On nonlinear and quasiliniear elliptic functional differential equations

We consider nonlinear elliptic 
functional differential equations. The corresponding operator has the form of a product of nonlinear elliptic differential mapping and linear difference mapping. It were obtained sufficient conditions for solvability of the Dirichlet problem. A concrete example shows that a nonlinear differential--difference operator may not be strongly elliptic even if the nonlinear differential operator is strongly elliptic and the linear difference operator is positive definite. The analysis is based on 
the theory of pseudomonotone--type operators and linear theory of elliptic 
functional differential operators.


1.
Introduction. Nonlinear elliptic differential equations have been considered since the middle of the 20th century by many mathematicians (see [16,1,2,9]). Abstract nonlinear elliptic functional differential equations were studied in [5]. At that moment it was no explicit example of nonlinear elliptic functional differential equations. In the 80s-90s, the basis of the theory of linear elliptic functional differential equations was created, see [12,13]. This theory was applied for study of elliptic and parabolic functional differential equations with nonlinearity in lower order terms, see [10,14]. In this paper we study elliptic functional differential equations with nonlinearity in terms with higher order derivatives. We note that the essentially nonlinear elliptic functional differential operators equal to the product of the p-Laplacian and positive definite difference operators were considered in [15].
Let Q ⊂ R n be a bounded domain with a boundary ∂Q ∈ C ∞ , or Q = (0, d) × G, G ⊂ R n−1 be a bounded domain (with a boundary ∂G ∈ C ∞ if n ≥ 3). If n = 1 we denote Q = (0, d). We consider the problem with the boundary condition where f ∈ W −1 q (Q), 1/q + 1/p = 1, and 1 < p < ∞. Here functional differential operator A R is a composition of nonlinear differential operator A given by the Au and of linear difference operator R given by the formula where a h ∈ R, M ⊂ Z n is a finite set of vectors with integer coordinates (similarly we can consider the case of commensurable shifts). Section 2 deals with the properties of difference operator R Q , which is uniquely defined by operator R and domain Q.
In Section 3, we prove solvability of problem (1.1)- (1.4). For this we study such properties of operator A R as psuedo-monotonocity, coercivity, and property (S + ).
In Section 4, we formulate the solvability conditions for 2m-order nonlinear elliptic functional differential problems (1.1)-(1.2) with Section 5 is devoted to second order quasilinear elliptic functional differential equations. We assume that coefficients of differential operator are differentiable and satisfy algebraic condition of strong ellipticity. Section 6 consists of examples. We show that a product of positive definite difference operator and strongly elliptic differential operator can form functional differential operator that does not satisfy the strong ellipticity condition. For linear theory such situation is not the case.
2. Difference operators. We consider properties of difference operators in the spaces L p (Q) and W k p (Q), 1 < p < ∞. Here W k p (Q) is the Sobolev space of functions u ∈ L p (Q) having all generalized derivatives D α u ∈ L p (Q) (|α| ≤ k) with the form If p = 2, these properties were studied in [12,13]. For 1 < p < ∞, corresponding generalizations were obtained in [15].
Denote by M the additive group generated by the set M. Let Q r be the open connected components of the set Q \ ( h∈M (∂Q + h)).
Definition 2.1. The set Q r is called a subdomain. The family R of all subdomains Q r , r = 1, 2, . . . , is called the decomposition of the domain Q.
It is easy to see that the set R is at most countable and, in addition, r ∂Q r = h∈M (∂Q + h) ∩ Q and r Q r = Q.
For any subdomain Q r1 and arbitrary vector h ∈ M , either there exists Q r2 such that Q r2 = Q r1 + h, or Q r1 + h ⊂ R n \ Q, see Lemma 7.2 in [13,Ch.2,Sec.7]. Thus, the family R can be decomposed into disjoint classes as follows: subdomains Q r1 , Q r2 ∈ R belong to the same class if Q r2 = Q r1 + h for some h ∈ M . We denote the subdomains Q r by Q sl , where s is the number of class and l is the number of subdomain in the sth class. Obviously, each class consists a finite number N = N (s) of subdomains Q sl and N (s) ≤ ([diamQ] + 1) n . The set of classes may be either finite or countable, see Secton 7 in [13]. Let us illustrate these decompositions by two examples.
, and Q = (0, 3) × (0, 1). For domain Q, we define three subdomains from one class that correspond to the operator R Q : The second class of subdomains contains and Consider the properties of the difference operator R : is the operator of extension of functions from L p (Q) by zero to R n \ Q and P Q : Recall that the operator R is nonlocal. Translation by a vector h can map a point x ∈ Q into R \ Q. Therefore, boundary conditions (1.2) for differential difference equation (1.1) are specified not only on the boundary ∂Q but on the whole set R n \ Q. To take into account boundary conditions (1.2) in the difference operator R , we introduce the operator I Q . Thus, a function u(x) defined on Q is replaced 872 OLESYA V. SOLONUKHA by the function (I Q u)(x) defined on the whole space R n . After the action of the operator R, we again obtain a function defined on the whole space R n . The operator P Q is introduced to obtain the restriction of the function (RI Q u)(x) onto Q.
Denote by L p ( l Q sl ) the subspace of functions in L p (Q) that vanish outside of l Q sl (l = 1, . . . , N (s)). We introduce a bounded operator P s : Let us introduce the isomorphism of the reflexive Banach spaces where l = 1, . . . , N = N (s) and the vector h sl is such that Q s1 +h sl = Q sl (h s1 = 0), is the operator of multiplication by the matrix In order to illustrate this lemma, we construct the matrices R s for Examples 2.2 and 2.3.
Example 2.8. In Example 2.2, we have one class of subdomains with three elements. Then the matrix R 1 corresponding to operator R Q has the form ON NONLINEAR AND QUASILINIEAR ELLIPTIC FDE'S 873 Example 2.9. In Example 2.3, we have two classes of subdomains with three and two elements. The matrices R 1 and R 2 corresponding to operator R Q have the form Since Q is a bounded domain, by virtue of (2.4), a number of different matrices R s is finite if the coefficients a h are constants. Let n 1 denote this number, and let R sν denote all different matrices R s (ν = 1, . . . , n 1 ). . Let the coefficients a h be constants, and k ≥ 0 be integer. Then, . Let the coefficients a h be constants. Then, for all u ∈ L p (Q) such that u ∈ W m p (Q sl ) (s = 1, 2, . . .; l = 1, . . . , N (s)), we have R Q u ∈ W m p (Q sl ) and Moreover, if det R sν = 0 (ν = 1, . . . , n 1 ), then R −1 Q u ∈ W m p (Q sl ) and Here the constants c 1 , c 2 > 0 are independent of s and u.
Corollary 2.13. It is easy to see that 3. General criteria of solvability for elliptic functional differential equations. Since Lebesgue spaces and Sobolev spaces are reflexive Banach ones for 1 < p < ∞, then in order to formulate general properties of operators we introduce some abstract reflexive Banach space X. We denote its topological dual space by X * and the corresponding duality by ·, · : X × X * → R.
Definition 3.1. A mapping A : X → X * is called demicontinuous, if it is continuous from the strong topology of X into the weak topology of X * . A mapping A : X → X * is called semicontinuous, if for any u, v, w ∈ X the function λ → A(u + λv), w is continuous from R to R.

874
OLESYA V. SOLONUKHA Definition 3.2. A mapping A : X → X * has property (S + ), if for arbitrary sequence u n → u weakly in X and sequence Au n → w weakly in X * such that lim n→∞ Au n , u n − u ≤ 0, (3.1) we have w = A(u) and u n k → u strongly in X for some subsequence {u n k } ⊂ {u n }.
Definition 3.3. A mapping A : X → X * is called pseudomonotone, if for arbitrary u n → u weakly in X such that condition (3.1) holds, we have lim n→∞ Au n , u n − ξ ≥ Au, u − ξ ∀ξ ∈ X.
i.e. for any u, y ∈W 1 p (Q) In order to formulate the necessary conditions for solvability of functional-differential equations we remind the classical conditions from theory of elliptic differential equations with operator of pseudomonotone-type. Let A i be such that the following conditions hold: I) Integrability condition: the functions A i satisfy the Caratheodory conditions, i.e. A i (·, ξ) are measurable for a.a. ξ ∈ R n+1 , and A i (x, ·) are continuous for a.a.
x ∈ Q; moreover, for a.a. x ∈ Q and for all ξ ∈ R n+1 we have The differential operator A :W 1 p (Q) → W −1 q (Q) given by the formula and satisfying conditions I)-III) is bounded, demicontinuous, pseudomonotone, and coercive, moreover, it has the property (S + ), see [8,4,11] for example. In order to describe conditions for the functions A i (x, u, ∂ 1 u, . . . , ∂ n u) and for the matrises We denote the elements of ζ by ζ li , where ζ ·i is the ith column of ζ, and ζ l· is the We assume that for any s the coefficients A i , i = 0, 1, . . . , n and the matrices R s satisfy the following conditions: (A0) Nondegeneracy condition: the coefficients a h ∈ R are constants and detR sν = 0 (ν = 1, . . . , n 1 ). (A1) Integrability condition: the functions A i satisfy the Caratheodory conditions, i.e. A i (x, ξ) are measurable in x for any ξ ∈ R n+1 and continuous in ξ ∈ R n+1 for a.a. x ∈ Q; moreover, for a.a. x ∈ Q and for all ξ ∈ R n+1 , we have where c 1 > 0, h ∈ L q (Q). (A2) Ellipticity condition: for any s, for a.a. x ∈ Q s1 , and for all ζ, η ∈ R N (s)×(n+1) such that η l0 = ζ l0 and η = ζ the following estimate holds: (A3) Coercivity condition: for any s, for a.a. x ∈ Q s1 , and for all ζ ∈ R N (s)×(n+1) , there exist c 2 > 0 and c 3 , c 4 ∈ R such that 1≤l≤N (s) 1≤i≤n Ellipticity condition (A2) is the modification of ellipticity condition II) for difference operator. In contrast to differential operators, see condition II), in ellipticity condition for functional-differential operators we must consider the connections of coordinates corresponding to the values of functions in different subdomains of the sth class. For this purpose in ellipticity condition (3.4) we use the weights in the form of matrix R −1 s . Theorem 3.6. Let conditions (A0), (A1), (A3) hold. Then the operator A R : given by formula (3.2) is bounded, demicontinuous, and coercive.
Proof. By Lemmas 2.11 and 2.12 operator R Q : [8,4,11]. Thus, the operator A R = AR Q : is demicontinuous as composition of linear bounded operator R Q and demicontinuous bounded operator A.
. Then we can use equality (2.7): (3.6) Here and further we sum over i = 1, . . . , n, if the opposite is not specified.
, from equalities (3.6), Lemmas 2.10, 2.11 and theorem on the equivalent norms inW 1 where constants k 1 , k 2 , k 3 do not depend on u; moreover, k 1 > 0. By Lemma 2.12 and condition (A1), where constants k 4 , k 5 do not depend on u. Using well-known formula ab ≤ where constants k 6 , k 7 do not depend on u. Thus, which is defined for any ζ, η such that ζ l0 = η l0 . Analogous function was proposed in [7]. For the convenience of the reader we prove its property completely.
Lemma 3.7. For any κ, C, C 1 > 0, there exists a positive function c(x) such that on the bounded set the following estimate holds: Here c(x) > 0 is defined for a.a. x ∈ Q s1 , it depends on κ, C, C 1 only, and it does not depend on ξ and η.
By ellipticity condition (A2) h(1) > 0. We introduce ξ 0 = η + ζ 0 . In this notation (1) for any χ ≥ 1. This result may be written in the following form: There exists the minimum of function c(x) on the closed bounded set U κ . Moreover, the value of this minimum is strongly positive because two last arguments of function H s (x, η +ζ 0 , η) are different and estimate (3.4) holds.
is pseudomonotone and has property (S + ).
Proof. We assume that u j → u weakly in X =W 1 p (Q) and Since the imbedding W 1 p (Q) ⊂ L p (Q) is compact, there exists a subsequence {u j k } that converges in L p (Q) to a function u. Without loss of generality we assume that u j k = u j . Using Lemma 2.5, condition (A1), and the Lebesgue theorem on a limit under integral, we have In the second relation we again consider the sequence {u j } instead of some subsequence. Let us rewrite (3.9) The first term of (3.9) I 1j tends to zero as j → ∞, since it is a product of convergent sequence in L q (Q) and weakly convergent to zero sequence in L p (Q). The second term of (3.9) I 2j tends to zero as j → ∞, since it is a product of weakly convergent sequence in L q (Q) and convergent to zero sequence in L p (Q). Hence, by virtue of (3.8), we have 0 ≥ lim j→∞ (I 1j + I 2j + I 3j ) = lim j→∞ I 3j . (3.10) Now we consider the third term of (3.9). Let w = R Q u, w j = R Q u j . By Lemma 2.10, there exists a bounded inverse operator R −1 Q . Then we can use equality (2.7): Then we obtain Then we continue estimate (3.10) using equality (3.11): By Lemma 3.7 the values of H s (x, ζ j , ζ) are nonnegative. Using countable additivity and positivity of the Lebesgue integral, we get that

This is possible if and only if
. We proved that A R has property (S + ). Therefore from Remark 3.4 it follows that the operator A R is pseudomonotone.
Unfortunately, if nonlinear differential operator satisfies conditions I)-III) we can not propose general condition on difference operator R Q such that A R satisfies conditions (A0) − (A3). We will illustrate this by examples, see Section 6. Now we consider the solvability of the Dirichlet problem Definition 3.9. A function u ∈W 1 p (Q) is called a generalized solution of the problem (3.12), (3.13), if the following integral identity holds for any ξ ∈W 1 p (Q): Proof. Conditions (A0) − (A3) guarantee that operator A R is bounded, demicontinuous and pseudomonotone, see Theorem 3.6 and Theorem 3.8, then problem (3.12),(3.13) has at least one generalized solution, see Theorem II.2.7 [8].
Let U be a set of generalized solutions of problem (3.12),(3.13). By Theorem 3.6, the operator A R is coercive. Hence the set U is bounded inW 1 p (Q). Therefore there is a sequence {u j } ⊂ U such that u j → u weakly inW 1 p (Q). We prove that u ∈W 1 p (Q) is a generalized solution of problem (3.12),(3.13). Since {u j } are solution of (3.12),(3.13), and u j → u weakly inW 1 p (Q), we have lim By virtue of Theorem 3.8 the operator A R has property (S + ). Hence A R u = f . Therefore the function u ∈W 1 p (Q) is a generalized solution of problem (3.12),(3.13). We have proved that the set of solutions of problem (3.12),(3.13) is weakly compact.
Note that conditions (A0) − (A3) are sufficient but not necessary for solvability of problem (3.12), (3.13). In [15] the problem with p-Laplacian was considered and conditions on R s was obtained. These conditions guarantee that the problem with p-Laplacian has at least one generalized solution.
4. Solvability of 2m-th order nonlinear functional differential equations. For the convenience of readers, in this section we formulate the results only. Proofs are similar to the ones in the previous section.
Denote by N 0 the number of all α such that |α| ≤ m. The differential operator for arbitrary ξ ∈W m p (Q). We denote by ζ the matrix of order N 0 × N (s) with column ζ ·α , |α| ≤ m. Let ζ l,α be the elements of this matrix and ζ l· be the lth line of ζ. Assumptions on A R : (A0 m ) Nondegeneracy condition: The coefficients a h ∈ R are constants and detR sν = 0 (ν = 1, . . . , n 1 ). (A1 m ) Integrability condition: The functions A α satisfy the Caratheodory conditions, i.e. A α (x, ξ) is measurable in x ∈ Q for any ξ ∈ R N0 and continuous in ξ for a.a. x ∈ Q; moreover, there exists a constant c 1 > 0 such that (A2 m ) Ellipticity condition: For arbitrary s, for a.a. x ∈ Q s1 , and for any ζ, η ∈ R N (s)×N0 such that η = ζ and ζ lβ = η lβ for |β| ≤ m − 1 the following estimate holds 1≤l≤N (s) |α|=m (A3 m ) Coercivity condition: For arbitrary s, for a.a. x ∈ Q s1 , and for any ζ ∈ R N (s)×N0 , there exist c 2 > 0, and c 3 , c 4 ∈ R such that Conditions (A0 m )-(A3 m ) are similar to corresponding conditions for a second order operator, see Section 3. (Ω), given by the formula
for any u, y ∈W 1 p (Q). Let us remind that in linear case a product of strongly elliptic operator A and positive definite operator R Q is strongly elliptic too. Thus, we show that in quasilinear case a composition of strongly elliptic operator A that satisfies algebraic condition of strong ellipticity, see [4], and symmetric positive definite operator R Q has useful properties such that the corresponding equation has a generalized solution.
We consider symmetric positive definite matrices R s . Then there exists symmetric positive definite matrices Theorem 5.2. Let p ∈ [2, ∞), and let {R s } be the set of matrices that correspond to operator R Q . We suppose that, for any s, the matrices R s are symmetric and positive definite. Moreover, we assume that the operator has differentiable coefficients A i (x, ξ) with respect to ξ and satisfies algebraic ellipticity condition: for any s, for a.a. x ∈ Q s1 , ζ ∈ R N (s)×n , and η ∈ R N (s)×n , 1≤m,l≤N (s) 1≤i,j≤nt where c 5 > 0 does not depend on x, ζ, and η, and for a.a. x ∈ Q and any ξ ∈ R n , c 6 > 0 does not depend on x and ξ. Then operator A R :W 1 p (Q) → W −1 q (Q) given by is strongly elliptic.
Recall that the differential operator A :W 1 p (Q) → W −1 q (Q) given by is strongly elliptic if for almost all x ∈ Q this operator satisfies the algebraic condition of strong ellipticity: 1≤i,j≤n ]. Thus estimate (5.2) is a generalization of the algebraic condition of strong ellipticity for nonlinear differential operator. The coefficients t lm andt lm of the matrices T s = √ R s andT s = R −1 s corresponding to the difference operator R Q allow to consider a contribution of different subdomains Q sm .
Integrating by parts and substituting ∇R Q u = ∇v + ∇w, ∇R Q y = ∇v, we obtain Then, using (2.7), we have By virtue of (2.2), we can rewrite I 1 in the following form From differentiability of A i we obtain Using the rules of the matrix multiplication, the left factor in this scalar product can be written as following: ; l = 1, · · · , N (s); j = 1, · · · , n).

(5.4)
Now we can use property (5.2): Using the well known estimate we obtain that Hence, Since the matricesT s and R s are nondegenerate, by virtue of (2.6), we have Strong ellipticity is proved.
Theorem 5.4. Let p ∈ [2, ∞), for any s, the symmetric and positive definite matrices R s correspond to R Q . Moreover, let operator A be given by the formula We assume that functions A i (x, ξ) are differentiable with respect to ξ and satisfy algebraic ellipticity condition: for any s, for a.a. x ∈ Q s1 , ζ ∈ R N (s)×n , and η ∈ R N (s)×n , 1≤m,l≤N (s) 1≤i,j≤nt for a.a. x ∈ Q and any ξ ∈ R n+1 , c 6 > 0 does not depend on x and ξ. Then the operator A R = AR Q :W 1 p (Q) → W −1 q (Q) is demicontinuous, pseudomonotone, and coercive. Moreover, it has property (S + ).
2. We show that the operator A R has property (S + ). Clearly, Using condition (5.6), similarly to Theorem 5.2 we can show that the operator A u R :W 1 p (Q) → W −1 q (Q) is strongly elliptic. For the convenience of readers, we shall present the proof.
Let w = R Q (u − y), v = R Q y, where u, y ∈W 1 p (Q). By Lemma 2.10, there exists a bounded inverse operator R −1 Integrating by parts and substituting ∇R Q u = ∇v + ∇w, ∇R Q y = ∇v, we obtain Then, using (2.7), we have By virtue of (2.2), we can rewrite I 3 in the following form From differentiability of A i we obtain Let us consider the integrand of Using the rules of the matrix multiplication, left vector in this product can be written as following:T , (∇v + τ ∇w)(x + h sl ))dτ . Using notation (5.4), by property (5.6) we obtain Using the well known estimate Hence, Since the matricesT s and R s are nondegenerate, by virtue of (2.6), we have Hence the operator A u R is strongly elliptic. Therefore it is pseudomonotone and has property (S + ).
Let us show that A R has property (S + ). We assume that y m → y weakly in W 1 p (Q). Then, passing to a subsequence and using the imbedding theorem, we have y m → y in L p (Q). For continuous operator R Q , we have that R Q y m → R Q y weakly in W 1 p (Q) and R Q y m → R Q y in L p (Q). At the same time, A i (·, R Q y m , ∇R Q y) → A i (·, R Q y, ∇R Q y) in L q (Q) by continuity of A i , see the proof of Theorem 3.8. Then we have lim Therefore, repeating the above arguments, we derive Thus the operator A R has property (S + ).
3. Let us show that A R is coercive. Clearly, For the first term, we have estimate (5.8): For the second term, from the continuity of A i and inequality (5.7) we obtain Using boundedness of R Q (see estimate (2.5)) and the well-known formula ab ≤ (µa) p p + b q µ q q , we can find constants c 13 , c 14 such that c 13 < c 12 and . We have proved that the operator A R is coercive. 6. Examples. In this section, we consider examples with strongly elliptic differential operators. We formulate conditions for a difference operator, under which Theorem 3.10 holds. We also demonstrate that even in the case of symmetric positive definite matrices the equation can have several solutions. It will be considered an example of nonsymmetric difference operator. We shall demonstrate that in this example the condition of ellipticity is not fulfilled. Example 6.1. Let p = 4, Q = (0, 2) × (0, 1), and let We consider the problem where w ∈ C 2 (Q) is a given function.
For domain Q, we define two subdomain from one class that correspond to the operator R Q : x 2 The matrix R 1 corresponding to R Q has the form This matrix is nondegenerate, if |γ| = 1.
Obviously, condition (A1) holds. It is sufficient to show that conditions (A2) and (A3) are fulfilled. Note that The left part of ellipticity condition (A2) for A R is given by the formula: 1≤i≤2 1≤l≤2 We introduce the new variables: . Note that c > 0 and d > 0. We have This expression is positive for any a, b, c > 0, d > 0, if We have proved, that if |γ| < 1, then for any a, b ∈ R \ {0}, c > 0, and d > 0 Hence, in (6.3) we have the sum of positive summands, if |γ| < 1. Ellipticity condition (3.4) is true if |γ| < 1.
In the following examples, the nonlinear differential operator is strongly elliptic, while the symmetric part of difference operator corresponds to positive definite matrix. In linear case these conditions are sufficient for differential-difference operator to be strongly elliptic. Hence a corresponding linear problem has a unique generalized solution, see [13,Chap.II,§9]. For nonlinear problem it is not true. In particular, nonlinear differential-difference operator can be not strongly elliptic, moreover, ellipticity condition (A2) can be false for them. Ru(x 1 , x 2 ) = u(x 1 , x 2 ) + γu(x 1 + 1, x 2 ).
We consider properties of the operator A R = AR Q , if Au(x) = − 1≤i≤2 ∂ i (∂ i u(x)) 3 .
Hence, for any γ there exists the pair of vectors (ξ, η) such that ellipticity condition is not fulfilled for this pair.