weak solutions that whose norms in X are less than σ. Hence, we have the conclusion.
Because, from (α 2 ) we have and since k 1 τ > θ, we get and so Hence, multiplying by 1 pc p we obtain

Remark 2.
In applying Theorem 1, it is enough to know as explicit upper bound of the constant c. To be precise, we can use formula (2.1) as constant c the righthand term of the formula in page 393, so that the constant k 1 in Theorem 1 is numerically well determined.
We now present a particular case of Theorem 1, in which the function f has separated variables.
Corollary 1. Let f 1 ∈ L 1 (Ω) and f 2 ∈ C(R) be two functions. Put F (t) = t 0 f 2 (ξ) dξ for all t ∈ R, and assume that there exist four positive constants θ, τ , η and γ with k 1 τ > θ, γ < p such that where k 1 is given in (2.1). Then, for each ) and a positive real number σ such that, for each λ ∈ Λ 2 , the problem (2.2) admits at least three weak solutions in X whose norms are less than σ.
For simplicity, we fix Ω = (a, b) for a, b ∈ R and x 0 ∈ Ω. Taking into account that, in this situation, c = (b−a) we have the following result: Assume that there exist three positive constants θ, τ and γ with ( b−a 2(r2−r1) ) 1 2 τ > θ, γ < 2 and a function μ ∈ L 1 ([a, b]) + such that for almost every x ∈ (a, b) and for all t ∈ R.
Then, for each and a positive real number σ such that, for each λ ∈ Λ 2 , the problem (2.5) admits at least three classical solutions in X whose norms are less than σ.