Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth

In this paper, we consider the Dirichlet problem for some nonlinear elliptic problems such as − div ([a (x) + |u|q] ∇u) + u = H (x, u, ∇u) + f, x ∈ Ω, u ∈ H1 0 (Ω) , (1) under the following assumptions:Ω is a bounded open subset of R, whereN ≥ 3, 0 < q < 1, and f ∈ Lm with m ≥ 2 and a : Ω 󳨀→ R is a measurable function satisfying the following conditions: α ≤ a (x) ≤ β, (2) for almost every x ∈ Ω, where α and β are positive real constants. H(x, s, ξ) is a Carathéodory-type function satisfying to:

The purpose of the present paper is to study the same kind of lower order terms as in problem (4) in the case of an elliptic operator with unbounded coefficients such as (1).
Another motivation for studying these problem arises from the calculus of variations in the case where 0 ≤  ∈   (Ω) with  ≥ /2 and where  ∈ (0, 1), which is considered by Puel in [5].
We point out that in [6] the authors considered (, ) as a bounded function and where  ∈ (0, 1].The function (, ) : Ω × R → R  2 is symmetric, measurable with respect to  and continuous with respect to  with the following uniform ellipticity condition: for  ∈ Ω, and  ∈ R, We shall prove the following main results on existence and regularity of solutions for problem (1).Theorem 1.Let α = min{1, }.Assuming that the functions  and  satisfy ( 2) and ( 3) then, if  belong to   (Ω), with there exists a distributional solution  ∈  1,1 0 (Ω) of problem (1) such that Furthermore, any solution of the problem (1) belongs to  1 0 (Ω).
In the next result, we consider the case where  has a high summability.
Theorem 2. Let α = min{1, }, and assume that ( 2) and (3) hold true.If  the solution given by Theorem 1 and  belongs to   (Ω), with The rest of the paper is organized as follows: Section 2 is devoted to give some a priori estimates for the approximated problem associated with problem (1); while in Section 3, we give the detailed proofs of Theorems 1 and 2.

The Approximated Problem
In this section, we use the hypotheses (2) and (3) and we suppose that where α = min{1, } holds true.To prove Theorem 1 and Theorem 2, we will use the following approximating problems associated with problem (1): where and By the results of [2,4] there exists a weak solution for every  ∈  1 0 (Ω) ∩  ∞ (Ω).The following lemma will be very useful, as it gives us an a priori estimate on the summability of the solutions to problems (13).

Lemma 3. If
Thus, joining the terms involving the gradient, we get Using ( 12) we deduce that and the Hölder inequality on the right hand side yields which implies (17).16), and we obtain
Lemma 5. Let   be the sequence of solutions to problems (13) and let the function  given by Remark 4. Then   strongly converges to  in   (Ω).Moreover ∇  strongly converges to ∇ in As before, we first choose  such that the second integral is small, uniformly with respect to , and then the measure of  small enough such that the first term is small.The almost everywhere convergence of   to  and Vitali's theorem imply that   strongly converges to  in   (Ω).
For the second convergence, we will follow the same technique as in [1] (see also [7]).Let ℎ,  > 0. In the sequel  will denote a constant independent of , ℎ, .Let us consider  ℎ [  −   ()] as a test function in problems (16).Then, Moreover, thanks to the   (Ω) convergence of   , the second integral in (32) converges (as n diverges) to a positive number.Thus, it yields to thus, yielding where () denote any quantity that vanishes as  diverges.
Hence, by Hölder's inequality, we deduce that Fix, now,  > 0 there exist ℎ 0 such that, for ℎ < ℎ 0 , we have Thanks to the weak convergence of   in  1 0 (Ω) and the absolute continuity of the integral, there exists  0 independent from  such that, for  >  0 , we have In addition, by Dunford Pettis Theorem, we deduce that there exists (ℎ, ) such that, for  > (ℎ, ), we have We can write Using (37), (39), and (40), for ℎ < ℎ 0 and  > (ℎ, ), we have This proves the strong convergence of ∇  to ∇ in  1 (Ω)  .
The following lemma yields some a priori estimate on {  }.
Proof.For every  > 1, we take [ The next result will be used in the proof of Theorem 2.

Proof of the Main Results
We are now ready to prove the main result of this paper.We first observe that condition (9) implies (12).Hence the results of the previous section hold true.In order to prove the result, we have to pass to the limit in (16).To this aim, let  be a function in  1 (R) such that where Observe that, by (9),  is positive, increasing, and it verifies We will use, for  > 0 and  ∈ R, to define a test function.Remark that   ≥ 0, −−1 ≤   () ≤  + 1 and First of all, note that the a.e.convergence of ∇  (see Lemma 5), Remark 6, and (20) imply both that and where  is defined in (59).
The proof of the result will be achieved in two steps.

Finally, interchanging
and − we conclude that