Wave equation in higher dimensions – periodic solutions

We discuss the solvability of the periodic-Dirichlet problem for the wave equation with forced vibrations xtt(t, y)− ∆x(t, y) + l(t, y, x(t, y)) = 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity. To this effect we derive a new dual variational method.


Introduction
The aim of the paper is to look for solutions and their regularity to the following problem x tt (t, y) − ∆x(t, y) + l(t, y, x(t, y)) = 0, t ∈ R, y ∈ (0, π) n , x(t, y) = 0, y ∈ ∂(0, π) n , t ∈ R, x(t + T, y) = x(t, y), t ∈ R, y ∈ (0, π) n . (1.1) The study of time periodic solutions to (1.1), typically with T = 2π, has a long history.First papers (nonlinear l) concerned the case when l = f with | | sufficiently small and f (t, y, •) strongly monotone (see a survey [26] and also [4,19]).To prove the existence results there a variant of the Lyapunov-Schmidt method together with the theory of monotone operators were used.The case when f is only monotone, using similar method and combining them with Schauder fixed point theorem was considered in [13].In [31] Rabinowitz used his saddle point theorem in critical point theory together with a Galerkin argument to prove the existence of weak solution for nonlinearity l being of C 1 and sublinear at infinity.That paper has initiated a large literature devoted to the use of various techniques of modern critical point theory in the study of semilinear wave equations (see [15,34] and the references therein).
The strongly monotone and weakly monotone nonlinearities were considered also e.g. in [14,21,25].In all the quoted papers the monotonicity assumption (strong or weak) is the key property for overcoming the lack of compactness in the infinite dimensional kernel of equation x tt (t, y) − x yy (t, y) = 0 (periodic-Dirichlet solutions).We underline that, in general, the weak solutions obtained in [32] are only continuous functions.Concerning regularity, Brézis and Nirenberg [14] proved -but only for strongly monotone nonlinearities -that any L ∞ -solution of (1.1) is smooth, even in the nonperturbative case = 1, whenever the nonlinearity l is smooth.On the other hand, very little is known about existence and regularity of solutions if we drop the monotonicity assumption on the forcing term l.Willem [38], Hofer [21] and Coron [16] have considered the class of equations (1.1) where l(t, x, u) = g(u) + h(t, x) and g(u) satisfies suitable linear growth conditions.In [16] for the autonomous case h ≡ 0, is proved, for the first time, existence of nontrivial solutions for non-monotone nonlinearities.The case of l being a difference of two convex nonautonomous functions is investigated in [3]: the nonlinearity l ∈ C([0, π] × R 2 , R) has the form l(t, y, x) = λg(t, y, x) + µh(t, y, x) with λ, µ ∈ R, g superlinear in x, h sublinear in x, and both g, h are 2π-periodic in t and nondecreasing in x.The solutions to (1.1) are obtained using variational method.The special form of l allows to control the levels of the weak limits of certain Palais-Smale sequences as the functional corresponding to (1.1) does not satisfy the Palais-Smale condition (compare also the references in [3]).In the paper [7] existence and regularity of solutions of (1.1) (with l = f ) are proved for a large class of non-monotone forcing terms f (t, y, x), including, for example: f (t, y, x) = ±x 2k + x 2k+1 + h(t, y), f (t, y, x) = ±x 2k + f (t, y, x) with fx (t, y, x) ≥ β > 0. The proof is based on a variational Lyapunov-Schmidt reduction, minimization arguments and a priori estimate methods.
It is interesting that arithmetical properties of the ratio α = T/π play an important role in the solvability of the periodic-Dirichlet problem (1.1) over [0, T] × (0, π) n .The main reason is that the nature of the spectrum of the corresponding linear problem x tt (t, y) − x yy (t, y) + g(t, y) = 0 (1.2) depends in an essential way on the arithmetical nature of α.It has already been pointed out by Borel in [10] that there exist numbers α, satisfying some arithmetical conditions, such that the linear problem (1.2) need not have a solution in the class of analytic functions, if g is analytic.
Later Novak [28] proved even more: that there exist irrationals α and functions g in L 2 that the equation (1.2) does not have any generalized periodic-Dirichlet solution.References on these questions can be found in [35].The papers which treat the nonlinear problem of (1.1) consider in most cases only one dimensional space variable i.e. n = 1, autonomous nonlinearities (l = l(x) or lastly some cases of l = l(y, x)) and in all cases only the irrational numbers with bounded partial quotients (see e.g.[2,9,17,18] and the references therein).Kuksin [22] (see also [23]) and Wayne [37] (compare also [36]), were able to find, extending in a suitable way KAM techniques, periodic solutions in some Hamiltonian PDE's in one spatial dimension under Dirichlet boundary conditions.As usual in KAM-type results, the periods of such persistent solutions satisfy a strong irrationality condition, as the classical Diophantine condition, so that these orbits exist only on energy levels belonging to some Cantor set of positive measure.
The main limitation of this method is the fact that standard KAM-techniques require the linear frequencies to be well separated (non resonance between the linear frequencies).To overcome such difficulty a new method for proving the existence of small amplitude periodic solutions, based on the Lyapunov-Schmidt reduction, has been developed in [18].Rather than attempting to make a series of canonical transformations which bring the Hamiltonian into some normal form, the solution is constructed directly.Making the ansatz that a periodic solution exists one writes this solution as a Fourier series and substitutes that series into the partial differential equation.In this way one is reduced to solve two equations: the so called (P) equation, which is infinite dimensional, where small denominators appear, and the finite dimensional (Q) equation, which corresponds to resonances.Due to the presence of small divisors the (P) equation is solved by a Nash-Moser Implicit Function Theorem.Later on, this method has been improved by Bourgain to show the persistence of periodic solutions in higher spatial dimensions [12].The first results on the existence of small amplitude periodic solutions for some completely resonant PDE's as (1.1) have been given in [24], for the specific nonlinearity l(x) = x 3 , and in [1] when l(x) = x 3 + h.o.t.The approach of [1] is still based on the Lyapunov-Schmidt reduction.The (P) equation is solved, for the strongly irrational frequencies ω ∈ W γ , where through the Contraction Mapping Theorem.Next, the (Q) equation, infinite dimensional, is solved by looking for non degenerate critical points of a suitable functional and continuing them, by means of the Implicit Function Theorem, into families of periodic solutions of the nonlinear equation.The case of higher space dimension is investigated in [2].In [8] is proved, assuming only that the nonlinearity l satisfies l(0 the existence of a large number of small amplitude periodic solutions of (1.1) with fixed period.
The aim of this paper is to consider the case n ≥ 2 with T being irrational numbers such that α = T/π has not necessary bounded partial quotients in its continued fraction and nonautonomous nonlinearity l.Moreover we show some relation between the type of number α, the regularity of nonlinearity of l and the regularity of the solution to (1.1), which is treated for the first time.To the knowledge of the authors, the above problem with α having unbounded partial quotients is also considered for the first time (except some special cases in [18]).To this effect we modify Theorem 6.3.1 from [35] to the case of higher dimension in (1.1).Next we develop our own critical point theorem basing on the type of irrational frequencies α to build a set on which the minimum of suitable functional is considered.That means first we define a functional of convex type (l is then monotone only) and using duality properties of convex analysis we prove existence and regularity of solution to (1.1) as a minimum of the functional on a suitable defined set depending on the type of irrational frequency.Next we consider similarly as in [3] l being the difference of two monotone functions but with different properties, and again the new functional corresponding to this l is considered on a new defined set depending on a new irrational frequency to which we apply the former result (with one monotone function!) and develop duality for that functional.We do not apply any known critical point tools.As the last step we investigate a certain form of l being a special combination of a finite number of increasing functions to which we apply induction method (with respect to the number of functions) and use the obtained result for difference of two monotone functions.Such an approach to (1.1) is different from all cited above.We would like to stress that the sets on which we minimize our functionals depend strictly on the type of an irrational frequency and the type of a nonlinearity.This means that for a given fixed irrational frequency and nonlinearity our theorems may not assert an existence to (1.1).They assert only that for some type of nonlinearity there exists an irrational frequency for which (1.1) has a solution.
More precisely we shall study (1.1) by variational method, i.e. we shall consider (1.1) as the critical points of the functional: where L x = l, Ω = (0, π) n , defined on U 1 = H 1 per ((0, T);H 1 0 (Ω)).First we consider L(t, y, •) convex, next L(t, y, •) is a difference of convex functions (but more general case than in [3]) and lastly L(t, y, •) as a special finite combination of convex functions.Moreover we use different definition of α (see T below).Our purpose is to investigate (1.1) by studying critical points of functional (1.3) using in an essential way the form of l and the irrationality α.To this effect we apply approach which is based on ideas developed in [20] (r = 2 and n = 1, see below).Our aim is to find a nonlinear subsets X of U 1 and to study modifications of (1.3) just only on X.The main difficulty in our approach is just the construction of the final set X which depend on the irrational frequency.Moreover we give clear relation between type of r, type of nonlinearity l and irrationality α (see below).We assume that T T = πα, α is such that α 2 is irrational and satisfies α 2 − p/d ≥ cd −r for all p, d ∈ N with some constant c > 0 for a fixed r ≥ 2.
We would like to stress that if r > 2 then we admit α being real algebraic number of degree greater than 2 as well as having unbounded partial quotients -on several properties of such numbers see e.g.[33].We only mention that the case of (0, π) n being of dimension n is a little bit more complicated than n = 1, as some numbers |q| are irrational.However even in the case of n = 1 the assumption T is interesting, usually it is assumed then that |α − p/d| ≥ cd −2 .We must underline that T means, in particular, that we do not consider irrationals of the type α = √ n, n ∈ N (see [27] for deep discussion on that case).In order to give a reader an insight what does condition T mean let us recall some fundamental facts from number theory.Let α 2 = [a 0 , a 1 , a 2 , . . .] (a 0 , a 1 , a 2 , . . .integers) be the continued fraction decomposition of the real number α 2 [33].The integers a 0 , a 1 , a 2 , . . .are the partial quotients of α 2 and the rationals p n d n = [a 0 , a 1 , a 2 , . . ., a n ] with p n , d n relatively prime integers, called the convergent of α 2 , are such that for every rational p/d, such a constant c(α) must satisfy 0 < c(α) < 1/ √ 5. α 2 is badly approximated if and only if the partial quotients in its continued fraction expansion are bounded: |a n | ≤ K(α), n = 0, 1, 2, . . .There are continuum many badly approximated numbers, and there exist continuum many numbers which are not badly approximated.The set of irrational numbers with bounded partial quotients coincides with the set of numbers of constant type, which are the numbers α 2 such that d dα 2 ≥ 1 r for some real number r ≥ 1 and all integers d > 0, where b denotes the distance between the irrational number b and the closest integer.
By a classical theorem of Lagrange all real quadratic irrationals have bounded partial quotients.It follows from results of Borel [10] and Bernstein [6] that the set of all irrational numbers having bounded partial quotients is a dense, uncountable and null subset of the real line.Examples of transcendental numbers having bounded partial quotients are given by for n ≥ 2 an integer.Examples of transcendental numbers with unbounded partial quotients are given by or ζ (3).For π 2 we have for all ε > 0 and d sufficiently large.
The famous Roth's Theorem states that if α 2 is an algebraic number, i.e. a root of a polynomial f (X) = a e X e + a e−1 X e−1 + • • • + a 0 (a i integers), of degree e ≥ 2, then for an arbitrary fixed ε > 0 and all rationals p/d with sufficiently large d the following inequality holds: ( If α 2 is of degree 2 then by Liouville's Theorem we have inequality (1.4).For no single α 2 of degree ≥ 3 we do not know whether (1.4) holds.It is very likely (see [33]) that in fact (1.4) is false for every such α 2 , i.e. that no such α 2 is badly approximated, or, put differently, that such α 2 has unbounded partial quotients in its continued fraction.
From the above we infer that the set of α satisfying T is nonempty, in the following sense: there exists α irrational and r ≥ 2 satisfying T with some constant c > 0 (compare (1.5))!
Let L ⊂ Z n be the lattice of the integers vectors k we define as square root of ∑ j,k |k| 9r−6 g j,k 2 , i.e.
, where To formulate our main results we need a modification of Theorem 6.3.1 from [35] for the case of higher dimension periodic-Dirichlet boundary conditions (1.1) and stronger regularity, i.e. the following Proposition 2.1.Let g ∈ H 9 2 r−3 .Then there exists x ∈ U 5 2 r−1 being a unique solution to x(t, y) = 0, t ∈ R, y ∈ ∂Ω, g j,k is as in (2.1) and such that x with B 2 = (2α + 1) 5r−2 α 4 c −2 and C 2 = 1 9 α 5r−2 independent of g, where α and c are defined in T.
Remark 2.2.Notice that in different way to one dimension case (n = 1) the right hand side of (2.2) has to be more regular in space variable than existing solution of it -even for r = 2.This fact will have influence for necessary regularity assumptions for our nonlinear equation (1.1).

Remark 2.3.
Let us notice that constants B and C are determined by α and c.Everywhere below constants B and C will always denote those occurring in (2.4) and (2.5).

Assumptions M concerning equation (1.1).
M Let F 1 , F 2 , . . ., F n of the variable (t, y, x) and a function G of the variable (t, y) be given.F 1 , F 2 , . . ., F n are measurable with respect to (t, y) in [0, T] × Ω for all x in R and are continuously differentiable and convex with respect to x in R and satisfy for some Let j 1 , . . ., j n−1 be a sequence of numbers having values either −1 or +1.Assume that our original nonlinearity (see (1.1)) has the form Define the set where and some D 0 n .
Remark 2.4.The Assumptions M and M' look very cumbersome.However the aim of them is only to ensure that the set Xn FG is nonempty -we seek at it critical points (see theorem below).Of course, we could state less cumbersome assumptions but then they have to be much stronger to imply nonemptiness of Xn FG .In fact that is the price we pay for looking for new types of critical points.

Theorem 2.5 (Main theorem). Under Assumptions M, M' there exists x ∈ Xn
FG such that J (x) = inf x∈ Xn FG J (x) and x is a solution to (1.1).Now we can formulate theorem which gives us additional informations on solutions to (1.1) important in classical mechanics.This theorem is absolutely new for problem (1.1).
The proofs of the theorems are given in Sections 3, 4.They consist of several steps.First we prove Proposition 2.1.Next we prove Theorem 2.5 (Main theorem).First for the nonlinearity l consisting only of one function x and then by an induction for the general case.

Proof of Proposition 2.1
We shall consider a more general case of Proposition 2.1, namely the case for U q = H q−2r+2 per ) and H q =H 0 per (R; H q (Ω)), q ≥ 9 2 r − 3. The norm • H q of g ∈ H q we define as square root of ∑ j,k |k| 2q g j,k 2 , i.e.
, where We prove stronger regularity case, i.e. the following.
Proof of Proposition 3.1.Our reasoning is inspired by the proof of Theorem 6.3.1 from [35], but now for the case of higher dimension periodic-Dirichlet boundary conditions (1.1) and a stronger regularity result.We know that x ∈ L 2 ((0, T);L 2 (Ω)) belongs to U q if and only if where The square root of (3.5) defines a norm in U q .Similarly for g ∈ H q ⊂ L 2 ((0, T);L 2 (Ω)) we have By our assumption T we can write a solution x of the problem (2.2) in the form (2.3).This function belongs to U q since ∑ j,k with B q some constant independent of g, defined later.This inequality is a direct consequence of the relation To prove it let us put We confine ourselves to the estimation on the set ∑ 2 (the other cases are more simply) -we apply assumption T, i.e.
Hence we get also the estimation (3.1) with B 2 q = (2α + 1) 2q−4r+4 α 4 c −2 .To obtain the estimation (3.2) we rewrite it for our case and show that: We note that it is true if Again we show that only in the set ∑ 2, : Remark 3.3.In order to get Proposition 2.1 it is enough to put in Proposition 3.1 q = 9 2 r − 3.
Proof of Corollary 3.2.We follow the same way as in the proof of Proposition 3.1.We have for Substituting (3.6) and (3.10) in (3.3) gives We can write a solution x of the problem (3.3) in the form This function belongs to U with A q some constant, defined later, independent of g and x such that x To prove it let us put We confine ourselves to the estimation on the set ∑ 2 (the other cases are more simply) -thus: Hence we get also the estimation (3.4) with A 2 q = 1 16 .
4 Proof of the existence of solutions and their regularity for problem (1.1)

Simple case -one function: l = F x
First consider another equation and corresponding to it functional defined on U 1 = H 1 per ((0, T);H 1 0 (Ω)).Observe, that (4.1) is the Euler-Lagrange equation for the action functional J F .For that problem we assume the following hypotheses: G1 F (t, y, x) is measurable with respect to (t, y) in (0, T) × Ω for all x in R, continuously differentiable and convex with respect to the third variable in R for a.e.(t, y) ∈ (0, T) × Ω.
Remark 4.1.Let us notice that, except convexity, restrictions for F 1 x are not strong, they are rather natural.
Remark 4.2.The convexity assumption of F (t, y, •) is strong.For example x 7 is nonconvex.To overcome that problem (at least partially) we study in last section the case of l = j where F i are convex and j i takes values {−1, +1}.Then x 7 = x 8 + x 7 + x 4 − x 8 − x 4 is equal to difference of two convex functions x 8 + x 7 + x 4 and x 8 + x 4 .This case will be considered as a next step.
Exploiting the definition of the set XF and Proposition 2.1 we prove the following lemma.Lemma 4.3.Let x ∈ X F and v be a solution of the periodic-Dirichlet problem for v tt (t, y) − ∆v(t, y) = −F x (t, y, x(t, y)) a.e. on (0, T) × Ω. (4.4) Next we get the following estimations Hence we get where v is a solution of the periodic-Dirichlet problem (4.4).By Lemma 4.3 and the definition of X F , H(X F ) is bounded in U 5 2 r−1 , it is contained in XF .Moreover the limit of weakly convergent sequence, in U Next define the set X Fd : an element (p, q) ∈ H 1 ((0, T)×Ω) × H 1 ((0, T)×Ω) belongs to X Fd provided that for each x ∈ XF there exist x ∈ X F such that for a.e.(t, y) ∈ (0, T) × Ω p (t, y) = x t (t, y) and p t (t, y) − div q (t, y) = −F x (t, y, x(t, y)) with q (t, y) = ∇ x (t, y) .
As the sets XF , X F are nonempty therefore the set X Fd is nonempty.
The dual functional to (4.2) is usually taken as where F * is the Fenchel conjugate of F with respect to third variable and J F D : X Fd → R.
We will look at relationships between the functional J F and J F D on the set X F and X Fd respectively: Variational Principle at extreme points.It relates the critical values of both functionals and provides the necessary conditions that must be satisfied by the solution to problem (4.1).Now we state the simple result of the paper which is existence theorem for particular case of (1.1) i.e. problem (4.1).Theorem 4.6.Assume G1-G3.Then there exists x ∈ XF such that and the following system holds x t (t, y) = p (t, y) , (4.7) ∇x (t, y) = q (t, y) , (4.8) p t (t, y) − div q (t, y) = −F x (t, y, x (t, y)) .(4.9) This result is new however it has strong assumption for nonlinearity F x : convexity of F(t, y, •).Our aim is to relax them.But first we illustrate, by an example, a case of the above theorem.
Example 4.8.We can consider also the case with F(t, y, x) = x 6 + x 5 + x 2 .Then we take F 1 (t, y, x) = x 6 + x 5 + x 2 + x and F 2 (t, y) = −1.Thus the theorem for that case asserts that there exists nontrivial solution to (4.1).

The auxiliary results
By G1-G3, definition of XF , mean value theorem we get the following lemma.
Lemma 4.10.The functional J F attains its infimum on XF i.e. inf x∈ XF J F (x) = J F (x), where x ∈ XF .
Proof.By the definition of the set XF and Lemma 4.9 we see that the functional J F is bounded below on XF .We denote by x j a minimizing sequence for J F in XF .This sequence has a subsequence which we denote again by x j converging weakly in U 5 2 r−1 and strongly in U 1 , hence also strongly in L 2 ((0, T) × Ω; R) to a certain element x ∈ U 5 2 r−1 .Moreover x j is also convergent almost everywhere.Thus by construction of the set XF , we observe that x ∈ XF .

Hence lim inf
Thus inf

Proof of Theorem 4.6
Let x ∈ XF be such that J F ( x) = inf x∈ XF J F (x).This means that there exists an x ∈ X F ⊂ XF such that p(t, y) = xt (t, y) (4.10) and pt (t, y) = div q (t, y) − F x (t, y, x (t, y)) , (4.11) for a.a.(t, y) ∈ (0, T) × Ω where q is given by q(t, y) = ∇ x(t, y).(4.12) By the definitions of J F , J F D , relations (4.10), (4.11) and the Fenchel-Young inequality it follows that Therefore we get that and so Thus we have equality J F ( x) = J F D ( p, q).It implies The last means that x (t, y) ( pt (t, y) − div q (t, y)) dydt Hence and the standard convexity arguments and (4.11) we obtain two equalities p(t, y) = xt (t, y) , pt (t, y) = div q (t, y) − F x (t, y, x (t, y)) .

Simple case -one function: l = −F x
A similar theorem to Theorem 4.6 is true for the problem x tt (t, y) − ∆x(t, y) − F x (t, y, x(t, y)) = 0, and corresponding to it functional defined on U 1 with same hypotheses G1-G3 and the sets XF , XF .Really, Lemmas 4.3-4.10 are still valid as sign of F does not change their proofs.The set Now by Corollary 3.2 X −F is nonempty.The set X −Fd is defined: an element (p, q) ∈ H 1 ((0, T)×Ω) × H 1 ((0, T)×Ω) belongs to X −Fd provided that for each x ∈ XF , there exist x ∈ X −F such that for a.e.(t, y) ∈ (0, T) × Ω q (t, y) = ∇x (t, y) and p t (t, y) − div q (t, y) = −F x (t, y, x(t, y)) with p (t, y) = xt (t, y) .
As the set X −F is nonempty therefore the set X −Fd is nonempty.The dual functional to (4.14) is now Hence following the same way as in the proof of the above theorem we get for (4.13) the following theorem.

The case of nonlinearity F − G
Now we consider more complicated problem i.e.
x tt (t, y) − ∆x(t, y) − G x (t, y, x(t, y)) + F x (t, y, x(t, y)) = 0, and corresponding to it functional defined in U 1 .The similar case of L (l = L x ) being a difference of two convex nonautonomous functions is investigated in [3]: the nonlinearity l ∈ C([0, π] × R 2 , R) has the form l(t, y, x) = λg(t, y, x) + µh(t, y, x) with λ, µ ∈ R, g superlinear in x, h sublinear in x and both g, h are 2π-periodic in t and nondecreasing in x.
For problem (4.18) we assume the following hypotheses: GG1 F (t, y, x) and G(t, y, x) are measurable with respect to (t, y) in (0, T) × Ω for all x in R and continuously differentiable and convex with respect to the third variable in R for a.a.(t, y) ∈ (0, T) × Ω. (t, y) → F (t, y, 0) − G(t, y, 0) is integrable on (0, T) × Ω.

GG2
There exist constants D, E, G and x ∈ H GG3 F (t, y, x) and G(t, y.x) satisfy G3.
Proof.Let us fix any v ∈ XFG and notice that x ∈ U 5 2 r−1 defining X FG is a solution of the equation which is an Euler-Lagrange equation to the functional Notice that Ĵ(x) is the strictly convex, thus weakly lower semicontinuous in U 5 2 r−1 and X FG is weakly compact (since XFG weakly compact) in U 5 2 r−1 .Thus Ĵ attains its unique minimum in U 5 2 r−1 and the minimizer x m of it belongs to X FG .Next define a functional dual to Ĵ (in the sense of convex analysis): considered in H 1 ((0, T)×Ω), where and G * (t, y, •) is Fenchel conjugate of G(t, y, •).We see that ĴD ( p) attains its minimum at p m such that p m t (t, y) = ∆v (t, y) − F x (t, y, v(t, y)) and using the standard tools of convex analysis we see next that Ĵ( x m ) = − ĴD ( p m ).Moreover, minimizer x m to Ĵ satisfies (4.22).Following in the same way as in the proof of Corollary 3.2 we can write a solution x ∈ X FG in a form where Then since x ∈ U 5 2 r−1 we have also estimation as in Corollary 3.2, namely x Thus X FG ⊂ XFG .
By Lemma 4.13 the set X FGd is nonempty.The dual functional to (4. 19) is then taken as where G * is the Fenchel conjugate of G with respect to third variable and Remark 4.14.Let us observe that if x is a solution to (4.18) then by Lemma 4.13 it has to belong to XFG .
Analogously as in the case of the functional J F we prove the following lemma.
Example 4.17.We show, how to use, the above theorem to solve the nonconvex superlinear problem: Assume n = 2 and let α be such that α 2 is an algebraic number of degree 3. Thus α 2 satisfies (1.6) with e.g.ε = 1/2.Then α 2 satisfies condition in T with c = 1 and r = 5/2.Let us put Then assume F(t, y, x) = x 6 + x 5 + x 2 + x cos 1 2 y and G(t, y, x) = x 6 + x 2 .F(t, y, •) and G(t, y, •) are convex.Thus take for x such an x(•, •) > 0 that x ∈ H 100 and F x ( x) then assumptions GG1-GG3 are satisfied, so by the above theorem there exists x ∈ XFG (nontrivial) being a solution to (4.27).

More general case -proof of the main theorem
Let us consider now a sequence of convex (with respect to third variable) functions F 1 , F 2 , . . ., F n of the variables (t, y, x) and a function G of the variable (t, y).Let j 1 , . . ., j n be a sequence of numbers having values either −1 or +1.Let us assume that our original nonlinearity (see (1.1)) has the form To prove the existence of solution to (1.1) with nonlinearity l just defined we use an induction argument.To this effect let us put and consider the problem x tt (t, y) − ∆x(t, y) + l n−1 (t, y, x(t, y)) + G(t, y) = 0, t ∈ R, y ∈ Ω, x(t, y) = 0, y ∈ ∂Ω, t ∈ R, x(t + T, y) = x(t, y), t ∈ R, y ∈ Ω.For that problem we assume the following hypotheses: G n−1 1 F 1 , F 2 , . . . ,F n−1 are measurable with respect to (t, y) in (0, T) × Ω for all x in R and continuously differentiable and convex with respect to the third variable in R for a.e.(t, y) ∈ (0, T) × Ω. (t, y) → l n−1 (t, y, 0) + G(t, y) is integrable on (0, T) × Ω, G (•, •) ∈ H 9 2 r−3 .

G n−1 2
There exist constants D n−1 , E n−1 , F and x ∈ H

Define the following set
By G n−1 2 the set X n−1 is nonempty.
and some D 0 n−1 .
Analogously as Lemma 4.3 one can prove the following lemma.
where v is a solution of the periodic-Dirichlet problem (4.34).Similarly as in the former cases we notice that H(Xn−1 ) ⊂ H   G n 1 Assume G n−1 1-G n−1 3. Let F n is measurable with respect to (t, y) in (0, T) × Ω for all x in R and continuously differentiable and convex with respect to the third variable in R for a.e.(t, y) ∈ (0, T) × Ω.
Define the set and some D 0 n .
Similarly as in Lemma 4.12, using now hypothesis IH, one can prove the following lemma.
Lemma 4.20.Define in X n FG the map X n

Conclusions
The nonlinear terms in problems of type (1.1), at the beginning of investigations, were monotone functions or sublinear at infinity (see a survey [26]).Next step was the paper of [3] where nonlinear term was a difference of two monotone functions.In this paper we extend nonlinearity to be a finite linear combination of monotone functions.In many papers concerning problem (1.1) we can observe that monotonicity of nonlinearity is essential to prove existence of solution to (1.1).The open problem appears: whether the nonlinearity l can be of the form l = f + g where f is monotone function and g only continuous.It was already pointed in [10] that arithmetical properties of the ratio α = T/π play an important role in a solvability of the periodic-Dirichlet problem (1.1) (see also interesting discussion on that problem in [35]).There is only a few papers which treat that problem in case when T is irrational number such that α = T/π has not necessary bounded partial quotients in its continued fraction with nonlinear l.The case with spatial dimension n ≥ 2 and r > 2 has not been almost investigated.We have proved, for n ≥ 2 and r ≥ 2, that if α satisfies assumption T then with l being a finite linear combination of monotone functions, problem (1.1) has a solution in H

5 2 5 2
r−1 and let x ∈ U 5 2 r−1 be such that x U r−1 ≤ B9 2 r−3 W, for some W > 0. Then there exists x ∈ U 5 2 r−1 being a unique solution to

9 2 r− 3 . 5 2
Hence by Proposition 2.1 there exists a unique solution v ∈ U r−1 of the periodic-Dirichlet problem for the equation (4.4) satisfying

Remark 4 . 4 . 5 2 r− 1 . 5 2
Let us note that since v ∈ XF thus F x (v) ∈ H Therefore by Corollary 3.2 X F is nonempty and bounded in U r−1 by A

5 2 r 9 2 9 2 5 2 r− 1 ≤
−1 by B(E + G).Proof.Fix arbitrary x ∈ XFG .It follows that G x (x) ∈ H r−3 and G x (x) H r−3 ≤ G. Hence by Theorem 4.6 there exists a solution v of the periodic-Dirichlet problem for the equation (4.20) satisfying v U B(E + G).Thus the last relation implies that for an arbitrary x ∈ XFG there exists

Remark 4 . 19 . 5 2 r− 1 .
Let us note that since v ∈ Xn−1 thus l n−1 (v) ∈ H Therefore by Corollary 3.2 X n−1 is nonempty and bounded in U