BOUNDED WEAK SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS

In this work, we are concerned with a class of elliptic problems with both absorption terms and critical growth in the gradient. We suppose that the data belong to L m () with m > n/2 and we prove the existence of bounded weak solutions via L 1 - estimates. A priori estimates and Stampacchia's L 1 -regularity are our main ingredient.


Introduction
In this work, we intend to study the Dirichlet problem for some nonlinear elliptic equations whose model example is: where Ω is a bounded open set in R N , N > 2, ∆ denotes the Laplace operator and β is a continuous nonincreasing real function , with β ∈ L 1 (R).The real function a (x) is nonnegative and bounded in L ∞ (Ω).Under suitable conditions on the data, we shall study existence and regularity of solutions for problem (P ).
These kind of problems have been treated in a large literature starting from the classical references [18] and [19].Later, many works have been devoted to elliptic problems with lower order terms having quadratic growth with respect to the gradients (see e.g.[8], [9], [13], [15], [16], [17], [22] and the references therein).
The general problem (P ), though being physically natural, does not seem to have been studied in the literature.So for special situations, in the case where a = 0, β is constant and f = 0, this equation may be considered as the stationary part of equation Key words and phrases.Nonlinear elliptic equations, critical growth, absorption terms, existence, a priori estimates, weak solutions.EJQTDE, 2009 No. 10, p. 1 which appears in the physical theory of growth and roughening of surfaces.It is well known as the Kardar-Parisi-Zhang equation (see [16]).It presents also the viscosity approximation as → +∞ of Hamilton-Jacobi equations from stochastic control theory (see [21]).
For the simpler case where a = 0, β is a constant (we can assume β = 1 without loss of generality) and f ∈ L N 2 ; that is when (P ) of the form the problem has been studied in [17], where the change of variable v = e u − 1 leads to the following problem , it is proved there that (1.1) admits a unique solution in W 1,2 0 (Ω).
In the case where a = 0, f ∈ L q with q > N 2 , and β is a continuous nonnegative function satisfying supplementary conditions according to each situation, for instance 2 ) , a priori estimates have been proved in [1] and [6] to obtain existence and regularity results, while uniqueness have been shown in [2].
In this paper, to prove existence of bounded weak solutions for (P ), we assume that f ∈ L m , m > N 2 , β is a continuous real function nonincreasing with β ∈ L 1 (R) and a is a nonnegative bounded real function.We shall obtain a solution by an approximating process.Using a priori estimates and Stampacchia's L ∞ -regularity results we shall show that the approximated solutions converges to a solution of problem (P ).

Preliminaries and main results
In this section, we present some notations and assumptions.We also recall some concepts and results which will be used in our further considerations.We will refer the reader to the corresponding references.
EJQTDE, 2009 No. 10, p. 2 Throughout this paper Ω will denotes a bounded open set in R N with N > 2. We denote by c a positive constant which may only depend on the parameters of our problem, its value my vary from line to line.
For 1 ≤ q ≤ N we denote q * = N q N −q .Moreover, we denote N = N N −1 and its Sobolev conjugate by N 0 = N N −2 .
For k > 0 we define the truncature at level ±k as We also consider G k (s) = s−T k (s) = (|s| − k) + sign (s) .We introduce T 1,2 0 (Ω) as the set of all measurable functions u : For a measurable function u belonging to T 1,2 0 (Ω) , a gradient can be defined as a measurable function which is also denoted by ∇u and satisfies ∇T k (u) = ∇u χ [|u|<k] for all k > 0 (see e.g.[3]).
We are going to investigate the solution of the following nonlinear elliptic problem where Ω denotes a bounded open set in R N with N > 2. u denote a real function depending on x in R N .
We denote by γ the real function We assume that r > 1, and that Both functions β and a have to satisfy certain structural assumptions which are described by: (A) There exists a 0 such that a ≥ a 0 > 0 a.e in Ω and a ∈ L ∞ (Ω).(B) The real function β is continuous nonincreasing with β ∈ L 1 (R) .
Without loss of generality we assume β (0) = 0 EJQTDE, 2009 No. 10, p. 3 By a weak solution of problem (P ) we mean a function u, such that both functions β(u) |∇u| 2 and a(x)u|u| r−1 are integrable, and the following equality holds Then, under the assumptions (A) and (B) the problem (P ) has at least one solution which belongs to W 1,2 (Ω) ∩ L ∞ (Ω).

Fundamental estimates
3.1.Estimates on general problem.In this sections we prove some basic estimate for regular elliptic problem.The main tools for proving theorem 2.1 are a priori estimates together with compactness arguments applied to a sequences of bounded approximating solutions.We shall study the nonlinear elliptic equation Where B(x, s, ξ) = b (s, ξ) − a (x, s) |s| r−1 ; a (., .)and b (., .)are two functions satisfying the following hypothesis: (a 1 ) a (x, s) : Ω × R → R is measurable in x ∈ R N for any fixed s ∈ R and continuous in s for a.e.x. (a 2 ) There exists a constant c > 0 such that for all s and almost every x a (x, s) ≥ a (x) s + c.
(a 3 ) For any α > 0 the function Let us note that if u is a weak solution of (3.1), then it satisfy the following equality We will now prove the following basic results.If u is a weak solution of (3.1) we denote u k = T k (u).Then we have the following estimates: Proof.We consider for m > 1 the following function Taking ψ m (u) as test function in (3.1),where m is such that 0 < m − 1 < r −1, we obtain

Then we have
Since s|s| r−1 ψ(s) is nonnegative, then using the fact that EJQTDE, 2009 No. 10, p. 5 We get for all u Then we obtain Let u be a weak solution of problem (3.1).Then, for N > 2 and r > 1, one has ∇u ∈ L q (Ω) for any q, 1 ≤ q < N = N N − 1 and u ∈ L q * (Ω) where q * = qN N − q . Proof.
Proof.We define the following functions Taking ψ k,h (u) as a test function in (3.1), we obtain Applying monotone convergence theorem, we have Letting h tend to infinity in (3.10) and applying Lebesgue's dominated convergence theorem, we obtain Then, we obtain Therefore, we have which implies that (3.8) is satisfied.
To prove the second assertion, let us take as a test function in (3.1), we obtain The monotone Convergence Theorem yields Now, applying Lebesgue's dominated convergence theorem in (3.11), we obtain After simplifications we have Therefore, we get Finally, by Fatou's lemma, we deduce that Taking e γ(u h ) ψ k,h (u) as test function in (3.1), we obtain From Monotone Convergence Theorem, we have Letting h tend to infinity in (3.12) and applying Lebesgue's dominated convergence theorem, we obtain Then, we have Therefore from Fatou's lemma, we obtain and then we have Finally, this implies that (3.13)

3.2.
Estimates on the approximating solutions.This section is devoted to study the limiting process of the approximating problem.We consider the following sequence of problems which we denote by (P n ) : under the assumption a n and b n are two sequences of functions defined by From standard result by Leray and Lions (see e.g.[19]) there exist weak solutions, for problem (3.14), which we denote by It yields that, (3.15) EJQTDE, 2009 No. 10, p. 11 Hence the previous result of the precedent section can be applied.Using lemma (3.3) we deduce that there exist a constant c such that Applying Hölder's inequality, we obtain . Using Sobolev's imbedding theorem, we obtain for .
In this stage by using Stampacchia's L ∞ -regularity procedure (see [24]) we obtain that u n is bounded uniformly in L ∞ (Ω).That is where c > 0 is a constant that only depends on the parameters of the problem.
Using lemma 3.3 we obtain (3.16) After substraction, we obtain The equi-integrability of f n and B n (x, u n , ∇u n ) gives Let us now observe that by Hölder's inequality, we have (3.20)

Existence and regularity results
Let u n be the solution of the approximating problem.Then for all ψ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , we have From the construction of f n we have for n tending to + ∞.
From lemma (3.2) the solution u n is bounded independently on n in W 1,q (Ω) , for any q, 1 ≤ q < q 0 .Then, up to a subsequence ,that we denote again by u n , there exist u ∈ W 1,q (Ω) , for any q, 1 ≤ q < q 0 , such that u n converge to u weakly in W 1,q (Ω) , for any q, 1 ≤ q < q 0 .From Rellich-Kondrachov's theorem we have the almost every where convergence in Ω.That is u n → u weakly in W 1,q (Ω) for any q, 1 ≤ q < q 0 .From lemma 3.5 we have up to a subsequence u n , that (4.3) ∇u n → ∇u almost every where in Ω .
Since ∇u n is bounded in L q (Ω) for any q, 1 ≤ q < N , we have ∇u n → ∇u in L q (Ω) for any q, 1 ≤ q < N , and then we conclude that ∆u n → ∆u in L 1 (Ω) .Which conclude to the desired convergence result.

(4. 1 )
u n → u almost every where in Ω .an (x, u n ) → a(x, u) almost every where in Ω. Taking into account the equi-integrability of u n in L r (Ω) , it follows that of a n (x, u n )|u n | r−1 in L 1 (Ω) .Hence, we have(4.2) a n (x, u n )|u n | r−1 → a(x, u)|u| r−1 in L 1 (Ω) .