On the Existence Solutions for some Nonlinear Elliptic Problem

where Ω is an open and bounded subset of R (N > p), 0 < α ≤ 1, s ≥ 1 and f be in L(Ω) function. Problem (1.1) has been applied in chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [19,2,8,17] for detailed discussion. In this work, we are dealing with absorption zero order terms, that usually has a regularizing effect on the solutions to (1.1), by starting from measure data for regularity results on the Lebesgue scale (in [6,10]) and on the Marcinkiewicz one ( [4]). We refer the reader to [20,21,12] for another approach using results on elliptic and parabolic problems in the setting of Sobolev spaces. See also [1,3,15,16] for related topics. In [6] the authors studied the regularizing effect of the term u on the solution to the following classical problem


Introduction and Main Result
In this paper, we are concerned with the existence and regularity results for the positive solution to the following problem : where Ω is an open and bounded subset of R N (N > p), 0 < α ≤ 1, s ≥ 1 and f be in L 1 (Ω) function. Problem (1.1) has been applied in chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [19,2,8,17] for detailed discussion. In this work, we are dealing with absorption zero order terms, that usually has a regularizing effect on the solutions to (1.1), by starting from measure data for regularity results on the Lebesgue scale (in [6,10]) and on the Marcinkiewicz one ( [4]). We refer the reader to [20,21,12] for another approach using results on elliptic and parabolic problems in the setting of Sobolev spaces. See also [1,3,15,16] for related topics.
In [6] the authors studied the regularizing effect of the term u s on the solution to the following classical problem when the term u s is added in the left-hand side of (1.2), we obtain the following problem In [7] the authors study the following problem Recently in [14] the authors study the regularity of the solution the following problem and Our purpose is to establish the following result .
The paper is organized as follows: Section 2 is devoted to describing the approximated problems and we prove some properties that we need in the proof of our main results. Finally, Section 3, we shall give the complete proof of Theorem 1.2.

Preliminary results
For a fixed k > 0, we define the truncation functions T k : R → R and G k : R → R as follows T k (s) := max(−k; min(s; k)) and G k (s) := (|s| − k) + sign(s).
We will also use the following functions we will denote with R * the set R\{0}, with R + the set {t ∈ R s.t. t > 0}, with r * the Sobolev conjugate of 1 ≤ r < N , given by N r N −r , and with r ′ = r r−1 the Hölder conjugate of 1 < r < ∞ (if r = 1 we define r ′ = ∞, if r = ∞ we define r ′ = 1). Moreover, if no otherwise Specified, we will denote by c several positive constants whose value may change from line to line and, sometimes, on the same line. These values will only depend on the data (for instance c can depend on Ω, α, s, p) but they will never depend on the indexes of the sequences we will introduce.
On the Existence Solutions for some Nonlinear Elliptic Problem 3

Approximating problems
Let us consider the following approximating problems, There exists u n,k weak solution to (2.2), for each n, k ∈ N fixed (see [ [18], Theorem 2]). Moreover (Ω) as test function in (2.2) and using that G m (u n,k ) and T k (|u n,k | s−1 u n,k ) have the same sign of u n,k , we have taht and so we can proceed as in [22] to end up with u n,k ∈ L ∞ (Ω). Moreover the previous L ∞ estimate is independent from k ∈ N. Now by chossing u n,k as a test function in the weak formulation of (2.2), we obtain u n,k is bounded in W 1,p 0 (Ω) with respect to k for n ∈ N fixed . Since u n,k is bounded in L ∞ (Ω) independently on k, for each n ∈ N fixed we choose k n large enough in order to get the following scheme of approximation where u n ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) is given by u n,kn . As concerns the sign of u n , by chosing u − n := min(u n , 0) ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) as test function in (2.3), we obtain and so that u n ≥ 0 almost everywhere in Ω. Now we prove some local positivity property that will guarantee that the limit of the approximations (2.3) satisfies (1.6).

Proof:
We can prove that the sequence u n is nondecreasing in n ∈ N proceeding precisely as in [ [7], Lemma 2.2], namely taking (u n − u n+1 ) + := max(u n − u n+1 , 0) ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) as test function in the difference between the problem solved by u n and the one solved by u n+1 , so we will omit the details. To prove (2.4), we will instead use that u n ≥ u 1 ∀n ∈ N a.e. in Ω, (2.5) and we will apply the strong maximum principle to u 1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), that solves Then (2.4) follows from (2.5).

A priori estimates
Now we need some compactness results on the sequence of approximating solutions u n , at least up to subsequences.
On the Existence Solutions for some Nonlinear Elliptic Problem Then, we can deduce that and by letting ǫ → 0, we have It follows that Now, if q < p, by thanking to Hölder's inequality with exponents p q and p p−q , we obtain It is not difficult to verify that which achieve the proof of this proposition.

Proof of Theorem 1.2
Proof: Let u n be a solution to (2.2), then it follows from 2.2 that it is bounded in W 1,p 0 (Ω) with respect to n. Hence there exists a function u p ∈ W 1,p 0 (Ω) such that u n , up to subsequences, converges to u p in L r (Ω) for all r < pN N −p and weakly in W 1,p 0 (Ω). proposition 2.2 also gives that f n (x) (|u n | + 1 n ) α is bounded in L 1 loc (Ω) and clearly, |u n | s−1 u n is bounded in L 1 (Ω) with respect to n. Hence one can apply Theorem 2,1 of [5] which gives that ∇u n converges to ∇u p almost everywhere in Ω. Now we prove that u p satisfies 1.7 by passing to the limit in n every term in the weak formulation of (2.2) easily pass to the limit the first term in (2.2) with respect to n; hence we focus on the absorption term u s , which we show to be equi-integrable.Indeed if we test (2.2) with S η,k (u n )(defined in (2.1) where η, k > 0 and we deduce Which, observing that the first term on the left hand side is nonnegative and taking the limit with respect to η → 0,implies Which, since f n converges to f in L r (Ω), r ≥ 1 easily implies that u s n is equi-integrable and so it converges to u s p in L 1 (Ω).This is sufficient to pass to the limit in the second term of the weak formulation of (2.2). For what concerns the right hand side, using (2.4), we find Then, thanks to Lebesgue Theorem, we can pass to the limit also in the right hand side of the distributional formulation of (2.3). This concludes the proof.

Some comments on the regularizing effect
Firstly, it is easy to verify that, if (Ω) is the weaker assumption on the datum in order to find a priori estimates in W 1,p 0 (Ω) for the sequence of approximating solutions to problem below: it follows that, if we add the term u s , with s satisfying (3.2), in the left hand side of (3.3), we find a priori estimates in W 1,p 0 (Ω) for the sequence of approximating solutions also for less regular data. Furthermore, if f ∈ L 1 (Ω) and α < 1, the Sobolev space in which the sequence of approximating solutions to (3.3) is bounded is given by W 1, (Ω) (see [9,11,13] ). It is easy to verify that, if α < 1 and s > N + αp N − p , (3.4) then N (α+1) So we have another regularizing effect of the lower order term u s , with s such that (3.4) holds, on the a priori estimates for the approximating solutions.
Finally we recall that, if f ∈ L 1 (Ω) and s > N N −p , then the sequence of approximating solutions to the following problem:    −div |∇u| p−2 ∇u) + u s = f (x) in Ω, u > 0 in Ω, u = 0 on ∂Ω,