Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 65825, 15 pages doi:10.1155/2007/65825 Research Article On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with L 1 Data S. Bonafede and F. Nicolosi Received 24 January 2007; Accepted 29 January 2007 Recommended by V. Lakshmikantham We establish H ¨ older continuity of gener alized solutions of the Dirichlet problem, a sso- ciated to a degenerate nonlinear fourth-order equation in an open bounded set Ω ⊂ R n , with L 1 data, on the subsets of Ω where the behavior of weights and of the data is regular enough. Copyright © 2007 S. Bonafede and F. Nicolosi. This is an open access article distributed under the Creative Commons Attribution License, w hich permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we will deal with equations involving an operator A : ◦ W 1,q 2,p (ν,μ,Ω) → ( ◦ W 1,q 2,p (ν,μ,Ω))  of the form Au =  |α|=1,2 (−1) |α| D α A α  x, ∇ 2 u  , (1.1) where Ω is a bounded open set of R n , n>4, 2 <p<n/2, max(2p, √ n) <q<n, ν and μ are positive functions in Ω with properties precised later, ◦ W 1,q 2,p (ν,μ,Ω)istheBanachspace of all functions u : Ω → R with the properties |u| q ,ν|D α u| q ,μ|D β u| p ∈ L 1 (Ω), |α|=1, |β|=2, and “zero” boundary values; ∇ 2 u ={D α u : |α|≤2}. The functions A α satisfy growth and monotonicity conditions, and in particular, the following strengthened ellipticity condition (for a.e. x ∈ Ω and ξ ={ξ α : |α|=1,2}):  |α|=1,2 A α (x, ξ)ξ α ≥ c 2   |α|=1 ν(x)   ξ α   q +  |α|=2 μ(x)   ξ α   p  − g 2 (x), (1.2) where c 2 > 0, g 2 (x) ∈ L 1 (Ω). 2 Boundary Value Problems We will assume that the rig ht-hand sides of our equations, depending on unknown function, belong to L 1 (Ω). A model representative of the given class of equations is the following: −  |α|=1 D α  ν   |β|=1   D β u   2  (q−2)/2 D α u  +  |α|=2 D α  μ   |β|=2   D β u   2  (p−2)/2 D α u  =−| u| σ−1 u + f in Ω, (1.3) where σ>1and f ∈ L 1 (Ω). The assumed conditions and known results of the theory of monotone operators allow us to prove existence of generalized solutions of the Dir ichlet problem associated to our operator (see, e.g., [1]), bounded on the sets G ⊂ Ω where the behavior of weights and of the data of the problem is regular enough (see [2]). In our paper, following the approach of [3], we establish on such sets a result on H ¨ older continuity of generalized solutions of the same Dirichlet problem. We note that for one high-order equation with degener ate nonlinear operator satisfy- ing a strengthened ellipticity condition, regularity of solutions was studied in [4, 5] (non- degenerate case) and in [6, 7] (degenerate case). However, it has been made for equations with right-hand sides in L t with t>1. 2. Hypotheses Let n ∈ N, n>4, and let Ω be a bounded open set of R n .Letp, q be two real numbers such that 2 <p<n/2, max(2p, √ n) <q<n. Let ν : Ω → R + be a measurable function such that ν ∈ L 1 loc (Ω),  1 ν  1/(q−1) ∈ L 1 loc (Ω). (2.1) W 1,q (ν,Ω) is the space of all functions u ∈ L q (Ω) such that their derivatives, in the sense of distribution, D α u, |α|=1, are functions for which the following properties hold: ν 1/q D α u ∈ L q (Ω)if|α|=1; W 1,q (ν,Ω) is a Banach space w ith respect to the nor m u 1,q,ν =   Ω |u| q dx +  |α|=1  Ω ν   D α u   q dx  1/q . (2.2) ◦ W 1,q (ν,Ω)istheclosureofC ∞ 0 (Ω)inW 1,q (ν,Ω). Let μ(x):Ω → R + be a measurable function such that μ ∈ L 1 loc (Ω),  1 μ  1/(p−1) ∈ L 1 loc (Ω). (2.3) W 1,q 2,p (ν,μ,Ω) is the space of all functions u ∈ W 1,q (ν,Ω), such that their derivatives, in the sense of distribution, D α u, |α|=2, are functions with the following properties: S. Bonafede and F. Nicolosi 3 μ 1/p D α u ∈ L p (Ω), |α|=2; W 1,q 2,p (ν,μ,Ω) is a Banach space w ith respect to the nor m u=u 1,q,ν +   |α|=2  Ω μ   D α u   p dx  1/p . (2.4) ◦ W 1,q 2,p (ν,μ,Ω)istheclosureofC ∞ 0 (Ω)inW 1,q 2,p (ν,μ,Ω). Hypothesis 2.1. Let ν(x) be a measurable positive function: 1 ν ∈ L t (Ω)witht> nq q 2 −n , ν ∈ L t (Ω)witht> nt qt −n . (2.5) We put q = nqt/(n(1 + t) −qt). We can easily prove that a constant c 0 > 0 exists such that if u ∈ ◦ W 1,q (ν,Ω), the following inequality holds:  Ω |u| q dx ≤ c 0   suppu  1 ν  t dx  q/qt   |α|=1  Ω ν|D α u| q dx  q/q . (2.6) We set ν = μ q/(q−2p) (1/ν) 2p/(q−2p) . Hypothesis 2.2. ν ∈ L 1 (Ω). Hypothesis 2.3. There exists a real number r> q( q −1)/(q(q −1)(p −1) −q)suchthat 1 μ ∈ L r (Ω). (2.7) For more details about weight functions, see [8, 9]. Let Ω 1 be a nonempty open set of R n such that Ω 1 ⊂ Ω. Definit ion 2.4. It is said that G closed set of R n is a “regular set”if ◦ G is nonempty and G ⊂ Ω 1 . Denote by R n,2 the space of all sets ξ ={ξ α ∈ R : |α|=1,2} of real numbers; if a func- tion u ∈ L 1 loc (Ω) has the weak derivatives D α u, |α|=1,2 then ∇ 2 u ={D α u : |α|=1,2}. Suppose that A α : Ω ×R n,2 → R are Carath ´ eodory functions. Hypothesis 2.5. There exist c 1 ,c 2 > 0andg 1 (x), g 2 (x) nonnegative functions such that g 1 ,g 2 ∈ L 1 (Ω) and, for almost every x ∈ Ω,foreveryξ ∈ R n,2 , the following inequalities 4 Boundary Value Problems hold:  |α|=1  ν(x)  −1/(q−1)   A α (x, ξ)   q/(q−1) +  |α|=2  μ(x)  −1/(p−1)   A α (x, ξ)   p/(p−1) ≤ c 1   |α|=1 ν(x)   ξ α   q +  |α|=2 μ(x)   ξ α   p  + g 1 (x), (2.8)  |α|=1,2 A α (x, ξ)ξ α ≥ c 2   |α|=1 ν(x)   ξ α   q +  |α|=2 μ(x)   ξ α   p  − g 2 (x) . (2.9) Moreover, we will assume that for almost every x ∈ Ω and every ξ,ξ  ∈ R n,2 , ξ = ξ  ,  |α|=1,2  A α (x, ξ) −A α (x, ξ  )  ξ α −ξ  α  > 0. (2.10) Let F : Ω ×R → R be a Carath ´ eodory function such that (a) for almost every x ∈ Ω, the function F(x,·) is nonincreasing in R; (b) for every x ∈ Ω, the function F(·,s)belongstoL 1 (Ω). Let A: ◦ W 1,q 2,p (ν,μ,Ω)→( ◦ W 1,q 2,p (ν,μ,Ω))  be the operator such that for every u,v∈ ◦ W 1,q 2,p (ν, μ,Ω), Au,v=  Ω   |α|=1,2 A α  x, ∇ 2 u  D α v  dx. (2.11) We consider the following Dirichlet problem: (P) = ⎧ ⎨ ⎩ Au = F(x,u)inΩ D α u = 0, |α|=0,1, on ∂Ω. (2.12) Definit ion 2.6. A W-solution of problem (P)isafunctionu ∈ ◦ W 2,1 (Ω)suchthat (i) F(x,u) ∈ L 1 (Ω); (ii) A α (x, ∇ 2 u) ∈ L 1 (Ω), for every α : |α|=1,2; (iii) Au,φ=F(x, u), φ in distributional sense. It is well known that Hypotheses 2.1–2.3, 2.5,andassumptionsonF(x,s)implythe existence of a W-solution of problem (P) (see [1]). Moreover, a boundedness local result for such solution has been established in [2] under more restrictive hypotheses on data and weight functions. More precisely, the following holds (see [2, T heorem 5.1]). Theorem 2.7. Suppose that Hypotheses 2.1–2.3 and 2.5 are satisfied. Let q 1 ∈ (q, q(q − 1)/q), τ>q/(q −q 1 ). Assume that restrictions of the functions ν q 1 /(q 1 −q) , ν, g 1 , g 2 ,and|F(·, 0) | q 1 /(q 1 −1) on G belong to L τ (G),forevery“regularset”G. Then there exists uW-solution of problem (P) such that for every G, ess G sup|u|≤M G < + ∞,withM G positive constant depending only on known values. S. Bonafede and F. Nicolosi 5 3. Main result In the sequel of paper, G will be a “regular set.”Inordertoobtainourregularityresulton G, we need the following further hypotheses. Hypothesis 3.1. There exists a constant c  > 0 such that for all y ∈ ◦ G and for all ρ>0, with B(y,ρ) ⊂ ◦ G,wehave  ρ −n  B(y,ρ)  1 ν  t dx  1/t  ρ −n  B(y,ρ) ν τ dx  1/τ ≤ c  . (3.1) With regard to this assumption, see [3]. Hypothesis 3.2. There exist a real positive number σ and two real functions h(x)( ≥ 0), f (x)(> 0) defined on G,suchthat   F(x,s)   ≤ h(x)|s| σ + f (x), for almost every x ∈G and every s ∈ R. (3.2) Moreover, we assume that h(x), f (x) ∈ L τ (G), (3.3) with τ defined as above. Using considerations stated in [1], following the approach of [3], we establish the fol- lowing result. Theorem 3.3. Let all above-stated hypotheses hold and let conditions of Theorem 2.7 be satisfied. Then, the W-solution u of Dirichlet problem (P), essentially bounded on G,isalso locally H ¨ olderian on G. More prec isely, there exist positive constant C and λ (0 <λ<1) such that for every open set Ω  ,Ω  ⊂ ◦ G,andeveryx, y ∈Ω    u(x) −u(y)   ≤ C  d  Ω  ,∂ ◦ G  −λ |x − y| λ , (3.4) where C and λ depend only on c 1 , c 2 , c 0 , c  , n, q, p, t, τ, σ, M G , diamG, meas G, f  L τ (G) , h L τ (G) , g 1  L τ (G) , g 2  L τ (G) , ν L τ (G) ,and1/ν L t (Ω) . Proof. For every l ∈ N, we define the function F l : Ω ×R →R by F l (x, s) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ − l if F(x,0)−F(x,s) < −l, F(x,0) −F(x,s)if   F(x,0)−F(x,s)   ≤ l, l if F(x,0) −F(x,s) >l, (3.5) and the function f l : Ω → R by f l (x) = ⎧ ⎨ ⎩ F(x,0) if   F(x,0)   ≤ l, 0if   F(x,0)   >l. (3.6) 6 Boundary Value Problems By Lebesgue’s theorem and property (b) of F(x,s), we have that f l (x)goestoF(x,0) in L 1 (Ω). Next, inequalities (2.6), (2.8)–(2.10), property (a) of F(x,s), and known results of the theory of monotone operators (see, e.g., [10]) imply that for any l ∈ N, there exists u l ∈ ◦ W 1,q 2,p (ν,μ,Ω)suchthat  Ω   |α|=1,2 A α  x, ∇ 2 u l  D α v + F l  x, u l  v  dx =  Ω f l vdx, (3.7) for every v ∈ ◦ W 1,q 2,p (ν,μ,Ω). From considerations stated in [1, Section 3], we deduce that there exists a W-solution u of problem (P)suchthat u l −→ u a.e. in Ω. (3.8) Moreover, see proof of Theorem 2.7, ess G sup   u l   ≤ M G ,foreveryl ∈ N. (3.9) We set n = q 2 /(q −2p), a = (1/n)(q −n/t −n/τ). Let us fix y ∈ ◦ G, ρ>0andB(y,2ρ) ⊂ ◦ G.Letusput ω 1,l = ess B(y,2ρ) inf u l , ω 2,l = ess B(y,2ρ) supu l , ω l = ω 2,l −ω 1,l . (3.10) We will show that osc  u l ,B(y,ρ)  ≤  cω l + ρ a , (3.11) with c ∈]0, 1[ independent of l ∈ N. To this aim, we fix l ∈ N and we set Φ l =  |α|=1 ν   D α u l   q +  |α|=2 μ   D α u l   p , ψ(x) = ρ −an  1+ f (x)+h(x)+g 1 (x)+g 2 (x)+ν(x)  + ρ −q ν. (3.12) Obviously, we will assume that ω l ≥ ρ a (otherwise, it is clear that (3.11)istrue). (3.13) We int roduce now the following functions: F 1,l (x) = ⎧ ⎪ ⎨ ⎪ ⎩ 2eω l u l (x) −ω 1,l + ρ a if x ∈ B(y,2ρ), e if x ∈ Ω \B(y,2ρ); (3.14) S. Bonafede and F. Nicolosi 7 ϕ ∈ C ∞ 0 (Ω): 0 ≤ ϕ ≤ 1inΩ, ϕ = 0inΩ \B(y,2ρ) and satisfying   D α ϕ   ≤ cρ −|α| , |α|=1,2, (3.15) where the positive constant c depends only on n. Let us fix s>qand r ≥ 0anddefine v l =  lgF 1,l  r F q−1 1,l ϕ s , z l =− 1 2eω l  r  lgF 1,l  r−1 +(q −1)  lgF 1,l  r  F q 1,l ϕ s . (3.16) From Hypothesis 2.2 and (3.15), we have that v l ∈ ◦ W 1,q 2,p (ν,μ,Ω) and the next inequal- ities are true:   D α v l −z l D α u l   ≤ csϕ s−1  lgF 1,l  r F q−1 1,l ρ −1 if |α|=1a.e.inB(y,2ρ), (3.17)   D α v l −z l D α u l   ≤ 5q 2 s(r +1) 2  lgF 1,l  r F q−1 1,l ϕ s   |β|=1 |D β u l | 2  u l −ω 1,l + ρ a  2  +2nqs 2 c 2 ρ −2  lgF 1,l  r F q−1 1,l ϕ s−2 if |α|=2a.e.inB(y,2ρ). (3.18) Since u l (x) satisfies (3.7), for v =v l ,weobtain  Ω   |α|=1,2 A α  x, ∇ 2 u l  D α v l + F l  x, u l  v l  dx =  Ω f l v l dx. (3.19) From this, taking into account (3.9)andHypothesis 3.2,wehave  Ω  |α|=1,2 A α  x, ∇ 2 u l  D α v l dx ≤  3+M σ G   Ω  1+ f (x)+h(x)  v l dx. (3.20) Hence  Ω  |α|=1,2  A α  x, ∇ 2 u l  D α u l  − z l  dx ≤  3+M σ G   Ω  1+ f (x)+h(x)  v l dx + I 1 + I 2 , (3.21) where I i =  Ω  |α|=i   A α  x, ∇ 2 u l      D α v l −z l D α u l   dx, i =1,2. (3.22) Using Hypothesis 2.5 and definition of z l ,wehave (q −1)c 2 2eω l  Ω Φ l  lgF 1,l  r F q 1,l ϕ s dx ≤  3+M σ G   Ω  1+ f (x)+h(x)  lgF 1,l  r F q−1 1,l ϕ s dx +  Ω g 2 (x)  − z l  dx + I 1 + I 2 . (3.23) 8 Boundary Value Problems Note that F q−1 1,l ≤ (diamG) a  2eω l  q−1 ρ −aq , −z l ≤ (q −1)(r +1)  2eω l  q−1 ρ −aq ϕ s  lgF 1,l  r a.e. in B(y,2ρ), (3.24) consequently, from (3.23), we obtain c 2 2eω l  B(y,2ρ) Φ l  lgF 1,l  r F q 1,l ϕ s dx ≤ c 3 (r +1)  2eω l  q−1  B(y,2ρ) ρ −aq  1+ f (x)+h(x)+g 2 (x)  lgF 1,l  r ϕ s dx + I 1 + I 2 , (3.25) where c 3 = (q −1)(3 + M σ G )(diamG +1). Let us fi x |α|=1. Let  > 0, then, applying Young’s inequality and using (2.8)and (3.17), we establish I 1 ≤ c 1  2eω l  B(y,2ρ) Φ l F q 1,l  lgF 1,l  r ϕ s dx + c 1   2eω l  q−1  B(y,2ρ) ρ −aq g 1 (x)  lgF 1,l  r ϕ s dx +  1−q  2eω l  q−1 n(cs) q  B(y,2ρ) ρ −q ν  lgF 1,l  r ϕ s−q dx. (3.26) Let us fix |α|=2 and estimate I 2 . To this aim, it will be useful to observe that the follow ing equalities are true: p −1 p + 2 q + q −2p qp = 1, q −1 = p −1 p q +  q p −1  . (3.27) Moreover, ρ −aq−2p μ ≤ ρ −an ν + ρ −q ν in Ω. (3.28) Furthermore, due to (2.8), (3.18), and Young’s inequality, we have I 2 ≤ c 4  2eω l  B(y,2ρ) Φ l F q 1,l  lgF 1,l  r ϕ s dx +c 5  2eω l  q−1   1+ 1   n s n (r +1) n  B(y,2ρ)  ρ −an  g 1 (x)+ν(x)  + ρ −q ν  lgF 1,l  r ϕ s−q dx, (3.29) where c 4 depends only on c 1 , n, q;andc 5 depends only on c 1 , n, q, p, c,anddiamG. S. Bonafede and F. Nicolosi 9 From (3.25), (3.26), and (3.29), we get c 2 2eω l  B(y,2ρ) Φ l  lgF 1,l  r F q 1,l ϕ s dx ≤  c 1 + c 4   2eω l  B(y,2ρ) Φ l F q 1,l  lgF 1,l  r ϕ s dx +  2eω l  q−1 c 6 (r +1) n s n  1+ + 1   n+1  B(y,2ρ) ψ  lgF 1,l  r ϕ s−q dx, (3.30) where the constant c 6 depends only on c 1 , c, n, q, p, M G , σ,anddiamG. Setting  = c 2 2  c 1 + c 4  , (3.31) from the last inequality, we deduce  B(y,2ρ) Φ l  lgF 1,l  r F q 1,l ϕ s dx ≤ c 7  2eω l  q (r +1) n s n  B(y,2ρ) ψ  lgF 1,l  r ϕ s−q dx, (3.32) where the constant c 7 depends only on c 1 , c 2 , c, n, q, p, M G , σ,anddiamG. Now, if we choose ϕ such that ϕ = 1inB(y,(4/3)ρ), from (3.32), with r = 0ands = q +1,weget  B(y,(4/3)ρ)   |α|=1 ν   D α u l   q  F q 1,l dx ≤ c 7  2eω l  q (q +1) n  B(y,2ρ) ψdx. (3.33) Moreover,ifwetakein(3.32)insteadofϕ the function ϕ 1 ∈ C ∞ 0 (Ω) with the properties 0 ≤ ϕ 1 ≤ 1inΩ, ϕ 1 = 0inΩ \B(y,(4/3)ρ), ϕ 1 = 1inB(y,ρ), and |D α ϕ|≤cρ −|α| in Ω, |α|=1,2, we obtain that for every r>0ands>q,  B(y,2ρ)   |α|=1 ν   D α u l   q   lgF 1,l  r F q 1,l dx ≤ c 7  2eω l  q s n (r +1) n  B(y,2ρ) ψ  lgF 1,l  r ϕ s−q 1 dx. (3.34) We fix arbitrary r>0ands> q,andlet z l =  lgF 1,l  r/q ϕ s/q 1 . (3.35) By means of Hypothesis 2.1, we establish that z l ∈ ◦ W 1,q (ν,Ω)andfor|α|=1, ν   D α z l   q ≤ 2 q−1  r q  q  lgF 1,l  (r/q−1)q  F 1,l  q 1  2eω l  q   D α u l   q νϕ sq/q 1 +2 q−1  s q  q  lgF 1,l  rq/q ϕ (s/q−1)q 1 c q ρ −q ν. (3.36) 10 Boundary Value Problems Now, it is convenient to observe that q/(q − q 1 ) >nt/(qt −n), then τ>nt/(qt − n); moreover, ψ(x) ∈ L τ (G). From (3.34)and(3.36), we deduce  Ω ν   D α z l   q dx ≤c 8 s n (r+1) n+q   B(y,2ρ) ψ τ dx  1/τ   B(y,2ρ)  lgF 1,l  r(q/q)(τ/(τ−1)) ϕ (s/q−1)q(τ/(τ−1)) 1 dx  (τ−1)/τ , (3.37) where the constant c 8 depends only on c 1 , c 2 , c, n, q, p, M G , σ,anddiamG. We set θ =  q(τ −1) qτ , m = qτ τ −1 , (3.38) and for every r,s>0, we define I(r,s) =  B(y,2ρ)  lgF 1,l  r ϕ s 1 dx. (3.39) Consequently, last inequality can be rewritten in this manner:  Ω ν   D α z l   q dx ≤ c 8 s n (r +1) n+q   B(y,2ρ) ψ τ dx  1/τ  I  r θ , s θ −m  (τ−1)/τ . (3.40) Due to Hypothesis 2.1, I(r,s) =  B(y,2ρ) z q l dx ≤ c 0   B(y,2ρ)  1 ν  t dx  q/qt   |α|=1  Ω ν   D α z l   q dx  q/q . (3.41) Let us denote by  G the norm of (1 + f (x)+h(x)+g 1 (x)+g 2 (x)+ν(x)) in L τ (G). By simple computation, we have   B(y,2ρ) ψ τ dx  1/τ ≤ ρ −q   B(y,2ρ) ν τ dx  1/τ +  G ρ −an . (3.42) Now, it is convenient to observe that (q −n/t −n/τ)(q/q) = n(θ −1). Then, from (3.40)–(3.42), using Hypothesis 3.1,weget I(r,s) ≤ M(r + s) m ρ n(1−θ)  I  r θ , s θ −m  θ ,foreveryr>0, s>q, (3.43) where m = 2(q + n)q and the positive constant M depends only on c 1 , c 2 , c, c 0 , c  , n, q, p, t, 1/ν L t (Ω) , M G , σ,measG,diamG,and  G . [...]... [2] A Kovalevsky and F Nicolosi, On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L1 -data,” Fundamentalnaya I Prikladnaya Matematika, vol 12, no 4, pp 99–112, 2006 [3] A Kovalevsky and F Nicolosi, “Existence and regularity of solutions to a system of degenerate nonlinear fourth-order equations, ” Nonlinear Analysis Theory, Methods & Applications,... Skrypnik and F Nicolosi, On the regularity of solutions of higher-order degenerate nonlinear elliptic equations, ” Dopov¯d¯ Nats¯onal’no¨ Akadem¯¨ Nauk Ukra¨ni, no 3, pp 24–28, 1997 ı ı ı ı ıı ı [7] A Kovalevsky and F Nicolosi, On H¨ lder continuity of solutions of equations and variao tional inequalities with degenerate nonlinear elliptic high order operators,” in Problemi Attuali dell’Analisi e della Fisica... Methods & Applications, vol 61, no 3, pp 281–307, 2005 [4] I V Skrypnik, “Higher order quasilinear elliptic equations with continuous generalized solutions, ” Differential Equations, vol 14, no 6, pp 786–795, 1978 [5] S Bonafede and S D’Asero, “H¨ lder continuity of solutions for a class of nonlinear elliptic vario ational inequalities of high order,” Nonlinear Analysis Theory, Methods & Applications, vol 44,... S Bonafede and F Nicolosi 15 [10] J.-L Lions, Quelques M´thodes de R´solution des Probl`mes aux Limites non Lin´aires, Dunod, e e e e Paris, France, 1969 [11] I V Skrypnik, Nonlinear Elliptic Equations of Higher Order, Naukova Dumka, Kiev, Ukraine, 1973 [12] O A Ladyzhenskaya and N N Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY, USA, 1968 S Bonafede: Dipartimento... Ω, |α| = 0,1, on ∂Ω (4.10) By Theorem 2.7, we establish that there exists a W-solution u of problem (4.10), o bounded in every “regular set” G ⊂ Ω \{0}, and moreover, applying our result, H¨ lderian in every open set A : A ⊂ Ω \ {0} References [1] A Kovalevsky and F Nicolosi, “Existence of solutions of some degenerate nonlinear elliptic fourth-order equations with L1 -data,” Applicable Analysis, vol... della Fisica Matematica, pp 205–220, Aracne Editrice, Rome, Italy, 2000 [8] F Guglielmino and F Nicolosi, “W -solutions of boundary value problems for degenerate elliptic operators,” Ricerche di Matematica, vol 36, supplement, pp 59–72, 1987 [9] F Guglielmino and F Nicolosi, “Existence theorems for boundary value problems associated with quasilinear elliptic equations, ” Ricerche di Matematica, vol 37,... Press, New York, NY, USA, 1968 S Bonafede: Dipartimento di Economia dei Sistemi Agro-Forestali, Universit` delgi Studi a di Palermo, Viale delle Scienze, 90128 Palermo, Italy Email address: bonafedes@unipa.it F Nicolosi: Dipartimento di Matematica e Informatica, Universit` delgi Studi di Catania, a Viale A Doria 6, 95125 Catania, Italy Email address: fnicolosi@dmi.unict.it ... min(q − n/q, q/2), and let ν, μ be the restriction in Ω \ {0} of real functions |x|γ , |x|2pγ/q (4.1) S Bonafede and F Nicolosi 13 According to considerations stated in [3, Section 7], we have that functions ν, μ satisfy Hypotheses 2.1 and 2.3 Now, we will verify that ν(x) satisfies Hypothesis 3.1, for all t: nq/(q2 − n) < t < n/γ To ◦ ◦ this aim, let G ⊂ Ω \ {0} be a “regular set,” and fix y ∈ G, ρ >... a (3.56) B(y,ρ) and so Recall that we proved (3.11) under assumption (3.47) If (3.47) is not true, we take instead of F1,l the function F2,l : Ω → Rn such that F2,l = 2eωl (ω2,l − ul + a )−1 in B(y,2ρ), and arguing as above, we establish (3.11) again It is important to observe that the positive constant c11 depends only on c1 , c2 , c, c, c0 , c , n, q, p, t, 1/ν Lt (Ω) , MG , σ, diamG, and G , and... independent of l ∈ N Now from (3.11), taking into account [12, Chapter 2, Lemma 4.8], we deduce that there exist positive constant C and λ(< 1) depending on c11 and a but independent of l ∈ N such that ◦ −λ λ for every ρ ∈ 0,d y,∂G ◦ −λ λ for every ρ ∈ 0,d y,∂G osc ul ,B(y,ρ) ≤ C d y,∂G ρ , ◦ (3.57) This and (3.8) imply that osc u,B(y,ρ) ≤ C d y,∂G ρ , ◦ (3.58) The proof is complete 4 An example Let . 2002. [2] A. Kovalevsky and F. Nicolosi, On the sets of boundedness of solutions for a class of degen- erate nonlinear elliptic four th-order equations with L 1 -data,” Fundamentalnaya I Prikladnaya Matematika,. Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 65825, 15 pages doi:10.1155/2007/65825 Research Article On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear. Prikladnaya Matematika, vol. 12, no. 4, pp. 99–112, 2006. [3] A. Kovalevsky and F. Nicolosi, “Existence and regularity of solutions to a system of degenerate nonlinear fourth-order equations, ” Nonlinear Analysis.