Existence of renormalized solutions for some quasilinear elliptic Neumann problems


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                        <jats:tex-math>\left\{ {\matrix{   {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill  \cr    {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill  \cr  } } \right.</jats:tex-math>
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                  </jats:disp-formula> in the anisotropic variable exponent Sobolev spaces, where <jats:italic>A</jats:italic> is a Leray-Lions operator and <jats:italic>g</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>u</jats:italic>, ∇<jats:italic>u</jats:italic>), <jats:italic>f</jats:italic> (<jats:italic>x</jats:italic>, <jats:italic>u</jats:italic>, ∇<jats:italic>u</jats:italic>) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.</jats:p>


Introduction
Let Ω be a bounded open domain of I R N (N ≥ ) with smooth boundary ∂Ω. The study of various mathematical problems with isotropic variable exponent has received considerable attention in recent years. These problems are interesting in applications and raise many di cult and interesting mathematical problems. For example, in [20] the authors established the following stationary p(x)-curl systems arising in electromagnetism: |∇ × u| p(x)− × u × n = , u · n = on ∂Ω, (1.1) In addition in [8], Boccardo and Gallouët have considered the elliptic problem −div a(x, u, ∇u) = f in Ω, where a(x, u, ∇u) is a Carathéodory function, and the right-hand side f is a bounded Radon measure. They have proved the existence and some regularity results. For more results, we refer the reader to [7,13,14]).
In [5], Ben Cheikh Ali and Guibé have studied some quasilinear elliptic equations of the type, λ(x, u) − div (a(x, ∇u) + Φ(x, u)) = f in Ω, (a(x, ∇u) + Φ(x, u)) · n = on Γn , u = on Γ d , (1.2) where Au = −div (a(x, ∇u)) is a Leray-Lions type operator, and λ(x, s), Φ(x, s) are Carathéodory functions. They have proved the existence and uniqueness of renormalized solutions for this problem under some growth conditions. In the recent years, the interest of scientists has turned towards anisotropic elliptic and parabolic equations. This special interest mainly comes from their applications to the mathematical modeling for some physical processes in an anisotropic continuous medium (see. [2,19]). Di Nardo, et al. have considered in [11] the following nonlinear elliptic problems of the type with f ∈ L (Ω) and g ∈ Π N i= L p ′ i (Ω). They have proved the existence and uniqueness of renormalized solution in the anisotropic Sobolev spaces. In [4], the authors have studied the nonlinear elliptic Dirichlet problem where g(x, u, ∇u) and ϕ(u) are Carathéodory functions, and the data f is assumed to be in L (Ω). They have proved the existence of an entropy solution in the anisotropic variable exponent Sobolev spaces.
In this paper, we will study the existence of renormalized solutions for the following nonlinear elliptic problem   

Lemma 2.1. We have the following continuous and compact embedding
The proof of this lemma follows from the fact that the embedding W , p(·) (Ω) → W ,p (Ω) is continuous, and in view of the compact embedding theorem for Sobolev spaces.
De nition 2.1. Let k > , the truncation function T k (·) : I R → I R is given by and we de ne The proof of the Proposition 2.1 follows the usual techniques developed in [7] for the case of Sobolev spaces. For more details concerning the anisotropic Sobolev spaces, we refer the reader to [6] and [11].

Essential Assumptions
Let Ω be a bounded open set of I R N (N ≥ ) with smooth boundary ∂Ω.
We consider a Leray-Lions operator A that acts from W , p(·) (Ω) into its dual (W , p(·) (Ω)) ′ , de ned by the formula where a i (x, s, ξ ) : Ω × I R × I R N −→ I R N are Carathéodory functions (i.e. measurable with respect to x in Ω for every (s, ξ ) in I R × I R N and continuous with respect to (s, ξ ) in I R × I R N for almost every x in Ω) and verifying the following conditions : for almost every x ∈ Ω and all (s, ξ ) ∈ I R × I R N , where K i (x) ∈ L p ′ i (·) (Ω) are some positive functions and α, β > . As a consequence of ( . ) and the continuity of the functions a i (x, s, ·) with respect to ξ , we have The lower order terms g(x, s, ξ ), f (x, s, ξ ) : Ω × I R × I R N −→ I R are two Carathéodory functions which satisfy the following growth conditions (3.5) and where g (·) and f (·) are assumed to be two positive measurable functions in L (Ω) and d(·) : I R → I R + is a decreasing function that belongs to L (I R) ∩ L ∞ (I R) and we assume that ≤ r (x) < q(x) − and ≤ r i (x) < a. e. in Ω for any i = , , . . . , N. We consider the strongly nonlinear elliptic Neumann problem

Existence of renormalized Solutions
Let  such that u verifying the following equality for every φ ∈ W , p(·) (Ω) ∩ L ∞ (Ω) and for any smooth function S(·) ∈ W ,∞ (I R) with a compact support.

Proof of the Theorem 4.1
Step 1 : Approximate problems.
Let n ∈ I N * be large enough, and let < θ < Having in mind ( . ), ( . ) and ( . ), we conclude that For the second and third terms on the right-hand side of ( . ), using Young's inequality we have We have also (4.10) By combining ( . ) and ( . ) − ( . ), we conclude that Moreover, thanks to ( . ) we have On the one hand, thanks to Young's inequality we have where C is a positive constant that does not depend on k and n. Thus (T k (un))n is bounded in W , p(·) (Ω) uniformly in n, and there exists a subsequence still denoted (T k (un))n and v k ∈ W , p(·) (Ω) such that → v k strongly in L (Ω) and a.e in Ω. (4.15) Moreover, thanks to ( . ) it follows that and since q − > , it follows necessary that Now, we will show that (un)n is a Cauchy sequence in measure. For all λ > , we have Let ε > , using ( . ) we may choose k = k(ε) large enough such that On the other hand, thanks to ( . ) we have T k (un) → v k in L (Ω) and a.e. in Ω. Thus, we can assume that (T k (un))n is a Cauchy sequence in measure, and for all k > and ε, λ > , there exists n = n (k, ε, λ) such that By combining ( . ) − ( . ), we conclude that for any n, m ≥ n (ε, λ). It follows that (un)n is a Cauchy sequence in measure, then converges almost everywhere, for a subsequence, to some measurable function u. We conclude that (4. 19) In view of Lebesgue's dominated convergence theorem, we obtain Moreover, thanks to ( . ) we conclude that T k (un) k → strongly in L (Ω) and weak * in L ∞ (Ω).
Step 3 : Some a priori estimates.
Let h ≥ , in this section we will prove that : as a test function in the approximate problem ( . ), we have: In view of ( . ), ( . ), and ( . ), we conclude that (4.23) Concerning the two last terms on the right-hand side of ( . ), similarly to ( . ) and ( . ) we have   Step 4 : The equi-integrability of (|T n (u n )| q(x)− T n (u n )) n and ( n |u n | p− u n ) n . Firstly, thanks to ( . ) we have (4.34) It's clear that, there exists µ(η) > , such that for all E ⊂ Ω, we have Thus, the sequence (|Tn(un)| q(x)− un)n and ( n |un| p− un)n are equi-integrable, which conclude the proof of ( . ).

Step 4: Strong convergence of truncations.
Let h ≥ k ≥ , we denote by ε i (n), i = , , . . . , various real-valued functions of real variables that converge to as n tends to in nity, similarly for ε i (h) and ε i (n, h). In this step, we will prove the convergence of the sequence (D i un)n to D i u almost everywhere in Ω, for any i = , . . . , N. We set where γ = (θ − ) + d(·) L ∞ (I R) α , note that ψ ′ (s) − γ|ψ(s)| ≥ ∀s ∈ I R.
By taking v = ψ(T k (un) − T k (u))S h (un)φ(un)e B(|un|) as a test function in the approximate problem ( . ), we

x, Tn(un), D i un)D i un|ψ(T k (un) − T k (u))|φ(un)e B(|un|) dx.
In view of ( . ) and ( . ), we conclude that  For the rst term of the right-hand side of ( . ), we have ψ(T k (un) − T k (u)) weak * in L ∞ (Ω), and since f (x) and g (x) belong to L (Ω), it follows that

N i= Ω a i (x, Tn(un), D i un)(D i T k (un) − D i T k (u))ψ ′ (T k (un) − T k (u))S h (un)φ(un)e B(|un|) dx
Similarly, since r (x) ≤ q(x) − a. e. in Ω and thanks to ( . ), we conclude that In addition, Concerning the last term of the right-hand side of ( . ) and using ( . ), we obtain  h).

x, Tn(un), D i un)(D i T k (un) − D i T k (u))ψ ′ (T k (un) − T k (u))S h (un)φ(un)e B(|un|) dx
We have a i (x, s, ) = , and S h (un) = on the set {|un| ≤ h}. Moreover, ψ(T k (un) − T k (u)) have the same sign as un on the set {|un| > k}. Thus, in view of ( . ) and using Young's inequality, one has h). h).
Concerning the second term of the left-hand side of ( . ), we have (a i (x, T h (un), D i T h (un)))n is bounded in For the second term of the right-hand side of ( . ), in view of ( . ) we have T k (un) → T k (un) in L p i (·) (Ω), then a i (x, T k (un), ∇T k (u)) → a i (x, T k (u), ∇T k (u)) strongly in L p ′ i (·) (Ω), and since D i T k (un) tends to D i T k (u) weakly in L p i (·) (Ω), we obtain (4.49) Similarly, we have Concerning the last term of the left-hand side of ( . ), we have (|a i (x, T k (un), D i T k (un))|)n is bounded in (Ω), and since D i T k (u)|ψ(T k (un) − T k (u))| tends strongly to in L p i (·) (Ω) for any i = , . . . , N, it follows that (4.52) In view of Lebesgue dominated convergence theorem, we have T k (un) → T k (u) strongly in L p (Ω). Thus, by letting n then h tend to in nity, we deduce that Step 4 : The equi-integrability of (gn(x, un , ∇un))n and (fn(x, Tn(un), ∇un))n.
Step 5 : Passage to the limit    It remains to show that Gn is pseudo-monotone. Let (u k ) k∈I N be a sequence in W , p(·) (Ω) such that (Ω), Gn u k χn in (W , p(·) (Ω)) ′ , lim sup k→∞ Gn u k , u k ≤ χn , u .
In view of the compact embedding W , p(·) (Ω) → → L p (Ω), there exists a subsequence still denoted (u k ) k∈I N * such that u k → u in L p (Ω). As (u k ) k∈I N is a bounded sequence in W , p(·) (Ω), using the growth condition ( . ), it's clear that the sequence (a i (x, Tn(u k ), ∇u k )) k∈I N * is bounded in L p ′ i (·) (Ω), and there exists a function φ i ∈ L p ′ i (·) (Ω) such that Similarly, since (Hn(x, u k , ∇u k )) k∈I N * and (fn(x, Tn(u k ), ∇u k )) k∈I N * are bounded in L p ′ (Ω), then there exists two measurable functions ψn and ϕn in L p ′ (Ω), such that gn(x, u k , ∇u k ) −→ ψn and fn(x, Tn(u k ), ∇u k ) −→ ϕn weakly in L p ′ (Ω).  and gn(x, u k , ∇u k ) gn(x, u, ∇u) and fn(x, Tn(u k ), ∇u k ) fn(x, Tn(u), ∇u) in L p ′ (Ω), having in mind ( . ), we obtain χn = Gn u. Thus, the proof of the Lemma 4.1 is concluded.
As models example of applications for problem ( . ), we state the following model: