EXISTENCE OF SOLUTIONS TO NONLOCAL AND SINGULAR VARIATIONAL ELLIPTIC INEQUALITY VIA GALERKIN METHOD

In this article, we study the existence of solutions for nonlocal variational elliptic inequality M(kuk 2 ) u f(x; u) Making use of the penalized method and Galerkin approximations, we estab- lish existence theorems for both cases when M is continuous and when M is discontinuous.

For example, let us consider K = {v ∈ L 2 (Ω); a ≤ v(x) ≤ b a.e. in Ω} where −∞ ≤ a ≤ 0 ≤ b ≤ ∞.We define ρ(λ) as follows In this paper we study some questions related to the existence of solutions for the nonlocal elliptic variational inequality: where f ∈ H 1 0 (Ω) and M : R → R is a function whose behavior will be stated later.The main purpose of this work is establishing properties on M under which EJQTDE, 2005, No. 18, p. 1 problem (1.1), and its nonlinear counterpart, possesses a solution.This inequality has called our attention because the operator contains the nonlocal term M ( u 2 ) which poses some interesting mathematical questions.Also the operator L appears in the Kirchhoff equation, which arises in nonlinear vibrations, namely The mathematical aspects of this model were largely investigated.See, for example, Arosio-Spagnolo [2], Hazoya-Yamada [11], Lions [16], Pohozhaev [18].A survey of the results about the mathematical aspects of Kirchhoff model can be found in Medeiros-Limaco-Menezes [14].Unilateral problems for nonlinear operators of the Kirchhoff type were initially studied by Kludnev [12], Larkin [13], Medeiros-Milla Miranda [17] among others.Recently, Alves-Corrêa [1] focused their attention on problem related with (1.1) in case M (t) ≥ m 0 > 0, for all t ≥ 0, where m 0 is a constant.Among other things they studied the above M -linear problem (1.1) where M , besides the strict positivity mentioned before, satisfies the following assumption: The function H : R → R with In a previous paper, Menezes-Corrêa [8] proved a similar results by allowing M to attain negative values and M (t) ≥ m 0 > 0 only for t large enough.This is possible thanks to a device explored by Alves-de Figueiredo [3], who use Galerkin method to attack a non-variational elliptic system.The technique can be conveniently adapted to problems such as (1.1).In this way we improve substantially the existence result on the above problem mainly because our assumptions on M are weakened.Indeed, we may also consider the case in which M possesses a singularity.The methodology used in our proof consists in transforming, by penalty, the inequality (1.1) into a family of equations depending of a parameter > 0 and apply Galerkin's method.In the application of Galerkin's methods we use the sharp angle lemma(see Lions [15, p.53]).This paper is organized as follows: Section 2 is devoted to the study of (1.1) in the continuous case.In Section 3 the inequality (1.1) is studied in case M possesses a discontinuity.In Section 4 we analyze another type of variational inequality.

The M -linear Problem: Continuous Case
In this section we are concerned with the M -linear problem (1.1) where f ∈ H 1 0 (Ω) and M : R → R is a continuous function satisfying EJQTDE, 2005, No. 18, p. 2 (M1) There exists a positive number m 0 such that M (t) ≥ m 0 , for all t ≥ 0.
We have Theorem 2.1.Assume that and (M1) and (H1) hold.Then for any choice of 0 = f ∈ H 1 0 (Ω) the problem (1.1) admits at least one solution.The proof of Theorem 2.1 is given by the penalty method.In fact, let us represent by β the operator from L 2 (Ω) into L 2 (Ω) defined by β = I − P K , e.g., (βv . The operator β is monotone and Lipschitzian.The next result can be found in Haraux [10] p. 58.Lemma 2.2.Let g : R → R be a Lipschitzian and increasing function with The penalized problem associated to the problem (1.1), consists in given > 0, find u solution in Ω of the problem where f is given in H 1 0 (Ω).The existence of one solution of the penalized problem (2.2) is give by the Theorem 2.3.Assume that (M 1) hold.Then, for any 0 = f ∈ H 1 0 (Ω) there exists at least one u ∈ H 1 0 (Ω) ∩ H 2 (Ω), solution of the problem (2.2) .Proof.We employ the Galerkin Method by using the sharp angle lemma.Let us consider the Hilbertian basis of spectral objects (e j ) j∈N and (λ j ) j∈N for the operator −∆ in H 1 0 (Ω), cf.Brezis [5].We know that the eigenvectors (e j ) j∈N are orthonormal complete in L 2 (Ω) and complete in ξ j e j with real ξ j , 1 ≤ j ≤ m.
We will consider u m instead of u m .The approximate problem consists in finding a solution u m ∈ V m of the system of algebraic equations We need to prove that (2.3) has a solution u m ∈ V m .To this end, we will consider the vector η = (η i ) 1≤i≤m of R m defined by EJQTDE, 2005, No. 18, p. 3 Let ξ = (ξ i ) 1≤i≤m be the components of the vector u m of V m .The mapping P : R m → R m defined by P ξ = η is continuous.If we prove that P ξ, ξ ≥ 0 for ξ R n = r, with an appropriate r, it will follow by the sharp angle lemma that there exists a ξ in the ball B r (0) ⊂ R m such that P ξ = 0.This implies the existence of a solution to (2.3).In fact, we have Using (M1), Hölder and Poincaré inequalities and observing that (β(u m ), u m ) ≥ 0, we get We can consider r large enough that (P ξ, ξ) ≥ 0 for ξ R m = r.Then P ξ = 0 for some ξ ∈ B r (0), which implies that system (2.3) has a solution u m ∈ V m corresponding to this ξ.Thus, there is r does not depend on m and , such that which implies that Because ( u m 2 ) is a bounded real sequence and M is continuous one has for some t0 ≥ 0, and perhaps for a subsequence.
Theorem 3.1.Assume that (M1)-( M3) and (H1) hold.Then for any choice of f ∈ H 1 0 (Ω) the problem (1.1) admits at least one solution solution.The proof of Theorem 3.1 is given as in the proof of Theorem 2.1.We will formulate the penalized problem, associated with the variational inequality (1.1), as follows.Given > 0, find a function u ∈ H 1 0 (Ω) solution of the problem where f is given in H 1 0 (Ω).The existence of one solution of the penalized problem (3.1) is given by the Theorem 3.2.Assume that (M 1) − (M 3) hold.Then, for any 0 = f ∈ H 1 0 (Ω) there exists at least one u ∈ L 2 (Ω), solution of the problem (3.1) .
Proof.We first consider the sequence of functions M n : R → R given by for n > m 0 , where θ − δ n and θ + δ n , δ n , δ n > 0, are, respectively, the points closest to θ, at left and at right, so that We point out that, in this case, δ n , δ n → 0 as n → ∞.
Take n > m 0 and observe that the horizontal lines y = n cross the graph of M .Hence M n is continuous and satisfies (M1), for each n > m 0 .In view of this, for each n like above, there is EJQTDE, 2005, No. 18, p. 6 Taking ω = u n in the above equation one has and so perhaps for subsequences.We note that if (M n ( u n 2 ) converges its limit is different of zero.Suppose that ), for such n, and so which is a contradiction.On the other hand, if there are infinitely many n so that ) and we arrive again in a contradiction.
Consequently u n 2 → θ 0 = θ which implies that for n large enough Taking ω = −∆u n in (3.4), using(2.1)and arguments of compactness as in the proof of theorem 2.1, we obtain that The estimates (3.3) and (3.5) are sufficient to pass to the limit in the approximate equation and to obtain Follows of (3.9) and (3.10) that because v ∈ K and monotonicity of β.Hence As in the proof of Theorem 2.1, letting n → ∞, we obtain that u satisfies the conditions of Theorem 3.2.

Another Nonlocal Problem
Next, we make some remarks on a nonlocal problem which is a slight generalization of one studied by Chipot-Lovat [6] and Chipot-Rodrigues [7].More precisely, the above authors studied the problem where Ω ⊂ R N is a bounded domain, N ≥ 1, and a : R → (0, +∞) is a given function.Equation (4.1) is the stationary version of the parabolic problem Here T is some arbitrary time and u represents, for instance, the density of a population subject to spreading.See [6,7] for more details.In particular, [6] studies problem (4.1), with f ∈ H −1 (Ω), and proves the following result.where K and f are as before and 1 < q < 2N/(N − 2), N ≥ 3. When q = 2 we have the well known Carrier model.
The proof of Theorem 4.2 is given as in the proof of Theorem 2.1.We will formulate the penalized problem, associated with the variational inequality (4.2), as follows: Given > 0, find a function u ∈ H where f is given in H 1 0 (Ω).The existence of one solution of the penalized problem (4.3) is given by the