Multiple solutions of nonlinear partial functional differential equations and systems

We shall consider weak solutions of initial-boundary value problems for semilinear and nonlinear parabolic differential equations with certain nonlocal terms, further, systems of elliptic functional differential equations. We shall prove theorems on the number of solutions and find multiple solutions. These statements are based on arguments for fixed points of some real functions and operators, respectively, and existence-uniqueness theorems on partial differential equations (without functional terms).


Introduction
It is well known that mathematical models of several applications are functional differential equations of one variable (e.g.delay equations).In the monograph by Jianhong Wu [8] semilinear evolutionary partial functional differential equations and applications are considered, where the book is based on the theory of semigroups and generators.In the monograph by A. L. Skubachevskii [7] linear elliptic functional differential equations (equations with nonlocal terms and nonlocal boundary conditions) and applications are considered.A nonlocal boundary value problem, arising in plasma theory, was considered by A. V. Bitsadze and A. A. Samarskii in [1].
It turned out that the theory of pseudomonotone operators is useful to study nonlinear (quasilinear) partial functional differential equations (both stationary and evolutionary equations) and to prove existence of weak solutions (see [2,4,5]).
In [6] we considered some nonlinear elliptic functional differential equations where we proved theorems on the number of weak solutions of boundary value problems for such equations and showed existence of multiple solutions.
In the present work we shall consider nonlinear parabolic functional equations and systems of elliptic functional equations.By using ideas of [6]: arguments for fixed points of L. Simon certain real functions and operators, respectively, we shall prove theorems on the number of solutions of such problems and show existence of multiple solutions.
First we recall the definition of weak solutions of boundary value problems for the nonlinear (quasilinear) elliptic equation with (zero) Dirichlet boundary condition u(x) = 0 on ∂Ω.
Let Ω ⊂ R n be a bounded domain with sufficiently smooth boundary (e.g.∂Ω ∈ C 1 ), 1 < p < ∞.Denote by W 1,p (Ω) the usual Sobolev space of real valued functions with the norm Further, let V ⊂ W 1,p (Ω) be a closed linear subspace containing C ∞ 0 (Ω), V the dual space of V, the duality between V and V will be denoted by •, • .
Weak solutions of (1.1) are defined as functions u ∈ V satisfying for all v ∈ V where F ∈ V is a given element and V = W By using the theory of monotone operators, one can prove existence and uniqueness theorems on weak solutions of the above boundary value problems.Namely, consider the (nonlinear) operator A : V → V , defined by One can formulate sufficient conditions on functions a j which imply that the operator A : V → V is bijection, i.e. for arbitrary F ∈ V there exists a unique solution u ∈ V of the equation A(u) = F. (See [3,9].)Namely, these sufficient conditions are: (A1) the functions a j : Ω × R n+1 → R satisfy the Carathéodory conditions; (A2) there exist a constant c 1 and a function

holds.
A typical example, having this property, is operator A defined by p-Laplacian p : with p ≥ 2 and constant c 0 > 0. (Clearly, 2 u = u.)Now we remind the definition of weak solutions of initial-boundary value value problems for nonlinear parabolic differential equations (for simplicity) with homogeneous initial and boundary condition.Denote by L p (0, T; V) the Banach space of functions u : (0, The dual space of L p (0, T; V) is L q (0, T; V ) where 1/p + 1/q = 1.Weak solutions of (1.3) with zero initial and boundary condition is a function u ∈ L p (0, T; V) satisfying D t u ∈ L q (0, T; V ) and for all v ∈ V, almost all t ∈ [0, T]. (For p ≥ 2, u ∈ L p (0, T; V) and D t u ∈ L q (0, T; V ) imply u ∈ C([0, T]; L 2 (Ω)) thus the initial condition u(0) = 0 makes sense.)There are well-known conditions on functions a j which imply that the operator A : L p (0, T; V) → L q (0, T; V ) (defined in (1.4)) is bijection, so for arbitrary F ∈ L q (0, T; V ) there exists a unique weak solution u ∈ L p (0, T; V) of the problem (See [3,9].)A simple example for A is with a positive constant c 0 (here A is not depending on t).

Parabolic equations with real valued functionals, applied to the solution
First consider a semilinear parabolic functional equation of the form (i.e. the elliptic operator A in (1.4) is linear), where M : L 2 (0, T; V) → R is a given linear continuous functional, V ⊂ W 1,2 (Ω), k : R → R is a given continuous function, F 1 , F 2 ∈ L 2 (0, T; V ).Further, a jk , a 0 ∈ L ∞ (Ω), a jk = a kj and the functions a jk satisfy the uniform ellipticity condition for all ξ = (ξ 0 , ξ 1 , . . ., ξ n ) ∈ R n+1 , x ∈ Ω with some positive constants c 1 , c 2 .It is well known that in this case for all F ∈ L 2 (0, T; V ) there exists a unique weak solution u of denoted by u = B −1 F where B −1 : L 2 (0, T; V ) → L 2 (0, T; V) is a linear continuous operator.Consequently, u ∈ L 2 (0, T; V) is a weak solution of (2.1) if and only if This equality implies that ) Corollary 2.2.The number of weak solutions u of (2.1) (with homogeneous initial-boundary condition) equals the number of solutions λ of equation (2.5).E.g. assume that k ∈ C 1 (R) and the function h defined by has the property inf λ∈R h (λ) > 0 or sup λ∈R h (λ) < 0. Then for any F 2 ∈ L 2 (0, T; V ) the problem (2.1) has exactly one solution u.In this case the mapping L 2 (0, T; V ) → L 2 (0, T; V) which maps F 2 to u is continuous since h −1 : R → R is continuous.Further, assuming M(B −1 F 1 ) = 0, for arbitrary N = 0, 1, . . ., ∞ we can construct continuous functions k : R → R such that the initial-boundary value problem (2.1) has exactly N weak solutions, as follows.Let g : R → R be a continuous function having N zeros and define function k by the formula Then, clearly, equation (2.5) has N solutions.
Corollary 2.3.The number of solutions of (2.1), with fixed function k depends on F 2 (on the value of M(B −1 F 2 )).
This statement can be illustrated as follows.Let F 2 ∈ L 2 (0, T; V ) be fixed and consider µF 2 instead of F 2 with some parameter µ ∈ R. Then equation (2.5) has the form for µ > µ 0 the function g µ has no zeros and for µ = µ 0 has infinitely many zeros.E.g. let Consequently, for g µ has infinitely many zeros and for µ > µ 0 it has no zeros.It is not difficult to show that if + sin λ + 1 λ then for µ = µ 0 the function g µ defined by (2.8) has no positive zeros but for 0 < µ/µ 0 < 1 the function g µ has infinitely many zeros.Further, by using (2.5), if M(B −1 F 1 ) = 0, it is not difficult to construct continuous functions k such that for arbitrary . ., u N and for the function Then there exist ε > 0, δ > 0 (they are independent) such that F2 − F 2 L 2 (0,t;V ) < δ implies: for every j there exists a unique ũj ∈ L 2 (0, T; V) weak solution of (2.1) with the property ũj − u j L 2 (0,t;V) < ε, where on the right hand side of (2.1) F2 is instead of F 2 .Further, ũj depends continuously on F2 , belonging to δ neighborhood of F 2 .

L. Simon
Proof.Consider the function h defined by and apply the implicit function theorem to this function.Since there exist ε 0 , δ 0 > 0 such that for every fixed j, | c − c| < δ 0 implies that there exists a unique λj satisfying and λj depends continuously on c (in the δ 0 neighborhood of c = M(B −1 F 2 )).Hence we obtain the statement of Remark 2.4.
In this case the problem (2.1) with the right hand side F2 may have other solutions, too.(See the first example in Corollary 2.3.)Remark 2.5.The linear continuous functional M : L 2 (0, T; V) → R may have the form where K ∈ L 2 ((0, T) × Ω).In this case the value of solutions of the initial-boundary problem for (2.1) in some time t are connected with the values of u in all t ∈ [0, T].
Now consider nonlinear parabolic functional equations of the form where the nonlinear operator A has the form (1.4) and has the property l is a given positive continuous function and the numbers β, γ satisfy Theorem 2.6.A function u ∈ V is a weak solution of (2.10) with zero initial and boundary condition if and only if λ = M(u) satisfies the equation where B is defined by B(u) = D t u + A(u), i.e.B −1 (u) is the unique weak solution of (1.3) (with zero initial and boundary condition).
Proof.Define u µ by µ −β u with some positive number µ.Then thus a function u satisfies the equation if and only if Consequently, a function u ∈ V satisfies (2.10) in weak sense (i.e.(2.14 The solution of (2.16) is ũ = B −1 (F), therefore the weak solution of (2.10) is (2.17) Thus, if u satisfies (2.10) then λ = M(u) and u satisfy (2.13).
Conversely, if λ ∈ R is a solution of (2.13) then is a solution of (2.10) because so λ = M(u), further, u satisfies (2.10) in weak sense, since (2.10) holds if and only if Corollary 2.7.The number of weak solutions of (2.10) equals the number of roots of (2.13).
Further, assuming M[B −1 (F)] > 0, for arbitrary N = 1, 2, . . ., ∞ one can construct a continuous positive function l such that (2.10) has exactly N solutions, in the following way.Let g : R → R be a continuous function such that g(λ) + λ > 0 for all λ ∈ R and g has N real roots.Then for (2.10) has N weak solutions.
Remark 2.8.Let the functional l be fixed.Then the number of solutions of (2.10) depends on F. Similar examples can be constructed as in Corollary 2.3.
Remark 2.9.An example for functional M with property (2.12) is integral operator of the form

Parabolic equations with nonlocal operators
Now consider partial functional equations of the form where A is a uniformly elliptic linear differential operator (see (2.1) or (2.2)) and C : L 2 (0, T; V) → L 2 (0, T; V ) is a given (possibly nonlinear) operator.Clearly, u ∈ V satisfies (3.1) if and only if where G : L 2 (0, T; V) → L 2 (0, T; V is a given (possibly nonlinear) operator, i.e. u is a fixed point of G. Then Now we consider three particular cases for G.

1.
The operator G is defined by where [D t K(t, τ, x, y) + A x K(t, τ, x, y)]u(τ, y)dτdy + D t F(t, x) + A x F(t, x).(3.5) (A x K(t, τ, x, y) denotes the differential operator applied to x → K(t, τ, x, y).) Further, if 1 is an eigenvalue of the linear integral operator L with multiplicity N then (3.5) may have N linearly independent solutions.

Proof. Equation (3.5) is equivalent with
which implies Theorem 3.1 since for a solution u ∈ L 2 ((0, T) × Ω) of the last equation we have u ∈ L 2 (0, T; V), D t u ∈ L 2 (0, T; V ) by the assumption of the theorem.

Remark 3.2.
Similarly to the problems in the previous section, the value of solutions u of (3.5) in some time t, are connected with the values of u for t ∈ [0, T].

Now consider the case
where Let 1 be an eigenvalue and v ∈ L 2 (Ω) an eigenfunction of G then by assumption v ∈ V. Further, let τ ∈ C 1 [0, T] with the property τ(0) = 0 then functions u, defined by u(t, x) = τ(t)v(x) are weak solutions of (3.7) with 0 initial condition.Remark 3.4.In the case of equations (3.7) the value of solutions u in some t are connected with the values of u only in t. (Compare to Remarks 2.5 and 3.2.)

Now consider operators G of the form
where operator L is defined in (3.4) and its kernel has the same smoothness property, P : Here assume that 1 is not an eigenvalue of the integral operator L : L 2 ((0, T) × Ω) → L 2 ((0, T) × Ω).Thus the number of solutions of (3.9) equals the number of the roots of (3.10).
Proof.Equation (3.9) is fulfilled if and only if u is a solution, belonging to since by the properties of F, H and L, for such a solution u ∈ L 2 (0, T; V), and D t u ∈ L 2 (0, T; V) hold.Thus (3.9) is equivalent with u ∈ L 2 ((0, T) × Ω) and Consequently, (3.11) (and so (3.9)) is satisfied if and only if λ = Pu satisfies (3.10).
Remark 3.8.For fixed functions h, F the number of solutions of (3.9) depends on H by (3.10).It may happen that the number of solutions of the problem with µF (where µ is a real parameter) is 0 for µ > µ 0 and is some N (= 1, 2, . . ., ∞) for µ = µ 0 .(See Corollary 2.3.)Further, assuming that for the function ϕ defined by we have inf λ∈R ϕ (λ) > 0 or sup λ∈R ϕ (λ) < 0 then for any (sufficiently smooth) H the equation (3.9) has exactly one solution.

Systems of elliptic functional equations
First consider systems of semilinear elliptic functional differential equations of the form where A j : V → V are uniformly elliptic linear differential operators (V ⊂ W 1,2 (Ω)), F j , G j , H j ∈ V ; M, N : V → R are linear continuous functionals and l j , k j : R → R are continuous functions.Clearly, u, v are weak solutions of (4.1), (4.2) with homogeneous boundary conditions if and only if where λ 1 = Mu and λ 2 = Nv are roots of the algebraic system ) Proof.If u, v are solutions of (4.1), (4.2) then by (4.3), (4.4) Thus λ 1 = Mu and λ 2 = Nv satisfy (4.7), (4.8).
Theorem 4.4.Assume that the function χ defined by is strictly monotone and its range is R. Then λ 1 , λ 2 are solutions of the system (4.7),(4.8) if and only if λ 2 is the root of the equation and Consequently, the number of solutions of the system (4.1),(4.2) equals the number of λ 2 ∈ R roots of (4.12).Further, if N(A −1 2 G 2 ) = 0 then for arbitrary continuous functions k 1 , l 2 one can construct continuous functions k 2 such that (4.1), (4.2) has N (= 0, 1, . . ., ∞) solutions as follows.Let g be any continuous function for which ) (g is defined by (4.12)), is strictly monotone and its range is R then for any H 1 , H 2 ∈ V there is a unique solution of (4.1), (4.2) with zero initial and boundary condition.Now consider the following system of nonlinear elliptic functional differential equations: (4.15) where the nonlinear elliptic differential operators A j : V → V of the form (1.2) are bijections (V ⊂ W 1,p (Ω)) and have the property (2.11), i.e.
A j (µu) = µ p−1 A j (u) (µ > 0, p > 1) (4.17) (e.g.A j may have the form A j u = − p u + c j u|u| p−1 with constants c j > 0); k j , l j are given continuous functions, M, N : V → R are nonnegative continuous functionals with the property and F j ∈ V .
Corollary 4.8.The number of weak solutions of (4.15), (4.16) equals the number of roots of the algebraic system (4.23),(4.24).Theorem 4.9.Assume that the function χ defined by is strictly monotone and its range is R. Then λ 1 , λ 2 are solutions of (4.23), (4.24) if and only if λ 2 is a root of the equation and

.28)
Consequently, the number of roots of (4.27) equals the number of solutions of system (4.15),(4.16).Further, if N[A −1 2 (F 2 )] > 0 then for arbitrary continuous positive functions k 1 , l 2 we can construct positive continuous functions k 2 such that the system has N (= 0, 1, . . ., ∞) solutions, in the following way.Let g be a continuous function having N zeros with the property λ 2 + g(λ 2 ) > 0 for all λ 2 > 0. Then (4.15), (4.16) has N solutions if Proof.By the assumption of the theorem, χ is a continuous bijection between R and R, thus (4.23) is equivalent with (4.28), hence (4.24) is equivalent with (4.27).The further statements of the theorem can be proved similarly to the former theorems.

L. Simon
Finally, consider the system of semilinear elliptic functional differential equations  (By F j ∈ V and the assumption on smoothness of K and L, solutions u, v ∈ L 2 (Ω) of (4.31), (4.32) should belong to V.) Let  thus the number of solutions of (4.29), (4.30) equals the number of roots of (4.39).