ON THE EXISTENCE OF CONTINUOUS SOLUTIONS FOR NONLINEAR FOURTH-ORDER ELLIPTIC EQUATIONS WITH STRONGLY GROWING LOWER-ORDER TERMS

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1 ROCKY MOUNTAIN JOURNAL OF MATHMATICS Volume 47, Number 2, 2017 ON TH XISTNC OF CONTINUOUS SOLUTIONS FOR NONLINAR FOURTH-ORDR LLIPTIC QUATIONS WITH STRONGLY GROWING LOWR-ORDR TRMS MYKHAILO V. VOITOVYCH ABSTRACT. In this article, we consider nonlinear elliptic fourth-order equations with the monotone principal part satisfying the common growth and coerciveness conditions for Sobolev space W 2,p (), R n. It is supposed that the lower-order term of the equations admits arbitrary growth with respect to an unknown function and is arbitrarily close to the growth limit with respect to the derivatives of this function. We assume that the lower-order term satisfies the sign condition with respect to the unknown function. We prove the existence of continuous generalized solutions for the Dirichlet problem in the case n = 2p. 1. Introduction. Let n, m N, p R be numbers such that n 3, m 2 and p > 1. Let be a bounded open set of R n, and let W m,p () denote the Sobolev space with the norm ( u m,p = α m D α u p dx) 1/p, where the α = (α 1,..., α n ) is an n-dimensional multi-index with nonnegative integer components α i, i = 1,..., n, α = α α n, and where the D α u = α u/ x α 1 1 xα n n is a generalized derivative of order α. We denote by W m,p 0 () the closure of the set C0 () in 2010 AMS Mathematics subject classification. Primary 35B45, 35B65, 35J40, 35J62. Keywords and phrases. Nonlinear elliptic fourth-order equations, lower-order term, strong growth, Dirichlet problem, continuous solutions, L -estimate. This research is supported by a grant from the Ministry of ducation and Science of Ukraine, project No. 0115U and by a grant from the State Fund for Fundamental Research of Ukraine, project No. 0116U Received by the editors on August 6, 2015, and in revised form on June 3, DOI: /RMJ Copyright c 2017 Rocky Mountain Mathematics Consortium 667

2 668 MYKHAILO V. VOITOVYCH W m,p (). The norm ( u W m,p 0 () = α =m ) 1/p D α u p dx is equivalent to m,p in the Banach space W m,p 0 (). We consider the general 2mth order equation in the divergence form (1.1) ( 1) α D α A α (x, u,..., D m u) = 0, α m where x, u W m,p () is an unknown function, D k u = {D α u : α = k, k = 1,..., m. We assume that n = mp, for every multi-index α with α m, A α : R N m R is a Carathéodory function (N m is the number of all multi-indices α with α m), and for almost every x and every ξ R N m the following inequalities hold: (1.2) A α (x, ξ)ξ α c 1 ξ α p c 2 ξ β p β g(x), (1.3) α =m α =m A α (x, ξ) c 2 β m β <m ξ β p αβ + g α (x), α m. Here c 1 and c 2 are positive constants, p αβ = p 1 if α = β = m, p β 1, p αβ 0 and (1.4) p β < n/ β if β < m, p αβ < (n α )/ β if α + β < 2m, g, g α are nonnegative functions such that g L τ (), τ > 1, g α L τα (), τ α > n/(n α ). Under these assumptions, a generalized solution of equation (1.1) is a function u W m,p () such that, for every function v W m,p 0 (), { (1.5) A α (x, u,..., D m u)d α v dx = 0. α m

3 TH XISTNC OF CONTINUOUS SOLUTIONS 669 As is known, see for instance [5, Chapter 7], W m,p 0 () C k () if n < mp and 0 k < m n/p. In the case that n = mp, the embedding (1.6) W m,p () L φ () (L φ () denotes the Orlicz space generated by the function φ(t) = exp[ t p/(p 1) ] 1, t R [5, Chapter 7]) does not provide the boundedness of generalized solutions of equation (1.1). In this situation, Frehse [4] has established the boundedness of the arbitrary generalized solution u W m,p 0 () of equation (1.1), and the continuity of the solution has been proved by Skrypnyk [11, Chapter 2]. Hölder continuity of solutions was studied by Widman [17] and Solonnikov [12] at similar assumptions. Finally, in the case where n > mp, there exist examples of equations in the form (1.1) (1.3) with unbounded solutions, see [2, 10]. We also note that the existence of a generalized solution of equation (1.1) with growth condition (1.3), (1.4) can be set using the theory of monotone operators and additional assumptions on the coefficients. If, in condition (1.4) on p β and p αβ we replace the inequalities on the equalities, the above-mentioned results of Frehse and Skrypnyk cease to be valid. At the same time, using the method of [11, Chapter 2], Todorov [13] proved the continuity of every bounded generalized solution of equation (1.1) in the case where p β = n/ β if β < m, p αβ = (n α )/ β if β = 0 and α + β 2m, and for every multiindex α with α m, the coefficient A α admits an arbitrary growth with respect to an unknown function. Next we recall the precise formulation of the Todorov s result [13]. Theorem 1.1. Suppose that the coefficients of equation (1.1) satisfy the following conditions. (1) For every multi-index α with α m, A α : R Nm R is a Carathéodory function. (2) For almost every x and for every ξ R Nm the following inequalities hold: (a) A α (x, ξ)ξ α λ( ξ 0 ) ξ α p C( ξ 0 ) ξ β n/ β α =m α =m 1 β <m g(x), (b) A α (x, ξ) C α ( ξ 0 ) 1 β m ξ β (n α )/ β +g α (x), α m.

4 670 MYKHAILO V. VOITOVYCH Here, p > 1, n = mp; R + = [0, + ) and C, C α : R + R + are continuous nondecreasing functions, λ : R + (0, + ) is a continuous nonincreasing function; g, g α are nonnegative functions such that g L τ (), τ > 1, g α L τ α (), τ α > n/(n α ). Let u W m,p () L () be a generalized solution of equation (1.1), that is, for every function v W m,p 0 () L () equality (1.5) is true. Then the solution u is continuous at every interior point of the set. The question on existence of a bounded generalized solution of equation (1.1) under conditions (A), (B) is still open. In this article, we consider fourth order equations (m = 2) in the form (1.1) satisfying the condition A α 0 if α = 1, and all assumptions in [4] except for inequality (1.3) for the lower-order term A 0. Instead of this, we suppose a more general condition admitting, unlike [4, 11, 12, 17], an arbitrary growth of the term A 0 with respect to the function u (even stronger than the growth of the function φ(u) = exp[ u p/(p 1) ] 1) and a growth of A 0 with respect to the derivatives D α u, α = 1, 2, which is arbitrarily close to the limiting growth. This means that { A 0 (x, u, Du, D 2 u) a( u ) 1+ D α u n/ α ψ( D α u ) + g 0 (x), α =1,2 where a, ψ : R + R + are continuous functions, lim t + ψ(t) = 0. At the same time, it is supposed that the lower-order term A 0 satisfies the sign condition A 0 (x, u, Du, D 2 u)u 0. The main result of this article is a theorem on the existence and L -estimate of continuous generalized solutions of the Dirichlet problem for the equations under investigation. We remark that, in the situation n > mp results on the existence of bounded generalized solutions for nonlinear elliptic equations with natural growth lower-order terms were established, for instance, in [1, 3] (the case of second-order equations, m = 1) and in [14, 15, 16] (the case of high-order equations with strengthened coercivity, m 2). 2. Statement of the main result. Let n N, n 3, and let be a bounded open set of R n.

5 TH XISTNC OF CONTINUOUS SOLUTIONS 671 We shall use the following notation: C() is the set of continuous functions on, Λ is the set of all n dimensional multi-indices α such that α 2, N 1 (correspondingly N 2 ) is the number of all multi-indices α with α 1 ( α 2), N = N 2 N 1. If τ [1, + ], then τ is the norm in L τ (). For every measurable set we denote by (or by meas ) n-dimensional Lebesgue measure of the set. We set p = n/2 and note that, by (1.6) (m = 2) and Sobolev inequality, see for instance, [5, Theorem 7.10], for every λ 1 and for every function u W 2,p 0 (), (2.1) c λ,n, ( 1/λ ( u dx) λ α =1 ( c n ) 1/n D α u n dx D α u p dx) 1/p, where c λ,n, is a positive constant depending only on λ, n and, and c n is a positive constant depending only on n. Next, let c 1, c 2 > 0, let g 1 and g 2 be nonnegative summable functions on, and let p 0, p 1, p be arbitrary numbers satisfying the inequalities p 0 0, (2.2) (2.3) 0 p 1 < n, 1 p < p. For every α Λ with α = 2, let A α : R N 2 R be a Carathéodory function. We assume that, for almost every x and for every ξ R N 2, the following inequalities hold: (2.4) A α (x, ξ)ξ α c 1 ξ α p c 2 ξ p α g 1 (x), (2.5) A α (x, ξ) p/(p 1) c 2 { ξ 0 p 0 + α =1 ξ α p 1 + ξ α p + g 2 (x).

6 672 MYKHAILO V. VOITOVYCH Next, let g 3 and g 4 be nonnegative summable functions on, let b : R + R + and ψ : R + R + be continuous functions, (2.6) lim ψ(t) = 0, t + and let B : R N 2 R be a Carathéodory function such that, for almost every x and for every ξ R N 2, the following inequalities hold: { (2.7) B(x, ξ) b( ξ 0 ) 1 + ξ α n/ α ψ( ξ α ) + g 3 (x), α =1,2 (2.8) B(x, ξ)ξ 0 g 4 (x). Further, let τ > 1 and (2.9) f L τ (). We consider the Dirichlet problem (2.10) D α A α (x, u, Du, D 2 u) + B(x, u, Du, D 2 u) = f in, (2.11) D α u = 0, α = 0, 1, on. The following remark provides correctness of the definition of a generalized solution to problem (2.10), (2.11). Remark 2.1. By (2.1), (2.2), (2.5) and imbedding (1.6), for every u, v W 2,p 0 () and every α Λ with α = 2, the function A α (x, u, Du, D 2 u)d α v is summable on, and by (2.1) and (2.7), for every u, v W 2,p 0 () L (), the function B(x, u, Du, D 2 u)v is summable on. Moreover, it follows from (1.6) and (2.9) that, for every v W 2,p 0 (), the function fv is summable on. Definition 2.2. A generalized solution of problem (2.10), (2.11) is a function u W 2,p 0 () L () such that, for every function v

7 TH XISTNC OF CONTINUOUS SOLUTIONS 673 W 2,p 0 () L (), (2.12) A α (x, u, Du, D 2 u)d α v dx + B(x, u, Du, D 2 u)v dx = fv dx. The next theorem is the main result of the present article. Theorem 2.3. Suppose R n is open and bounded, with n 3. Suppose also that the assumptions in (2.2) (2.9) hold with p = n/2, and with the functions g 1, g 2, g 3, g 4 and f belonging to L τ (), τ > 1. Let M be a majorant for the norms g 1 τ, g 2 τ, g 4 τ and f τ, and let, for almost every x and for every η R N 1 and ζ, ζ R N, ζ ζ, the following inequality holds: (2.13) [A α (x, η, ζ) A α (x, η, ζ )](ζ α ζ α) > 0. Then there exists a generalized solution u 0 C() of problem (2.10), (2.11) such that u 0 C 1 where C 1 is a positive constant depending only on n, p, p 0, p 1,, c 1, c 2, τ and M. We will prove Theorem 2.3 in Section 3. First, we give some remarks and an example of functions satisfying conditions (2.4) (2.8) and (2.13). Remark 2.4. The proof of the existence of the solution u 0 is based on the consideration of a sequence of approximate problems for equations with bounded lower-order terms, obtaining the uniform boundedness of their solutions {u i and the subsequent limit passage. At the same time, solvability of the approximate problems is established using the results of [9] on solvability of equations with pseudomonotone operators. By virtue of condition (2.8) the proof of the uniform boundedness of the solutions {u i follows the proof of the boundedness of arbitrary generalized solution u W m,p 0 () for equation (1.1) in [4]. So we omit this proof here. The limit passage in the approximate problems is justified using ideas of [7, 8].

8 674 MYKHAILO V. VOITOVYCH Remark 2.5. Suppose that conditions (2.2) (2.7) and (2.9) hold with p = n/2 and with the functions g 1, g 2, g 3, f L τ (), τ > 1. Let u W 2,p () L () be a generalized solution of equation (2.10), that is, for every function v W 2,p 0 () L () equality (2.12) is true. Then, by Theorem 1.1, we have the inclusion u C(). xample 2.6. Let, for every α Λ with α = 2, A α : R N2 be the function defined by R ( A α (x, ξ) = β =2 ξ 2 β ) (p 2)/2 ξ α + ξ p 1 β. β 1 Then the functions {A α : α = 2 satisfy inequalities (2.4) and (2.5) (with the exponents p 0 = p 1 = ( p 1)p/(p 1)) and (2.13). Next, for every (x, ξ) R N2, we set B(x, ξ) = ξ 0 b 1 ( ξ 0 ) { 1 + ξ α n [ ln(2 + ξ α ) ] 1 + ξ α p [ ln ln(3 + ξ α ) ] 1 α =1 where b 1 is an arbitrary nonnegative continuous function on R +, for example b 1 (t) = exp(t λ ), λ > 0. Then the function B satisfies inequalities (2.7), (2.8) and ψ(t) = [ ln ln(3 + t) ] 1, t R Proof of Theorem 2.3. Step 1. Suppose that conditions (2.2) (2.9) and (2.13) are satisfied with p = n/2 and with the functions g 1, g 2, g 3, g 4, f L τ (), τ > 1. Let M be a majorant for g 1 τ, g 2 τ, g 4 τ and f τ. By c i, i = 3, 4,..., we shall denote positive constants, depending only on n, p, p 0, p 1,, c 1, c 2, τ and M. For every i N, we define the function B i : R N 2 R by B i (x, ξ) = B(x, ξ) 1 + B(x, ξ) /i, (x, ξ) RN 2.

9 TH XISTNC OF CONTINUOUS SOLUTIONS 675 (3.1) (3.2) Obviously, for every i N and for every (x, ξ) R N 2, B i (x, ξ) i, B i (x, ξ)ξ 0 g 4 (x), { (3.3) B i (x, ξ) b( ξ 0 ) 1 + α =1,2 ξ α n/ α ψ( ξ α ) + g 3 (x). From (2.1) (2.5), (2.13), (3.1) and embedding (2.9) and the results of [9] on solvability of equations with pseudomonotone operators, it follows that, if i N, then there exists a function u i W 2,p 0 () such that, for every function v W 2,p 0 (), (3.4) A α (x, u i, Du i, D 2 u i )D α v dx Observe that, for every i N, (3.5) u i W 2,p 0 () c 3. + B i (x, u i, Du i, D 2 u i )v dx = fv dx. In fact, fixing an arbitrary i N and putting into (3.4) the function u i instead of v, we obtain A α (x, u i, Du i, D 2 u i )D α u i dx This, along with (2.4) and (3.2), implies that { c 1 D α u i p dx c 2 + B i (x, u i, Du i, D 2 u i )u i dx = fu i dx. { D α u i p dx + fu i dx + (g 1 + g 4 ) dx.

10 676 MYKHAILO V. VOITOVYCH From this inequality, estimating the first addend on the right-hand side by means of Hölder s and Young s inequalities and (2.1), (2.3), and the second addend by means of Hölder s, Young s inequalities and (2.1), we deduce (3.5). Taking into account inequalities (3.1), (3.2) and (3.5), inclusions g 1, g 2, g 4, f L τ (), τ > 1, and using the reasoning of [4], we establish that, for every i N, (3.6) u i c 4. By virtue of (3.5) and the compactness of the embedding W 2,p 0 () W 1,λ 0 () with λ < n, there exist an increasing sequence {i j N and a function u 0 W 2,p 0 () such that (3.7) (3.8) (3.9) u ij u 0 weakly in W 2,p 0 (), u ij u 0 almost everywhere in, D α u ij D α u 0 almost everywhere in, if α = 1. Now, from (3.6) and (3.8) we deduce the estimate (3.10) u 0 c 4. Step 2. For every i N, we set Φ i = [ Aα (x, u i, Du i, D 2 u i ) A α (x, u i, Du i, D 2 u 0 ) ] (D α u i D α u 0 ). Let us demonstrate that (3.11) lim j Φ ij dx = 0. Let j N. Since u ij u 0 W 2,p 0 (), by virtue of (3.4), we have Φ ij dx = f(u ij u 0 ) dx (3.12) B ij (x, u ij, Du ij, D 2 u ij )(u ij u 0 ) dx { A α (x, u ij, Du ij, D 2 u 0 )(D α u ij D α u 0 ) dx.

11 TH XISTNC OF CONTINUOUS SOLUTIONS 677 The integrals on the right-hand side of (3.12) tend to zero as j. In fact, by (3.6) and (3.8), we have (3.13) lim f(u ij u 0 ) dx = 0. j Next, we fix an arbitrary ε > 0. By virtue of (2.6) and the non-negativeness of the function ψ, there exists the number K > 1 depending only on ψ and ε such that (3.14) 0 ψ(t) < ε if t > K. We set b = max s [0, c4] b(s). By (3.3) and (3.6), we have B ij (x, u ij, Du ij, D 2 u ij )(u ij u 0 ) dx B ij (x, u ij, Du ij, D 2 u ij ) u ij u 0 dx (3.15) b D α u ij n/ α ψ( D α u ij ) u ij u 0 dx α =1,2 + (g 3 + b) u ij u 0 dx. Let α Λ, α = 1, 2 and ψ K = max t [ K,K] ψ(t). Using (3.6), (3.10) and (3.14), we obtain D α u ij n/ α ψ( D α u ij ) u ij u 0 dx = { D α u ij n/ α ψ( D α u ij ) u ij u 0 dx (3.16) D α u ij K + { D α u ij n/ α ψ( D α u ij ) u ij u 0 dx D α u ij >K K n ψ K u ij u 0 dx + 2c 4 ε D α u ij n/ α dx. From (3.15), (3.16), (3.5) and (2.1) we deduce the inequality B ij (x, u ij, Du ij, D 2 u ij )(u ij u 0 )dx

12 678 MYKHAILO V. VOITOVYCH (g 3 + b + bψ K K n N 2 ) u ij u 0 dx + 2c 4 c 5 b ε. From this and (3.6), (3.8) and an arbitrary choice of ε, it follows that (3.17) lim B ij (x, u ij, Du ij, D 2 u ij )(u ij u 0 ) dx = 0. j By virtue of (3.8) and (3.9), for every α Λ with α = 2, we have (3.18) A α (x, u ij, Du ij, D 2 u 0 ) A α (x, u 0, Du 0, D 2 u 0 ) a.e. in. From (2.2), (2.5) and (3.7), the property of absolute continuity of the Lebesgue integral and the compact embeddings W 2,p 0 () W 1,λ 0 () with λ < n and W 2,p 0 () L κ () with 1 κ < +, it follows that { (3.19) lim sup A α (x, u ij, Du ij, D 2 u 0 ) p/(p 1) dx = 0, 0 j N. Using (3.18), (3.19) and the convergence theorem of Vitali, we establish the following assertion: if, α Λ and α = 2, then A α (x, u ij, Du ij, D 2 u 0 ) A α (x, u 0, Du 0, D 2 u 0 ) strongly in L p/(p 1) (). From this and (3.7), it follows that { (3.20) lim A α (x, u ij, Du ij, D 2 u 0 )(D α u ij D α u 0 ) dx = 0. j Now, the validity of equality (3.11) follows from (3.12), (3.13), (3.17) and (3.20). Step 3. We now demonstrate that, for every α Λ with α = 2, (3.21) D α u ij D α u 0 in measure. For this purpose, we introduce some auxiliary functions and sets. Let Φ : R be the function defined by Φ(x) = inf j N Φ ij (x). Then Φ is an infimum of countably many measurable functions, and

13 TH XISTNC OF CONTINUOUS SOLUTIONS 679 hence measurable (see [6, Section 20]); moreover, by virtue of (2.5), we have (3.22) Φ L 1 (). Further, let for every x, A x : R N 1 R N R N R be the function such that, for every triplet (η, ζ, ζ ) R N1 R N R N, A x (η, ζ, ζ ) = [ Aα (x, η, ζ) A α (x, η, ζ ) ] (ζ α ζ α). Since, for every α Λ with α = 2, A α is a Carathéodory function and for almost every x and for every η R N 1 and ζ, ζ R N, ζ ζ, inequality (2.13) holds, there exists a set of measure zero such that (i) for every x \ the function A x is continuous in R N 1 R N R N ; (ii) for every x \, η R N1 and ζ, ζ R N, ζ ζ, we have A x (η, ζ, ζ ) > 0. For every θ > 0, σ > 0 and for every ν > σ, we set G θ,σ,ν = { (η, ζ, ζ ) R N 1 R N R N : ζ α ζ α σ, ζ α ν, ζ α ν, η α θ. vidently, for every θ > 0, σ > 0 and for every ν > σ, the set G θ,σ,ν is nonempty, closed and bounded in R N 1 R N R N. Let, for every θ > 0, σ > 0 and for every ν > σ, µ θ,σ,ν : R be the function such that min A x if x \, µ θ,σ,ν (x) = G θ,σ,ν 0 if x. Using properties (i) and (ii), we establish that, if θ > 0, σ > 0 and ν > σ, then (3.23) µ θ,σ,ν (x) > 0 for every x \.

14 680 MYKHAILO V. VOITOVYCH Now, we pass to the immediate proof of assertion (3.21). We fix σ > 0 and ε > 0. Using (2.1) and (3.5), we obtain that, for every θ > 0, ν > 0 and for every i N, { ( ) θ meas D α u i θ D α u i dx c 6, { ν meas D α u i ν { { D α u i θ D α u i ν ( Therefore, there exist θ > 0 and ν > max(1, σ) such that { sup meas j N D α u ij θ ε, (3.24) { sup meas j N { meas For every j N, we set D α u ij ν ε, D α u 0 ν ε. ) D α u i dx c 7. { j = D α u ij θ, D α u ij ν, D α u 0 ν, D α u ij D α u 0 σ. Let j N and x j \. We have D α u ij θ, D α u ij ν, D α u 0 ν, D α u ij D α u 0 σ.

15 TH XISTNC OF CONTINUOUS SOLUTIONS 681 Hence, (u ij (x), Du ij (x), D 2 u ij (x), D 2 u 0 (x)) G θ,σ,ν. Then, by virtue of the definition of µ θ,σ,ν and A x, we have µ θ,σ,ν (x) Φ ij (x), and hence, µ θ,σ,ν (x) Φ(x), which together with (3.23) yields (3.25) Φ > 0 almost everywhere in j \. Now, taking into account (2.13), we conclude that for every j N, Φ dx Φ ij dx Φ ij dx. j j This and (3.11) imply that lim j j Φ dx = 0. Hence, taking into account (3.22) and (3.25) and applying [7, Lemma 5], we deduce that (3.26) lim j meas j = 0. Obviously, for every j N, { meas D α u ij D α u 0 σ { + meas j=1 { meas D α u ij > ν { + meas From this and (3.24) and (3.26), we infer (3.21). D α u ij > θ D α u 0 > ν + meas j. We remark that, in the proof of assertion (3.21) we used some ideas of [7, 8]. Step 4. We now prove that the following assertions hold: (iii) for every function v W 2,p 0 (), lim sup 0 A α (x, u ij, Du ij, D 2 u ij )D α v dx = 0, ; j N

16 682 MYKHAILO V. VOITOVYCH (iv) for every function v W 2,p 0 () L (), lim sup B ij (x, u ij, Du ij, D 2 u ij )v dx = 0,. 0 j N In fact, let j N, v W 2,p 0 (), and let be an arbitrary measurable set. Using Hölder s inequality for sums and integrals along with (2.1), (2.2), (2.5) and (3.5), we obtain A α (x, u ij, Du ij, D 2 u ij )D α v dx [ [ c 8 [ { { ] (p 1)/p A α (x, u ij, Du ij, D 2 u ij ) p/(p 1) dx ] 1/p D α v dx p { dx] 1/p D α v p. This and the property of absolute continuity of Lebesgue integral imply that assertion (iii) holds. Now, let v W 2,p 0 () L () and ε > 0. By analogy with (3.15) and (3.16), we establish that B ij (x, u ij, Du ij, D 2 u ij )v dx b D α u ij n/ α ψ( D α u ij ) v dx α =1,2 + (g 3 + b) v dx, D α u ij n/ α ψ( D α u ij ) v dx K n ψ K v dx + ε v D α u ij n/ α dx, α = 1, 2.

17 TH XISTNC OF CONTINUOUS SOLUTIONS 683 From the last two inequalities and (2.1) and (3.5), it follows that B ij (x, u ij, Du ij, D 2 u ij )v dx (g 3 + b + b ψ K K n N 2 ) v dx + εc 5 b v. This and the property of absolute continuity of the Lebesgue integral and an arbitrary choice of ε imply that assertion (iv) holds. Using (3.8), (3.9), (3.21), assertions (iii) and (iv) and the convergence theorem of Vitali, we establish that, for every function v W 2,p 0 (), { lim A α (x, u ij, Du ij, D 2 u ij )D α v dx j { = A α (x, u 0, Du 0, D 2 u 0 )D α v dx, and, for every function v W 2,p 0 () L (), B ij (x, u ij, Du ij, D 2 u ij )v dx = B(x, u 0, Du 0, D 2 u 0 )v dx. lim j From this and (3.4), it follows that, for every function v W 2,p 0 () L (), A α (x, u 0, Du 0, D 2 u 0 )D α v dx + B(x, u 0, Du 0, D 2 u 0 )v dx = fv dx. The properties obtained of the function u 0 allow us to conclude that u 0 is a generalized solution of problem (2.10), (2.11). By Remark 2.5, this solution is continuous at every interior point of the set. Theorem 2.3 is proved. Acknowledgments. I wish to thank the Referee for useful suggestions.

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