Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 105, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR DEGENERATE ELLIPTIC EQUATIONS IN WEIGHTED SOBOLEV SPACES AHARROUCH BENALI, BENNOUNA JAOUAD Abstract. We study the existence of solutions for the nonlinear degenerated elliptic problem − div a(x, u,∇u) = f in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in RN , N ≥ 2, a is a Carathéodory function having degenerate coercivity a(x, u,∇u)∇u ≥ ν(x)b(|u|)|∇u|p, 1 < p < N , ν(·) is the weight function, b is continuous and f ∈ Lr(Ω). 1. Introduction In this article we prove the existence of solutions for some nonlinear elliptic equations with principal part having degenerate coercivity. The model case is −div (ν(·)|∇u|p−2∇u (1− |u|)α ) = f in Ω, u = 0 on ∂Ω, (1.1) with Ω a bounded open subset of RN , N ≥ 2, p > 1, α ≥ 0, ν(·) is weight function defined on Ω and f a measurable function on whose summability we will make different assumptions. It is clear from the above example that the differential operator is defined on W 1,p 0 (Ω, ν), but that it may not be coercive on the same space as u near to 1. Because of this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied. We consider the nonlinear degenerate elliptic problem A(u) = −div(a(x, u,∇u)) = f in Ω, u = 0 on ∂Ω, where, Ω is a bounded open subset of RN , N ≥ 2, 1 < p < N , and a : Ω×R×RN → RN is a Carathéodory function, such that the following assumption holds a(x, s, ξ).ξ ≥ ν(x)b(|s|)|ξ|p, for almost every x in Ω, for every (s, ξ) ∈ R× RN , with b(|s|) = 1/(1− |s|)α, (1.2) 2010 Mathematics Subject Classification. 35J70, 46E30, 35J85. Key words and phrases. Nonlinear degenerated elliptic operators; weighted Sobolev space; monotony and rearrangement methods. c©2020 Texas State University. Submitted December 4, 2019. Published October 12, 2020. 1 2 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 under various assumptions on f . As stated before, due to assumption (1.2), the operator A may not be coercive on W 1,p 0 (Ω, ν), when the solutions approach the critical values ±1. To overcome this difficulties, we will reason by approximation, cutting by means of truncatures the nonlinearity a(x, s, ξ) in order to get coercive differential operator on W 1,p 0 (Ω, ν), and give a sense to the equation when the solutions near to ±1 and to manage the set {x ∈ Ω : |u(x)| = 1}. For the case ν(·) being a constant, the existence of solutions to problem (1.1) is proved in [11], when f a measurable function on whose summability have make different assumptions, the analogous problems was treated by many other authors. See, for example, [3, 4, 9, 10, 8] where problems such as − div ( 1 (1± |u|)α |∇u|p−2∇u ) = f, are considered. This article is organized as follows: In section 2, we recall some preliminaries on Weighted Sobolev spaces and properties of rearrangement. In section 3, we first prove the propositions that we will use to prove some a priori estimates of the solutions, then we prove the existence of weak and entropy solution with respect to the summability of f . 2. Preliminaries Assumptions. Let b : [0, l[→ (0,∞), with l > 0, be a continuous function such that lim s→l− b(s) = +∞ . (2.1) We define A(s) = ∫ s 0 b(t) 1 p−1 dt, for s ∈ [0, l), A(l−) = lim s→l− ∫ s 0 b(t) 1 p−1 dt = +∞. We study Dirichlet problems of the form −div a(x, u,∇u) = f in Ω, u = 0 on ∂Ω, (2.2) where Ω is a bounded open set in RN , N ≥ 2, 1 < p < N , and a : Ω×(−l, l)×RN → RN , is a Carathéodory function and ν : Ω→ R+ satisfies the following assumptions: a(x, s, ξ) · ξ ≥ b(|s|)ν(x)|ξ|p, ν ∈ Lr(Ω), r ≥ 1, ν−1 ∈ Lt(Ω), t ≥ N, 1 + 1 t < p < N(1 + 1 t ). (2.3) for a.e. x ∈ Ω, for all s ∈ (−l, l) and all ξ ∈ RN ; |a(x, s, ξ)| ≤ ν(x)[h(x) + b(|s|)|ξ|p−1], (2.4) for a.e. x ∈ Ω, for all s ∈ (−l, l), for all ξ ∈ RN , and h ∈ Lp′(Ω, ν); (a(x, s, ξ)− a(x, s, ξ′)) · (ξ − ξ′) > 0, (2.5) for a.e. x ∈ Ω, for all s ∈ (−l, l) and all ξ ∈ RN , ξ 6= ξ′. Moreover, f is a measurable function on whose summability we will make several assumptions. For stating existence results in the next section, we need some classes of solutions. EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 3 Definition 2.1. We say that u ∈W 1,p 0 (Ω, ν) is a weak solution to problem (2.2) if∫ Ω a(x, u,∇u) · ∇ϕdx = ∫ Ω fϕ dx, ∀ϕ ∈W 1,p 0 (Ω, ν). (2.6) Definition 2.2. A measurable function u ∈ W 1,p 0 (Ω, ν) is an entropy solution to problem (2.2) if |u| ≤ l a.e. in Ω (2.7) and for all 0 < k < l,∫ Ω a(x, u,∇u) · ∇Tk(u− ϕ) dx ≤ ∫ Ω fTk(u− ϕ) dx, (2.8) for any ϕ ∈W 1,p 0 (Ω, ν) ∩ L∞(Ω) such that ‖ϕ‖L∞(Ω) < l − k. Weighted Sobolev spaces. Let 1 ≤ p < N , and ν : Ω→ R be a weight function, i.e. a function which is measurable and positive almost everywhere in Ω. The weighted Lebesgue spaces Lp(Ω, ν) is defined as Lp(Ω, ν) = { u : measurable, real-valued function, ∫ Ω ν(x)|u(x)|p dx <∞ } . which is a Banach space (uniformly convex and hence reflexive if p > 1) equipped with the norm ‖u‖Lp(Ω,ν) = (∫ Ω ν(x)|u(x)|p dx )1/p . By W 1,p(Ω, ν) we denote the completion of the space C1(Ω) with respect to the norm ‖u‖W 1,p(Ω,ν) = ‖u‖Lp(Ω,ν) + ‖|∇u|‖Lp(Ω,ν). Moreover we denote by W 1,p 0 (Ω, ν) the closure of C1(Ω) in W 1,p(Ω, ν) which is normed by ‖u‖W 1,p 0 (Ω,ν) = ‖|∇u|‖Lp(Ω,ν). We denote by W−1,p′(Ω, 1/ν) the dual space of W 1,p 0 (Ω, ν); for more details see [16]. Rearrangement properties. We recall some definitions about decreasing re- arrangement of functions. Let Ω be a bounded open set of RN and u : Ω → R a measurable function. Definition 2.3. The distribution function of u is defined as µu(t) = |{x ∈ Ω : |u(x)| > t}|, t ≥ 0. The function µu is decreasing and right continuous. Definition 2.4. The decreasing rearrangement of u is defined as u∗(s) := sup{t ≥ 0 : µu(t) > s}, s ≥ 0. The function u∗ is the generalized inverse of µu. We recall that∫ Ω |u|p dx = p ∫ +∞ 0 tp−1µu(t)dt, for p ≥ 1 . (2.9) Then the Lp-norm, for 1 ≤ p < +∞, is invariant with respect to rearrangement, that is, ‖u‖Lp(Ω) = ‖u∗‖Lp[0,|Ω|]. 4 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 Moreover, if u ∈ L∞(Ω), by definition u∗(0) = ess supΩ |u|. For more details about rearrangements we refer the reader to [6, 13, 18]. We recall that a measurable function u : Ω→ R belongs to the Marcinkiewicz space Mp(Ω) (or weak-Lp) if the distribution function µu satisfies µu(t) ≤ c tr , ∀t > 0, for some constant c. We observe that the above condition is equivalent to u∗(s) ≤ c s1/r , ∀s > 0, and we define ‖u‖Mp(Ω) = sup s>0 u∗(s)s 1/r. We observe that the Marcinkiewicz spaces are “intermediate” between Lebesgue spaces. Indeed, it is not difficult to show that Lp(Ω) ⊂Mp(Ω) ⊂ Lq(Ω), for 1 ≤ q < p. Now, we give a sense to the gradient of a function u ∈ L1(Ω) such that the truncates of u are Sobolev functions. Lemma 2.5 ([7]). For each measurable function u : Ω→ R such that for every k > 0 the truncated function Tk(u) belong to W 1,1 loc (Ω), there exists a unique measurable function v : Ω→ RN such that ∇Tk(u) = vχ|u| 0 and 1 ≤ γ < +∞. Let ψ a non-negative measur- able function on (0,+∞). Then the∫ +∞ 0 ( t−λ ∫ t 0 ψ(s)ds )γ dt t ≤ c ∫ +∞ 0 (t1−λψ(t))γ dt t , , (2.11)∫ +∞ 0 ( tλ ∫ +∞ t ψ(s)ds )γ dt t ≤ c ∫ +∞ 0 (t1+λψ(t))γ dt t . (2.12) Also we shall need the following proposition of weak approximation (see [5]). Let u ∈W 1,p 0 (Ω), and for s ∈ [0, |Ω|], let G(s) be a measurable subset of Ω such that |G(s)| = s s1 < s2 ⇒ G(s1) ⊂ G(s2) G(s) = {x ∈ Ω : |u(x)| > t} if s = µ(t). For a given a function ϕ ∈ L1(Ω), we set φ(s) = d ds ∫ G(s) ϕ(x) dx. EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 5 Lemma 2.8 ([5]). If ϕ ∈ Lp(Ω) with p > 1, then there exists a sequence (ϕ(s))n, such that ϕ∗n(s) = ϕ∗(s) and ϕn ⇀ φ weakly in Lp(0, |Ω|). 3. Main result The following Proposition gives a sufficient condition for the gradient of a func- tion to belong to some Marcinkiewicz space, These are the generalized results of [7] in the Weighted Sobolev spaces W 1,p 0 (Ω, ν). Proposition 3.1. Let 1 < p < N , and u ∈ T 1,p 0 (Ω, ν) be such that∫ {|u| 0. Then u ∈ Mp1(Ω) where p1 = p](1− λ/p). More precisely, there exists a c such that meas{|u| > k} = meas{x ∈ Ω : |u(x)| > k} ≤ ck−p1 . Proof. For k > 0, from (2.3), we have ‖Tk(u)‖p] ≤ c1‖∇Tk(u)‖Lp(ν) ≤ c1kλ/p. For 0 < ε ≤ k, we have {x ∈ Ω : |u| > ε} = {x ∈ Ω : |Tk(u)| > ε}. Hence meas{|u| > ε} ≤ ( ‖Tk(u)‖p] ε )p ] ≤ c1kλp ]/pε−p ] . Setting ε = k, we obtain meas{|u| > ε} ≤ c1k−p1 , where p1 = p](1− λ/p). � Proposition 3.2. Let 1 < p < N , and u ∈ T 1,p 0 (Ω, ν) be such that∫ {|u| 0. Then ν1/p∇u ∈Mp2(Ω) where p2 = pp1/(λ+ p1). More precisely, there exists a c such that meas{ν1/p|∇u| > h} ≤ ch−p2 . Proof. For k, h > 0. Set φ(k, α) = meas{ν(x)|∇u|p > α, |u| > k}. From Proposi- tion 3.1 we have φ(k, 0) ≤ c1k−p1 . Using that the function α 7→ φ(k, α) is non-increasing, for k, λ > 0 we obtain φ(0, α) ≤ 1 α ∫ α 0 φ(0, s)ds = 1 α ∫ α 0 φ(0, s) + φ(k, 0)− φ(k, 0)ds ≤φ(k, 0) + 1 α ∫ α 0 φ(0, s)− φ(k, 0)ds ≤φ(k, 0) + 1 α ∫ α 0 φ(0, s)− φ(k, s)ds. (3.1) Since φ(0, s)− φ(k, s) = meas{ν(x)|∇u|p > s, |u| < k} we have 1 α ∫ α 0 φ(0, s)− φ(k, s)ds = 1 α ∫ |u| k} ≤ ch−pp1/(λ+p1) � 3.1. A priori estimate. Let ε be positive and sufficiently small. We consider the problem −div aε(x, uε,∇uε) = fε in Ω, uε = 0 on ∂Ω, (3.3) where aε(x, s, ξ) = a(x, Tl−ε(s), ξ), with x ∈ Ω, s ∈ R and ξ ∈ RN and fε ∈ L∞(Ω). We use some classical results (see, for example [1, 2]) to assure that problem (3.3) has at least one solution uε ∈ W 1,p 0 (Ω, ν) ∩ L∞(Ω). Then, we define bε(t) = b(Tl−ε(t)) for all t ∈ [0,+∞), and Aε(s) = ∫ s 0 bε(r) 1/(p−1)dr. First, we prove an integral inequality for weak solutions of problem (3.3). Proposition 3.3. Let uε be a weak solution of (3.3). Then Aε(u ∗ ε(s)) ≤ CN ∫ |Ω| s r−p ′/N ′ [D(r)]p ′/p (∫ r 0 f∗ε (σ)dσ )p′/p dr, s ∈ [0, |Ω|], (3.4) where D : [0, |Ω|]→ R is a measurable function such that∫ |uε|>y ν−t(x) dx = ∫ µ(y) 0 (D(r))t dr. Proof. Let φ = Th(uε − Tθ(uε)) be a test function in (3.3). Then we have 1 h ∫ θ<|uε|≤θ+h b(|uε|)ν(x)|∇uε|p dx ≤ ∫ |uε|>θ |f | dx Applying Hardy-Littlewood inequality and passing to the limit on h to 0, we obtain b(θ) ( − d dθ ∫ |uε|>θ ν(x)|∇uε|p dx ) ≤ ∫ µuε(θ) 0 f∗ε (s)ds. (3.5) On the other hand by Hölder inequality, we obtain − d dθ ∫ |uε|>θ |∇uε| dx ≤ ( − d dθ ∫ |uε|>θ ν(x)|∇uε|p dx )1/p × ( − d dθ ∫ |uε|>θ ν−p ′/p(x) dx )1/p′ ≤ ( − d dθ ∫ |uε|>θ ν(x)|∇uε|p dx )1/p × ( − d dθ ∫ |uε|>θ ν−t(x) dx )1/r1p ′ (−µ′uε(θ)) 1/r2p ′ . (3.6) where 1/r1 + 1/r2 = 1 and p′r1/p = t. By Lemma 2.8, since ν−1 ∈ Lt(Ω), t > 1 there exists D ∈ Lt([0, |Ω|]) such that − d dθ ∫ |uε|>θ ν−t(x) dx = −µ′uε(θ)[D(µuε(θ))] t. EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 7 Then inequality (3.6), becomes − d dθ ∫ |uε|>θ |∇uε| dx ≤ ( − d dθ ∫ |uε|>θ ν(x)|∇uε|p dx )1/p × ( (−µ′uε(θ)) 1/p′ [D(µuε(θ))] t/r1p ′ ) . (3.7) From isoperimetric inequality and Fleming-Rishel formula (see [15]), it follows that CNb(θ) 1/p(µuε(θ)) 1/N ′ ≤ ( − d dθ ∫ |uε|>θ ν(x)|∇uε|p dx )1/p × ( (−µ′uε(θ)) 1/p′ [D(µuε(θ))] t/r1p ′ b(θ)1/p ) , (3.8) which by (3.5) gives b(θ)1/(p−1) ≤ CN (µuε(θ)) −p′/N ′(−µ′uε(θ))[D(µuε(θ))] t/r1 (∫ µuε(θ) 0 f∗ε (s) ds )p′/p integrating between 0 and u∗(s) we obtain A(u∗(s)) ≤ CN ∫ u∗(s) 0 [ (µuε(θ)) −p′/N ′(−µ′uε(θ))[D(µuε(θ))] t/r1 × (∫ µuε (θ) 0 f∗ε (s)ds )p′/p] dθ, (3.9) which gives the results. � Remark 3.4. Since 1 + 1 t < p < N(1 + 1 t ), and t ≥ N/p, we have qp′/p ≥ 1 and q/r′1 ≥ 1, where r1 = t(p − 1), which allows us to apply the Proposition 2.11 and Proposition 2.12 to prove estimation (3.10) and (3.11), below. Proposition 3.5. Let uε be a solution of (3.3). (a) If 1 < r < tN/(tp−N), then ‖(Aε(|uε|))q‖L1(Ω) ≤ c‖f‖ qp′/p Lr(Ω); (3.10) where q = rtN(p− 1)/(t(N − rp) + rN). (b) If r = 1, then ‖Aε(|uε|)‖MNt(p−1)/(N+t(N−p)) ≤ c‖f‖p ′/p L1(Ω)‖D‖ p′/p Lt[0,|Ω|] . (3.11) Proof. Case 1 < r < tN/(tp − N). Let us observe that Aε being monotone, by Proposition 3.3, properties of rearrangements, (2.12) and (2.11), we obtain ‖(Aε(|uε|))q‖L1(Ω) ≤ CN ∫ +∞ 0 [ ∫ |Ω| s r−p ′/N ′ [D(r)]p ′/p (∫ r 0 f∗(σ)dσ )p′/p dr ]q ds ≤ CN ∫ +∞ 0 [ ∫ |Ω| s r− p′r′1 N′ (∫ r 0 f∗(σ)dσ ) p′r′1 p dr ] q r′1 ds ≤ CN ∫ +∞ 0 [ s r′1 q ∫ |Ω| s r− p′r′1 N′ (∫ r 0 f∗(σ)dσ ) p′r′1 p dr ] q r′1 ds s ≤ CN ∫ +∞ 0 [ s ( r′1+q q − p′r′1 N′ ) p p′r′1 ∫ s 0 f∗(σ)dσ ] qp′ p ds s 8 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 ≤ CN ∫ +∞ 0 [ s ( r′1+q q − p′r′1 ′N ) p p′r′1 +1 f∗(s) ] qp′ p ds s ≤ CN ∫ +∞ 0 [ s ( r′1+q q − p′r′1 N′ ) p p′r′1 +1− p qp′ f∗(s) ] qp′ p ds, where qp′ p ≥ 1, p′r1 p = t, and CN a constant that vary from line to line. Since fε ∈Mr(Ω) we conclude that ‖(Aε(|uε|))q‖L1(Ω) ≤ CN ∫ +∞ 0 (f∗(s)) −rq( 1 r′1 − p′ N′+ p′ p )+ qp′ p ds ≤ CN‖f∗‖rLr([0,|Ω|]). (3.12) where r = −rq( 1 r′1 − p′ N ′ + p′ p ) + qp′ p , q = rtN(p− 1) t(N − rp) + rN . Case r = 1. By Proposition 3.3, and Hölder inequality, we have Aε(u∗(s)) ≤ CN ∫ |Ω| s r−p ′/N ′ [D(r)]p ′/p (∫ r 0 f∗(σ)dσ )p′/p dr ≤ CN‖D‖Lt[0,|Ω|] (∫ |Ω| s r − p′t(p−1) N′(tp−t−1) ) tp−t−1 t(p−1) ≤ CN‖D‖Lt[0,|Ω|]s 1− p′t(p−1) N′(tp−t−1) which implies the result. � Remark 3.6. Since p/N < 1 + 1 t , (see (2.3)), we have Ntp Nt(p− 1)−N + tp > 1. Proposition 3.7. Let uε be a solution of (3.3). (a) If Ntp Nt(p−1)−N+tp < r < tN tp−N , then ‖∇Aε(|uε|)‖Lp(Ω,ν) ≤ c1 . (3.13) (b) If max ( 1, tNp Nt(p− 1)p+ pt−N ) < r < tNp Nt(p− 1) + pt−N , then ‖∇Aε(|uε|)‖Lβ(Ω,νβ/p) ≤ c2, (3.14) where β = rNt(p−1)p rN+Ntp−ptr . (c) If 1 ≤ r ≤ max ( 1, tNp Nt(p− 1)p+ pt−N ) , then ‖ν1/p∇Aε(|uε|)‖Mβ(Ω) ≤ c3, (3.15) where β = rNt(p−1)p rN+Ntp−ptr . EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 9 Proof. Let uε is a solution of (3.3), by the definition of Aε we can use as test function v = [Th(Aε(|uε|)− Tθ(Aε(|uε|)] sign(uε) and obtain∫ θθ |fε| dx, (3.16) Case 1: Ntp Nt(p−1)−N+tp < r < tN tp−N . Passing to the limit in (3.16), we obtain d dθ ∫ Aε(|uε|)≤θ ν(x)|∇Aε(|uε|)|p dx ≤ ∫ µε(θ) 0 f∗ε (s)ds, (3.17) where we have denoted with µε(θ) the distribution functions of Aε(|uε|). Integrating (3.17) between 0 and +∞ and using a Hölder inequality, we have∫ Ω ν(x)|∇Aε(|uε|)|p dx ≤ ∫ +∞ 0 dθ ∫ µε(θ) 0 f∗ε (s)ds = ∫ |Ω| 0 Aε(u ∗ ε(s))f ∗ ε (s)ds ≤ ‖f‖Lr(Ω).‖Aε(|uε|)‖Lr′ (Ω). (3.18) We observe that if r is such that Nt Nt(p−1)−N+pt ≤ r < tN tp−N , by (3.10) the right- hand side of the above inequality is controlled by a constant depending on the norm of fε in Lr(Ω); so by (3.18) inequality (3.13) follows. Case 2: max ( 1, tNp Nt(p−1)p+pt−N ) < r < tNp Nt(p−1)+pt−N . Applying the Hölder inequality in (3.16) and reasoning as before, we obtain∫ Ω |∇Aε(|uε|)|βνβ/p(x) dx ≤ ∫ +∞ 0 (∫ µε(θ) 0 f∗ε (s)ds )β/p (−µ′ε(θ)) 1− βp dθ ≤ (∫ +∞ 0 (1 + θ)q(−µ′ε(θ))dθ )1− βp × (∫ +∞ 0 (1 + θ)q(1− p β ) (∫ µε(θ) 0 f∗ε (s)ds ) dθ )β/p . (3.19) By the properties of rearrangements, we can write the first integral on the right- hand side of (3.19) as∫ +∞ 0 (1 + θ)q(−µ′ε(θ))dθ = ∫ |Ω| 0 (1 +Aε(u ∗ ε)) qds, (3.20) and by (3.10) this quantity is bounded by a constant depending on the norm of fε in Lr(Ω). On the other hand, integrating by parts the second integral on the right-hand side of (3.19) we have∫ +∞ 0 (1 + θ)q(1− p β ) (∫ µε(θ) 0 f∗ε (s)ds ) dθ ≤ c ∫ |Ω| 0 f∗ε (s)[(1 +Aε(u ∗ ε)) (q(1− pβ )+1)]ds ≤ c‖fε‖Lr(Ω) [ ∫ |Ω| 0 [(1 +Aε(u ∗ ε)) q]ds ]1− 1 r . (3.21) 10 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 Applying again (3.10), by (3.19) it follows the estimate (3.14). Case 3: 1 ≤ r ≤ max ( 1, tNp Nt(p−1)p+pt−N ) . Integrating inequality (3.17) between 0 and k, we obtain∫ Aε(|uε|)≤k ν(x)|∇Aε(|uε|)|p dx ≤ ∫ k 0 dθ ∫ µε(θ) 0 f∗ε (s)ds. (3.22) If r = 1, from (3.22) we obtain∫ Aε(|uε|)≤k ν(x)|∇Aε(|uε|)|p dx ≤ k‖fε‖L1(Ω). by (3.11) and (2.3) we obtain the assertion. If 1 ≤ r ≤ max(1, tNp Nt(p−1)p+pt−N ), then by (3.10) it follows that Aε(|uε|) ∈ Mq(Ω), with q = rNt(p−1) tN+rN−ptr ; so we obtain∫ Aε(|uε|)≤k ν(x)|∇Aε(|uε|)|p dx ≤ ck1− q r′ by Proposition 3.2, we conclude the result. � Replacing ∇Aε(|uε|) by ∇uε the above estimates also hold; furthermore it follows that ∫ Ω ν(x)|∇uε|γ dx ≤ c, with γ < Nt(p−1) tN+N−t , where c is a constant depending on the L1(Ω) norm of fε. Using (3.5), the Tk(uε) are uniformly bounded in W 1,p 0 (Ω, ν) for any k > 0. Hence, there exists a function u ∈W 1,γ 0 (Ω, ν) such that uε → u a.e. in Ω, (3.23) and, for any k > 0, Tk(uε) ⇀ Tk(u) weakly in W 1,p 0 (Ω, ν). (3.24) Remark 3.8. Choosing k > l, we have uε ⇀ u weakly in W 1,p 0 (Ω, ν). (3.25) Indeed, let us suppose f ∈ L1(Ω). Using T2l(|uε|) − Tl(|uε|) as test function in (3.3), by (2.3) we obtain b(l − ε) ∫ Ω (T2l(|uε|)− Tl(|uε|))p ] dx ≤ l‖fε‖L1(Ω). Letting ε→ 0, from condition (2.1), we conclude that, for almost all x in Ω, |u| ≤ l, which give the result by (3.24). Next we prove a lemma needed for proving the existence result. Lemma 3.9. Let uε be a weak solution to problem (3.3). Suppose f ∈ L1(Ω), and let fε ∈ L∞(Ω) be such that fε → f in L1(Ω). Then ∇uε → ∇u a.e. in {|u| < l}. EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 11 Proof. We adapt the proof[presented in [11]. By Remark 3.8, we have uε → u in measure. We will prove that uε → u in measure on {|u| < m}. Let λ > 0 and η > 0 for 0 < k < l, and M > 0, we set E1 ={|u| < l} ∩ ({|∇uε| > M} ∪ {|∇u| > M} ∪ {|uε| > k} ∪ {|u| > k}), E2 ={|u| < l} ∩ {|uε − u| > η}, E3 ={|uε − u| ≤ η, |∇uε| ≤M, |∇u| ≤M, |uε| ≤ k, |u| ≤ k, |∇(uε − u)| ≥ λ} ∩ {|u| < l}. Observe that {|u| < l} ∩ {|∇uε| ≥ λ} ⊂ E1 ∪ E2 ∪ E3. Since uε and ∇uε are bounded in L1(Ω), for any σ > 0 we can fix M and k < l such that |E1| < σ/3 independently of ε. By the monotonicity Assumption (2.5), there exists a real valued function γ such that meas({x ∈ Ω : γ(x) = 0}) = 0, (a(x, s, ξ)− a(x, s, ξ′))(ξ − ξ′) ≥ γ(x), for any s ∈ (−l, l), ξ, ξ′ ∈ RN , |s| ≤ k, |ξ|, |ξ′| ≤ M , and |ξ − ξ′| ≥ λ. Denoting by χη the characteristic function of [0, η], we obtain∫ E3 γ(x) dx ≤ ∫ E3 [aε(x, uε,∇uε)− aε(x, uε,∇u)](∇uε − u) dx ≤ ∫ {|uε|≤k,|u|≤k} [( aε(x, uε,∇uε)− aε(x, uε,∇Tk(u)) ) × ( ∇uε − Tk(u))χη(|uε − Tk(u)| )] dx ≤ ∫ Ω [( aε(x, uε,∇uε)− aε(x, uε,∇Tk(u)) ) × ( ∇uε − Tk(u))χη(|uε − Tk(u)| )] dx ≤ ∫ Ω aε(x, uε,∇uε)(∇uε − Tk(u))χη(|uε − Tk(u)|) dx − ∫ Ω aε(x, uε,∇Tk(u)) · (∇uε − Tk(u))χη(|uε − Tk(u)|) dx := J1 − J2. For the term J1, using Tη(uε − Tk(u)), we have |J1| = ∣∣∣ ∫ Ω fεTη(|uε − Tk(u)|) dx ∣∣∣ ≤ η‖f‖L1(Ω). Choosing η > 0 such that k + η < l, there exists ε0 > 0 such that for all ε < ε0, aε(x, uε,∇Tk(u)) = a(x, uε,∇Tk(u)) in {x ∈ Ω : |uε − Tk(u)| ≤ η}; and since {x ∈ Ω : |uε − Tk(u)| ≤ η} ⊂ {x ∈ Ω : |uε| ≤ k + η} we obtain J2 = ∫ Ω a(x, uε,∇Tk(u)) · ∇Tη(uε − Tk(u)) dx = ∫ Ω a(x, Tk+η(uε),∇Tk(u)) · (∇Tk+η(uε − Tk(u)))χη(|uε − Tk(u)|) dx. 12 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 By (3.24), it follows that Tk+η(uε) ⇀ Tk+η(u) weakly in W 1,p 0 (Ω, ν), on the other hand |a(x, Tk+η(uε),∇Tk(u))| ≤ b(|Tk+η(uε|))ν(x)|∇Tk+η(u)|p−1 using Vitali’s theorem we have a(x, Tk+η(uε),∇Tk(u))→ a(x, Tk+η(u),∇Tk(u)) strongly in Lp ′ (Ω, ν−1/(p−1)). Letting ε and η tend to 0 respectively in J2, we obtain lim ε→0 ∫ Ω a(x, uε,∇Tk(u)) · ∇Tη(uε − Tk(u)) dx = ∫ Ω a(x, Tk+η(u),∇Tk(u)) · (∇Tk+η(u− Tk(u)))χη(|uε − Tk(u)|) dx, and lim η→0 ∫ Ω a(x, Tk+η(u),∇Tk(u)) · (∇Tk+η(u− Tk(u)))χη(|uε − Tk(u)|) dx = 0. For η small enough η‖f‖L1(Ω) < δ/2, by Kolmogorov theorem, we have |E3| < σ independently of ε. Fix η, by the fact that uε → u in measure, we choose ε1 such that |E2| < η for ε ≤ ε1. This implies that ∇uε → ∇u in measure in {|u| < l}, consequently ∇uε → ∇u a.e. in {|u| < l}. � We observe that since uε → u a.e. in Ω (see (3.23)), we have {x ∈ Ω : |u(x)| = l} = { x ∈ Ω : lim ε→0 ∫ |uε(x)| 0 bε(t) ≥ ∫ l 0 b(t) dt } . (3.26) Theorem 3.10. Let f be a function in Lr(Ω), with r > tN/(tp−N). Assume that (2.1)–(2.5) hold. Then there exists a weak solution u ∈W 1,p 0 (Ω, ν) of problem (2.2) such that ‖u‖L∞(Ω) < l. Proof. For fε = f with ε > 0. By classical results see for example [2, 1]) there exists a solution uε ∈ W 1,p 0 (Ω, ν) of the approximated problem (2.2). Estimate (3.4) implies Aε(‖uε‖L∞) ≤ C(f) = CN ∫ |Ω| 0 r−p ′/N ′ [D(r)]p ′/p (∫ r 0 f∗ε (σ)dσ )p′/p dr. (3.27) Since A is bijective in [0, l), we can take B = A−1(C(f)) and then we choose ε0 > 0 such that b(s) ≤ b(l− ε) for any s ∈ [0, B]. By definition of bε and Aε we have, for any ε < ε0, Aε(s) = A(s), s ∈ [0, B]. Moreover, being Aε increasing, it follows that, for any ε < ε0, Aε(s) ≤ C(f)⇔ s ∈ [0, B], so by (3.27) we obtain ‖uε‖L∞ ≤ B < l. EJDE-2020/105 DEGENERATE ELLIPTIC EQUATIONS 13 By (2.3) and Lemma 3.9, we have aε(x, uε1(x),∇uε1(x))→ a(x, u,∇u) strongly in Lp ′ (Ω, ν−1/(p−1)), fε → f strongly in L∞(Ω). Passing to the limit in the weak formulation of problem (3.3), we conclude that u is a weak solution of (2.2), which satisfies ‖u‖L∞(Ω) < l. � Theorem 3.11. Let f ∈ Lr(Ω), with Ntp Nt(p−1)−N+tp < r < tN tp−N . Under hypothesis (2.1)-(2.5), there exists a weak solution u ∈ W 1,p 0 (Ω, ν) of problem (2.2), such that meas({x ∈ Ω : |u(x)| = l}) = 0. Proof. Let uε ∈W 1,p 0 (Ω, ν) be a weak solution to the approximated problem (3.3). By Remark (3.8), we have uε → u a.e. in Ω, since A(l−) = +∞, (3.26) implies that Aε(|uε|)→ A(|u|) a.e. in Ω. (3.28) By (3.13) and (3.28), we obtain Aε(|uε|)→ A(|u|) weakly in W 1,p 0 (Ω, ν), (3.29) Since A(|u|) is bounded in L1(Ω) and meas({x ∈ Ω : |u(x)| = l}) = 0, by (2.3) we have aε(x, uε,∇uε)→ a(x, u,∇u) a.e. Ω. On the other hand by (2.3) and (3.13) |aε(x, uε,∇uε)| is bounded in Lp ′ (Ω, ν−1/(p−1)); passing to the limit in the weak formulation (3.3), we obtain∫ Ω a(x, u,∇u) · ∇ϕdx = ∫ Ω fϕ dx, for all ϕ ∈W 1,p 0 (Ω, ν). � Theorem 3.12. Let f ∈ Lr(Ω), with 1 ≤ r < Ntp Nt(p−1)−N+tp . Under hypothesis (2.1) − (2.5), there exists a solution u ∈ W 1,p 0 (Ω, ν) of problem (2.2), in the sense of Definition (2.2) such that meas({x ∈ Ω : |u(x)| = l}) = 0. Proof. Let uε be a weak solution of the approximate problem (3.3), by passing to the limit we can show that |u| < l a.e. in Ω. Take Tk(uε − ϕ), with ϕ ∈ W 1,p 0 (Ω, ν) ∩ L∞(Ω) as test function in (3.3) we obtain∫ |uε−ϕ|≤k a(x, Tl−ε(uε),∇uε) · ∇uε dx − ∫ |uε−ϕ|≤k a(x, Tl−ε(uε),∇uε) · ∇ϕdx = ∫ Ω fεTk(uε − ϕ) dx. (3.30) Since {|uε − ϕ|} ⊆ {|uε| ≤ k + ‖ϕ‖L∞(Ω) = M}, for 1 < k < l and ‖ϕ‖L∞(Ω) < l − k, we obtain M < l and consequently |a(x, TM (uε),∇TM (uε))| is bounded in Lp ′ (Ω, ν−1/(p−1)), and lim ε→0 ∫ |uε−ϕ|≤k a(x, Tl−ε(uε),∇uε) · ∇ϕdx = ∫ |u−ϕ|≤k a(x, u,∇u) · ∇ϕdx. (3.31) 14 B. AHARROUCH, J. BENNOUNA EJDE-2020/105 Moreover since fε strongly convergent to f in L1(Ω), and Tk(uε − ϕ) weakly* convergent to Tk(u− ϕ) in L∞(Ω), we have lim ε→0 ∫ Ω fεTk(uε − ϕ) dx = ∫ Ω fTk(u− ϕ) dx. (3.32) On the other hand a(x, Tl−ε(uε),∇uε) · ∇uε being non-negative, and almost every- where convergent to a(x, u,∇u) · ∇u, by Fatou’s lemma we conclude that lim inf ε→0 ∫ |uε−ϕ|≤k a(x, Tl−ε(uε),∇uε)·∇uε dx ≤ ∫ |u−ϕ|≤k a(x, u,∇u)·∇u dx. (3.33) Combining (3.31), (3.32) and (3.33) we obtain∫ Ω a(x, u,∇u) · ∇Tk(u− ϕ) dx ≤ ∫ Ω fTk(u− ϕ) dx, for all ϕ ∈W 1,p 0 (Ω, ν). � References [1] L. Aharouch, Y. Akdim, E. Azroul; Quasilinear degenerate elliptic unilateral problems, Ab- stract and Applied Analysis, 1 (2005) 11–31. 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Box 1796 Atlas Fez, Morocco Email address: jbennouna@hotmail.com 1. Introduction 2. Preliminaries Assumptions Weighted Sobolev spaces Rearrangement properties 3. Main result 3.1. A priori estimate References