Entire solutions with exponential growth for an elliptic system modeling phase-separation
aa r X i v : . [ m a t h . A P ] M a r ENTIRE SOLUTIONS WITH EXPONENTIAL GROWTH FOR AN ELLIPTICSYSTEM MODELING PHASE-SEPARATION
Nicola SoaveUniversit`a degli Studi di Milano Bicocca - Dipartimento di Matematica e ApplicazioniVia Roberto Cozzi 53, 20125 Milano, Italyemail: [email protected] ZilioPolitecnico di Milano - Dipartimento di Matematica “Francesco Brioschi”Piazza Leonardo da Vinci 32, 20133 Milano, Italyemail: [email protected]
Abstract.
We prove the existence of entire solutions with exponential growth for the semilinear ellipticsystem − ∆ u = − uv in R N − ∆ v = − u v in R N u, v > , for every N ≥
2. Our construction is based on an approximation procedure, whose convergence is ensuredby suitable Almgren-type monotonicity formulae. The construction of some solutions is extended tosystems with k components, for every k > Keywords: elliptic system, phase-separation, Almgren monotonicity formulae, entire solutions, expo-nential growth. 1.
Introduction and main results
In this paper we investigate the existence of entire solutions with exponential growth for the semilinearelliptic system(1.1) − ∆ u = − uv − ∆ v = − u vu, v > , in R (thus in R N for every N ≥ u, v ) is an entire solution to (1.1) and isglobally α -H¨older continuous for some α ∈ (0 , u and v is constant while the otheris identically 0. On the other hand, in [1] the authors show that there exists a nontrivial solution for thesystem of ODEs − u ′′ = − uv in R − v ′′ = − u v in R u, v > R , in the sense that there exists t ∈ R suchthat u ( t + t ) = v ( t − t ) for every t ∈ R , and has linear growth: there exists C > u ( t ) + v ( t ) ≤ C (1 + | t | ) ∀ t ∈ R . The paper [2] completes the study of the 1-dimensional problem with the proof of the uniqueness of thepositive 1-dimensional profile, up to translations and scalings. Always in [2], the authors construct entire solutions to (1.1) with algebraic growth for any integer rate of growth greater then 1; here and in therest of the paper we say that ( u, v ) has algebraic growth if there exist p ≥ C > u ( x ) + v ( x ) ≤ C (1 + | x | p ) ∀ x ∈ R N . The solutions constructed in [2] are not 1-dimensional, and are modeled on (we will be more preciselater, see Remark 1.2) the homogeneous harmonic polynomials ℜ ( z d ), for every d ≥
2. There is adeep relationship between entire solutions to (1.1) and harmonic functions; this relationship has beenestablished in [5, 9]. For instance, in case ( u, v ) has algebraic growth, it is possible to show that up to asubsequence, the blow-down family, defined by( u R ( x ) , v R ( x )) = R N − R ∂B R (0) u + v ( u ( Rx ) , v ( Rx )) , is uniformly convergent in every compact subset of R N , as R → + ∞ , to a limiting profile (Ψ + , Ψ − ),where Ψ is a homogeneous harmonic polynomial (see Theorem 1.4 in [2]).To conclude this bibliographic introduction, we have to mention that major efforts have been donerecently in order to prove classification results and in particular the 1-dimensional symmetry of solutionsto (1.1). This is motivated by the relationship between (1.1) and the Allen-Cahn equation, which has beenestablished in [1], and led the authors to formulate a De Giorgi’s-type and a Gibbons’-type conjecturefor solutions to (1.1); for results in this direction, we refer to [1, 2, 6, 7, 11].Motivated by the quoted achievements, we wonder if the system (1.1) has solutions with super-algebraicgrowth. We can give a positive answer to this question proving the existence of solutions with exponentialgrowth. In our construction we adapt the same line of reasoning introduced in the proof of Theorem 1.3of [2]. Therein, the authors proved the existence of solutions to (1.1) with the same symmetry of thefunction ℜ ( z d ) in any bounded ball B R (0) ⊂ R , with boundary conditions u = ( ℜ ( z d )) + , v = ( ℜ ( z d )) − on ∂B R (0). By means of suitable monotonicity formulae, they could pass to the limit for R → + ∞ obtaining convergence (up to a subsequence) for the previous family to a nontrivial entire solution. Inthis sense, their solutions are modeled on the harmonic functions ℜ ( z d ).Here, having in mind the construction of solutions with exponential growth, and recalling the relation-ship between entire solution of our system and harmonic functions, we start by consideringΦ( x, y ) := cosh x sin y. The first of our main results is the following.
Theorem 1.1.
There exists an entire solution ( u, v ) ∈ ( C ∞ ( R )) to system (1.1) such that u ( x, y + 2 π ) = u ( x, y ) and v ( x, y + 2 π ) = v ( x, y ) , u ( − x, y ) = u ( x, y ) and v ( − x, y ) = v ( x, y ) , the symmetries v ( x, y ) = u ( x, y − π ) u ( x, π − y ) = v ( x, π + y ) u (cid:16) x, π y (cid:17) = u (cid:16) x, π − y (cid:17) v (cid:18) x, π + y (cid:19) = v (cid:18) x, π − y (cid:19) hold, u − v > in { Φ > } and v − u > in { Φ < } , u > Φ + and v > Φ − in R , the function ( Almgren quotient ) r R (0 ,r ) × (0 , π ) |∇ u | + |∇ v | + 2 u v R { r }× [0 , π ] u + v is well-defined for every r > , is nondecreasing, and lim r → + ∞ R (0 ,r ) × (0 , π ) |∇ u | + |∇ v | + 2 u v R { r }× [0 , π ] u + v = 1 , XPONENTIAL ENTIRE SOLUTIONS 3 there exists the limit lim r → + ∞ R { r }× [0 , π ] u + v e r =: α ∈ (0 , + ∞ ) . Remark 1.2.
This solution is modeled on the harmonic function Φ, in the sense that it inherits thesymmetries of (Φ + , Φ − ) and has the same rate of growth of Φ. Remark 1.3.
Point 7) of the Theorem gives a lower and a upper bound to the rate of growth of thequadratic mean of ( u, v ) on { r } × [0 , π ] when r varies: Z { r }× [0 , π ] u + v ! = O ( e r ) as r → + ∞ . The domain of integration takes into account the periodicity of ( u, v ). The quadratic mean of ( u, v ) on { r } × [0 , π ] has exponential growth, and the rate of growth is the same of the function e r , which in turnshas the same rate of growth of Φ. Note that the coefficient 1 in the exponent of e r coincides with thelimit as r → + ∞ of the Almgren quotient defined in point 6). Remark 1.4.
With a scaling argument, it is not difficult to prove the existence of entire solutions withexponential growth of order λ for every λ > u λ ( x, y ) , v λ ( x, y )) = ( λu ( λx, λy ) , λv ( λx, λy ) . It is straightforward to check that ( u λ , v λ ) is still a solution to (1.1) in the plane, is πλ -periodic in y andis such that u λ ( x, y ) ≥ λ (cosh( λx ) sin( λy )) + and v λ ( x, y ) ≥ λ (cosh( λx ) sin( λy )) − . Moreover,(1.2) lim r → + ∞ R (0 ,r ) × ( , πλ ) |∇ u λ | + |∇ v λ | + 2 u λ v λ R { r }× [ , πλ ] u λ + v λ = λ, and lim r → + ∞ R { r }× [ , πλ ] u λ + v λ e λr = λα. One can consider the solution ( u λ , v λ ) as related to the harmonic function cosh( λx ) sin( λy ). This revealsthat there exists a correspondence { ( u λ , v λ ) : λ > } ↔ { sin( λx ) cosh( λy ) : λ > } . Due to the invariance under translations and rotations of problem (1.1), the family { ( u λ , v λ ) : λ > } can equivalently be related with the families of harmonic functions { cosh( λx ) [ C cos( λy ) + C sin( λy )] } or { [ C cos( λx ) + C sin( λx )] cosh( λy ) : λ > } , where C , C , C , C ∈ R .As observed in Remark 1.3, the limit of the Almgren quotient in (1.2) describes the rate of the growthof the quadratic mean of ( u λ , v λ ) computed on an interval of periodicity in the y variable. The previouscomputation reveals that for every λ > λ .This marks a relevant difference between entire solutions with polynomial growth and entire solutionswith exponential growth: while in the former case the admissible rates of growth are quantized (Theorem1.4 of [2]), in the latter one we can prescribe any positive real value as rate of growth.Remark 1.4 reveals that, starting from the solution found in Theorem 1.1, we can build infinitely-manyentire solutions with different exponential growth. However, noting that system 1.1 is invariant underrotations, translations and scalings, intuitively speaking they are all the same solution. We wonder ifthere exists an entire solution of (1.1) having exponential growth which cannot be obtained by that foundin Theorem 1.1 through a rotation, a translation or a scaling; the answer is affirmative. We denoteΓ( x, y ) := e x sin y. EXPONENTIAL ENTIRE SOLUTIONS
Theorem 1.5.
There exists an entire solution ( u, v ) ∈ ( C ∞ ( R )) to system (1.1) which enjoys points1), 3), 4) of Theorem 1.1; moreover for every r ∈ R (1.3) Z ( −∞ ,r ) × (0 , π ) |∇ u | + |∇ v | + u v < + ∞ , u > Γ + and v > Γ − in R , the function ( Almgren quotient ) r R ( −∞ ,r ) × (0 , π ) |∇ u | + |∇ v | + 2 u v R { r }× (0 , π ) u + v is well-defined for every r > , is nondecreasing, and lim r → + ∞ R ( −∞ ,r ) × (0 , π ) |∇ u | + |∇ v | + 2 u v R { r }× (0 , π ) u + v = 1 , there exist the limits lim r → + ∞ R { r }× [0 , π ] u + v e r =: β ∈ (0 , + ∞ ) and lim r →−∞ Z { r }× [0 , π ] u + v = 0 . Remark 1.6.
This solution is modeled on the harmonic function Γ. As explained in Remark 1.3, itis possible to obtain a family of entire solutions which is in correspondence with a family of harmonicfunctions.
Remark 1.7.
Note that the Almgren quotients that we defined in Theorem 1.1 and 1.5 are different.They are both different to the Almgren quotient which has been defined in [2].We can partially generalize our existence result to the case of systems with many components. To beprecise, given an integer k , we will construct a solution ( u , . . . , u k ) of(1.4) ( − ∆ u i = − u i P j = i u j u i > , i = 1 , . . . , k, in the whole plane R having the same growth and the same symmetries of Γ. Here and in the paper weconsider the indexes mod k . Theorem 1.8.
There exists an entire solution ( u , . . . , u k ) ∈ ( C ∞ ( R )) k to system (1.4) such that, forevery i = 1 , . . . , k , u i ( x, y + kπ ) = u i ( x, y ) , the symmetries u i +1 ( x, y ) = u i ( x, y − π ) u (cid:16) x, π y (cid:17) = u (cid:16) x, π − y (cid:17) hold, for every r ∈ R Z ( −∞ ,r ) × (0 ,kπ ) k X i =1 |∇ u i | + X ≤ i Theorem 1.9. Let ( u, v ) be a nontrivial solution of (1.1) in R which is π -periodic in y , and such thatone of the following situation occurs: ( i ) there holds lim r →−∞ Z { r }× [0 , π ] u + v = 0 , and d := lim r → + ∞ R ( −∞ ,r ) × (0 , π ) |∇ u | + |∇ v | + u v R { r }× [0 , π ] u + v < + ∞ . ( ii ) ∂ x u = 0 = ∂ x v on { a } × [0 , π ] for some a ∈ R , and d := lim r → + ∞ R ( a,r ) × (0 , π ) |∇ u | + |∇ v | + u v R { r }× [0 , π ] u + v < + ∞ . Then d is a positive integer, Z { r }× [0 , π ] u + v ! = O ( e dr ) as r → + ∞ , and the sequence ( u R ( x, y ) , v R ( x, y )) := 1 qR { r }× [0 , π ] u + v ( u ( x + R, y ) , v ( x + R, y )) converges in C ( R ) and in H ( R ) to (Ψ + , Ψ − ) , where Ψ( x, y ) = e dx ( C cos( dy ) + C sin( dy )) forsome C , C ∈ R . Notation. We will deal with functions defined in domains of type ( a, b ) × R , where a < b are extendedreal numbers ( a = −∞ and b = + ∞ are admissible). We will often assume that ( u , . . . , u k ) is kπ -periodicin y ; therefore, we can think to ( u , . . . , u k ) as defined on the cylinder C ( a,b ) := ( a, b ) × S k where S k = R / ( kπ Z ) . We will also denote Σ r := { r } × S k . In case b > a = − b , we will simply write C b instead of C ( − b,b ) tosimplify the notation. Plan of the paper. In section 2 we will prove some monotonicity formulae which will come useful in therest of the paper. We can deal with two types of solutions: solutions satisfying a homogeneous Neumanncondition defined in a cylinder C ( a,b ) with a > −∞ , or solutions defined in a semi-infinite cylinder oftype C ( −∞ ,b ) and decaying at x → −∞ . For the sake of completeness and having in mind to use somemonotonicity formulae in the proof of Theorem 1.8, we will always consider the case of systems with k components.The proof of Theorem 1.1 will be the object of section 3. It follows the same sketch of the proof ofTheorem 1.3 in [2]: we start by showing that for any R > u R , v R ) to (1.1) inthe cylinder C R , with Dirichlet boundary condition u R = Φ + and v R = Φ − on {− R, R } × [0 , π ] , EXPONENTIAL ENTIRE SOLUTIONS and exhibiting the same symmetries of (Φ + , Φ − ). In order to obtain a solution defined in the whole C ∞ ,we wish to prove the C loc ( C ∞ ) convergence of the family { ( u R , v R ) : R > } , as R → + ∞ . To show thatthis convergence occurs, we will exploit the monotonicity formulae proved in subsection 2.1. With respectto Theorem 1.3 of [2], major difficulties arise in the precise characterization of the growth of ( u, v ), points6) and 7) of Theorem 1.1.In section 4 we will prove Theorem 1.5. One could be tempted to try to adapt the proof of Theorem 1.1replacing Φ with Γ. Unfortunately, in such a situation we could not exploit the results of subsection 2.1;this is related to the lack of the even symmetry in the x variable of the function Γ (note that the functionΦ enjoys this symmetry). A possible way to overcome this problem is to work in semi-infinite cylinders C ( −∞ ,R ) and use the monotonicty formulae proved in subsection 2.2. But to work in an unbounded setintroduces further complications: for instance, the compactness of the Sobolev embedding and of sometrace operators, a property that we will use many times in section 3, does not hold in C ( −∞ ,R ) . Althoughwe believe that this kind of obstacle can be overcome, we propose a different approach for the constructionof solutions modeled on Γ, which is based on the elementary limitlim R → + ∞ Φ R ( x, y ) = Γ( x, y ) ∀ ( x, y ) ∈ R , where Φ R ( x, y ) = 2 e − R cosh( x + R ) sin y . We will prove the existence of a solution ( u R , v R ) of (1.1) in C ( − R,R ) with Dirichlet boundary condition u R = Φ + R and v R = Φ − R on {− R, R } × [0 , π ] , and exhibiting the same symmetries of (Φ + R , Φ − R ). Then, using again the results of section 2, we will passto the limit as R → + ∞ proving the compactness of { ( u R , v R ) } .Section 5 is devoted to the study of systems with many components. As in [2] the authors could provein one shot an existence theorem for 2 or k components (there are no substantial changes in the proofs),it is natural to wonder if here we can simply adapt step by step the construction carried on in section 3 or4, or not. Unfortunately, the answer is negative: following the sketch of the proof of Theorem 1.1, we canadapt most the results of sections 3 and 4 with minor changes, but in the counterpart of Proposition 3.1we cannot prove the pointwise estimate given by point 4). As a consequence, with respect to subsections3.2 and 4.2 we cannot show that the limit of the sequence ( u ,R , . . . , u k,R ) does not vanish. Note that,in the case of two components, this nondegeneracy is ensured precisely by the above pointwise estimate.As far as the case of k component in [2], we observe that they obtained nondegeneracy through theirCorollary 5.4, which is the counterpart of point ( i ) of our Corollary 2.5. But, while therein the estimateof the growth given by this statement is optimal, in our situation it does not provide any information; thisis related to the different expression of the term of rest in the Almgren monotonicity formula, Proposition2.4. This is why we have to use a completely different argument which is not based on the existence ofsolutions for the system of k components in bounded cylinders (or in semi-infinite cylinders), but rests onTheorem 1.6 of [2]. Roughly speaking, we will obtain the existence of a solution of (1.4) with exponentialgrowth as a limit of solutions of the same system having algebraic growth.The proof of Theorem 1.9 will be the object of section 6.We conclude the paper with an appendix, in which we state and prove some known results for whichwe cannot find a proper reference.2. Almgren-type monotonicity formulae Let k ≥ ( − ∆ u i = − u i P j = i u j u i > C ( a,b ) (this means that we assume from the beginning that ( u , . . . , u k ) is kπ -periodicin y ).In this section we will use many times the following general result: XPONENTIAL ENTIRE SOLUTIONS 7 Lemma 2.1. Let ( u , . . . , u k ) be a solution of (1.4) in C ( a,b ) . Then the function r Z Σ r k X i =1 |∇ u i | + X ≤ i In this subsection we are interested in solu-tions to (2.1) defined in C ( a,b ) (thus kπ -periodic in y ), with a > −∞ and b ∈ ( a, + ∞ ], and satisfying ahomogeneous Neumann boundary condition on Σ a , that is,(2.2) ∂ x u i = 0 on Σ a , for every i = 1 , . . . , k. Firstly, we observed that under this assumption Lemma 2.1 implies Lemma 2.2. Let ( u , . . . , u k ) be a solution of (2.1) in C ( a,b ) , such that (2.2) holds true. For every r ∈ ( a, b ) the following identity holds: Z Σ r k X i =1 |∇ u i | + X ≤ i The index sym denotes the fact that, as we will see, the quantities E sym and E sym arewell suited to describe the growth of the solution ( u , . . . , u k ) only if ( u , . . . , u k ) satisfies the (2.2), whichcan be considered as a symmetry condition. Indeed, under (2.2) one can extend ( u , . . . , u k ) on C (2 a − b,b ) by even symmetry in the x variable.By regularity, E , E and H are smooth. A direct computation shows that they are nondecreasingfunctions: in particular(2.3) H ′ ( r ) = 2 Z Σ r X i u i ∂ ν u i = 2 E ( r ) , where the last identity follows from the divergence theorem and the boundary conditions of ( u , . . . , u k ).Our next result consist in showing that also the ratio between E (or E ) and H is nondecreasing. EXPONENTIAL ENTIRE SOLUTIONS Proposition 2.4. Let ( u , . . . , u k ) be a solution of (2.1) in C ( a,b ) such that (2.2) holds true. The Almgren quotient N sym ( r ) := E sym ( r ) H ( r ) is well defined and nondecreasing in ( a, b ) . Moreover Z ra R Σ s P i Almgren quotient , too) N sym ( r ) := E sym ( r ) H ( r ) is well definedand nondecreasing in ( a, b ) , and N ′ ( r ) ≥ N ( r ) R C ( a,r ) P i Since ( u, v ) ∈ H ( C ( a,b ) ) is nontrivial, E and H are positive in ( a, b ) and bounded for r bounded.We compute, by means of Lemma 2.2 E ′ ( r ) = Z Σ r X i |∇ u i | + 2 X i Corollary 2.5. Let ( u , . . . , u k ) be a solution of (2.1) in C ( a,b ) such that (2.2) holds. ( i ) If N ( r ) ≥ d for r ≥ s > a , then H ( r ) e dr ≤ H ( r ) e dr ∀ s ≤ r < r < b,ii ) If N ( r ) ≤ d for r ≤ t < b , then H ( r ) e dr ≥ H ( r ) e dr ∀ a < r < r ≤ t. XPONENTIAL ENTIRE SOLUTIONS 9 Proof. We prove only ( ii ). Recalling that H ′ ( r ) = 2 E ( r ) (see (2.3)), we havedd r log H ( r ) = 2 N ( r ) ≤ d ∀ r ∈ ( a, t ] . By integrating, the thesis follows. (cid:3) The next step is to prove a similar monotonicity property for the function E . Our result rests onTheorem 5.6 of [2] (see also [1]), which we state here for the reader’s convenience Theorem 2.6. Let k be a fixed integer and let Λ > . Let L ( k, Λ) := min Z π k X i =1 ( f ′ i ) + Λ X ≤ i Having in mind to apply Theorem 2.6 on 2 π -periodic functions, note that the condition f ( π + t ) = f ( π − t ) can be replaced by f ( t + τ ) = f ( τ − t ) for any τ ∈ [0 , π ).For a fixed r ∈ ( a, b ), let us introduce ϕ ( r ; r ) := Z rr d sH ( s ) / . The function ϕ is positive and increasing in R + ; thanks to point ( i ) of Corollary 2.5 and to the mono-tonicity of N , whenever ( u, v ) is nontrivial ϕ is bounded by a quantity depending only H ( r ) and N ( r ).To be precise:(2.4) ϕ ( r ; r ) ≤ e N ( r ) r H ( r ) N ( r ) h e − N ( r ) r − e − N ( r ) r i . This, together with the monotonicity of ϕ ( · ; r ), implies that if b = + ∞ then there exists the limit(2.5) lim r → + ∞ ϕ ( r ; r ) < + ∞ . Lemma 2.8. Let ( u , . . . , u k ) be a solution of (1.1) in C ( a,b ) such that (2.2) holds. Let r ∈ ( a, b ) , andassume that (2.6) u i +1 ( x, y ) = u i ( x, y − π ) and u ( x, τ + y ) = u ( x, τ − y ) where τ ∈ [0 , kπ ) . There exists C > such that the function r E ( r ) e r e Cϕ ( r ; r ) is nondecreasing in r for r > r .Proof. Recalling the (2.3), we compute the logarithmic derivative(2.7) dd r log (cid:18) E ( r ) e r (cid:19) = − R Σ r P i ( ∂ ν u i ) + R Σ r ( ∂ y u i ) + 2 P i Let ( u , . . . , u k ) be a nontrivial solution of (2.1) in C ( a, + ∞ ) , and assume that (2.2) and (2.6) hold. If d := lim r → + ∞ N ( r ) < + ∞ , then d ≥ and lim r → + ∞ E ( r ) e r > . Proof. Let us fix r > a . Firstly, from the previous Lemma and the (2.5), we deduce that there existsthe limit l := lim r → + ∞ E ( r ) e r ≥ . Recalling that ϕ ( r ; r ) is bounded, it results E ( r ) e r ≥ e − Cϕ ( r ; r ) E ( r ) e r ≥ C > ∀ r > r , so that the value l is strictly greater then 0. Now, assume by contradiction that d = lim r → + ∞ N ( r ) < N implies N ( r ) ≤ d for every r > 0. Hence, from Corollary 2.5 we deduce H ( r ) e dr ≤ H ( r ) e dr ∀ r > r ⇒ lim sup r → + ∞ H ( r ) e dr < + ∞ ⇒ lim r → + ∞ H ( r ) e r = 0 , which in turns gives 0 < l = lim r → + ∞ E ( r ) e r = lim r → + ∞ N ( r ) lim r → + ∞ H ( r ) e r = 0 , a contradiction. (cid:3) XPONENTIAL ENTIRE SOLUTIONS 11 Solutions with finite energy in unbounded cylinders. In what follows we consider a solution( u , . . . , u k ) of (2.1) defined in an unbounded cylinder C ( −∞ ,b ) , with b ∈ R (the choice b = + ∞ isadmissible). In this setting we assume that ( u , . . . , u k ) has a sufficiently fast decay as x → −∞ , in thesense that(2.9) H ( r ) := Z Σ r k X i =1 u i → r → −∞ . First of all, we can show that under assumption (2.9) ( u , . . . , u k ) has finite energy in C ( −∞ ,b ) . Lemma 2.10. Let ( u , . . . , u k ) be a solution of (1.4) in C ( −∞ ,b ) , such that (2.9) holds. Then E unb ( r ) := Z C ( −∞ ,r ) k X i =1 |∇ u i | + X ≤ i Proof. Firstly, being a solution in C ( −∞ ,b ) , it results ( u , . . . , u k ) ∈ H loc ( C ( −∞ ,b ) ). Thus, under assump-tion (2.9), there exists C > H ( r ) ≤ C for every r < b .Let r < b . Let us introduce, for r > 0, the functional e ( r ) := Z C ( − r + r ,r X i |∇ u i | + X i As a byproduct of the previous Lemma, if ( u , . . . , u k ) solves the (1.4) in C ( −∞ ,b ) and(2.9) holds, then lim r →−∞ E ( r ) = 0 . Having in mind to recover the monotonicity formulae of the previous subsection in the present situation,we cannot adapt the proof of Lemma 2.2, where assumption (2.2) played an important role. However,we can obtain a similar result with a different proof. Lemma 2.12. Let ( u , . . . , u k ) be a solution to (1.1) in C ( −∞ ,b ) , such that (2.9) holds. Then Z Σ r X i = k |∇ u i | + X ≤ i 0, we introduce a cut-offfunction η ∈ C ∞ ( R ) such that η ( s ) = ( s ≤ r − ε s ≥ r. Since ηψ ∈ C ( −∞ , r ), k ηψ k C ( −∞ ,r ) < + ∞ and ηψ = 0 in a neighborhood of r , from (2.12) we deduce(2.13) Z C ( −∞ ,r ) X i ( ∂ x u i ) − X i |∇ u i | + X i 0, where the last identity follows from the regularity of ( u , . . . , u k ) and from the C -boundednessof ψ and η . Passing to the limit as ε → ψ ∈ C (( −∞ , r ]) such that k ψ k C (( −∞ ,r ]) < + ∞ it results Z C ( −∞ ,r ) X i ( ∂ x u i ) − X i |∇ u i | + X i Let ( u , . . . , u k ) be a solution to (1.4) in C ( −∞ ,b ) , such that (2.9) holds. Then H ′ ( r ) = 2 Z Σ r k X i =1 u i ∂ ν u i = 2 E ( r ) for every r < b . In particular, H is nondecreasing.Proof. For every s < r < b , the divergence theorem and the periodicity of ( u , . . . , u k ) imply that E ( r ) = E ( s ) + Z C ( s,r ) X i |∇ u i | + 2 X i 1) be a non negative cut-offfunction, even with respect to r = 0, such that η (0) = 1 and η ≤ − , η s ( x ) = η ( x − s );testing the equation (2.1) with u i η s in C ( s − ,s ) , we find Z C ( s − ,s ) ∇ u i · ∇ ( u i η s ) + u i X i = j u j η s = Z Σ s u i ∂ x u i Summing up for i = 1 , . . . , k , we obtain Z Σ s X i u i ∂ x u i = Z C ( s − ,s ) X i (cid:0) u i ∂ x u i η ′ s + |∇ u i | η s (cid:1) + 2 X i This is a consequence of the Poincar´e inequality Z C ( s − ,s ) u ≤ C Z Σ s u + Z C ( s − ,s ) |∇ u | ! ∀ u ∈ H ( C ( s − ,s ) )together with assumption (2.9) and the fact that E ( s ) → s → −∞ (see (2.14)). Thus, from the(2.16) we deduce that lim s →−∞ Z Σ s X i u i ∂ x u i = 0 , which in turns can be used in the (2.15) to obtain the thesis: E ( r ) = lim s →−∞ E ( s ) − Z Σ s X i u i ∂ x u i + Z Σ x X i u i ∂ ν u i ! = Z Σ x X i u i ∂ ν u i . (cid:3) In light of the previous results, the proof of the following statements are straightforward modificationof the proofs of Proposition 2.4, Corollary 2.5 and Lemmas 2.8 and 2.9. Proposition 2.14. Let ( u , . . . , u k ) be a solution of (2.1) in C ( −∞ ,b ) such that (2.9) holds. The Almgrenquotient N unb ( r ) := E unb ( r ) H ( r ) is well defined in ( −∞ , b ) and nondecreasing. Moreover, Z r −∞ R Σ s P i Let ( u , . . . , u k ) be a solution of (2.1) in C ( −∞ ,b ) such that (2.9) holds. ( i ) If N ( r ) ≥ d for r ≥ s , then H ( r ) e dr ≤ H ( r ) e dr ∀ s ≤ r < r < b,ii ) If N ( r ) ≤ d for r ≤ t < b , then H ( r ) e dr ≥ H ( r ) e dr ∀ r < r ≤ t. For a fixed r < b , let us introduce ϕ ( r ; r ) := Z rr d sH ( s ) / . The function ϕ is positive and increasing in R + ; thanks to point ( i ) of Corollary 2.15 and to the mono-tonicity of N , whenever ( u, v ) is nontrivial ϕ is bounded by a quantity depending only H ( r ) and N ( r ):(2.17) ϕ ( r ; r ) ≤ e N ( r ) r H ( r ) N ( r ) h e − N ( r ) r − e − N ( r ) r i . This, together with the monotonicity of ϕ ( · ; r ), implies that if b = + ∞ then there exists the limitlim r → + ∞ ϕ ( r ; r ) < + ∞ . Lemma 2.16. Let ( u , . . . , u k ) be a solution of (1.1) in C ( −∞ ,b ) such that (2.9) hold. Let r ∈ ( −∞ , b ) ,and assume that (2.18) u i +1 ( x, y ) = u i ( x, y − π ) and u ( x, τ + y ) = u ( x, τ − y ) where τ ∈ [0 , kπ ) . There exists C > such that the function r E ( r ) e r e Cϕ ( r ; r ) is nondecreasing in r for r > r . Lemma 2.17. Let ( u , . . . , u k ) be a nontrivial solution of (2.1) in C ∞ , and assume that (2.9) and (2.18) hold. If d := lim r → + ∞ N ( r ) < + ∞ , then d ≥ and lim r → + ∞ E ( r ) e r > . Remark 2.18. The achievements of this section hold true for solutions to ( − ∆ u i = − βu i P j = i u j u i > X i |∇ u i | + 2 X i Here we prove some monotonicity formulaefor harmonic functions of the plane which are 2 π periodic in one variable. In what follows, in the definitionof C ( a,b ) and Σ r we mean k = 2. The following results will come useful in section 6.Firstly, it is not difficult to obtain the counterpart of Lemma 2.1. Lemma 2.19. Let Ψ be an entire harmonic function in C ( a,b ) . Then the function r Z Σ r |∇ Ψ | − x is constant.Proof. We proceed as in the proof of Lemma 2.1: for a < r < r < b , we test the equation − ∆Ψ = 0with Ψ x in C ( r ,r ) and integrate by parts. (cid:3) In what follows we consider a harmonic function Ψ defined in an unbounded cylinder C ( −∞ ,b ) , with b ∈ R (the choice b = + ∞ is admissible). We assume that(2.19) H ( r ; Ψ) := Z Σ r Ψ → r → −∞ . Lemma 2.20. Let Ψ be a harmonic function in C ( −∞ ,b ) such that (2.19) holds true. Then ( i ) for every r ∈ R it results E unb ( r ; Ψ) := Z C ( −∞ ,r ) |∇ Ψ | < + ∞ ( ii ) it results (2.20) Z Σ r |∇ Ψ | = 2 Z Σ r ( ∂ x Ψ) Proof. In light of Lemma 2.19, it is not difficult to adapt the proof of Lemma 2.11 and obtain ( i ).As far as ( ii ), we can proceed as in the proof of Lemma 2.12 (note that, thanks to (2.19), it results H ′ ( r ; Ψ) = 2 E unb ( r ; Ψ)). (cid:3) Proposition 2.21. Let Ψ be a nontrivial harmonic function in C ( −∞ ,b ) , such that (2.19) holds true.The Almgren quotient N unb ( r ; Ψ) := R C ( −∞ ,r ) |∇ Ψ | R Σ r Ψ is nondecreasing in r . If N ( · ; Ψ) is constant for r in some non empty open interval ( r , r ) , then N ( r ; Ψ) is constant for all r ∈ R and there exists a positive integer d ∈ N such that N ( r ; Ψ) = d ; furthermore, Ψ( x, y ) = [ C cos( dy ) + C sin( dy )] e dx for some C , C ∈ R .Proof. The Almgren quotient is well defined, thanks to Lemma 2.20. To prove its monotonicity, wecompute the logarithmic derivative by means of the Pohozaev identity (2.20)( N unb ) ′ ( r ; Ψ) N unb ( r ; Ψ) = R Σ r |∇ Ψ | R C ( −∞ ,r ) |∇ Ψ | − R Σ r Ψ ∂ x Ψ R Σ r Ψ = 2 R Σ r | ∂ x Ψ | R Σ r Ψ ∂ x Ψ − R Σ r Ψ ∂ x Ψ R Σ r Ψ ≥ N unb ( r ; Ψ) is constant for r ∈ ( r , r ). By the previous computations it followsthat necessarily Z Σ r | ∂ x Ψ | Z Σ r Ψ = (cid:18)Z Σ r Ψ ∂ x Ψ (cid:19) for every r ∈ ( r , r ). Again from the Cauchy-Schwarz inequality, we evince that it must be ∂ x Ψ = λ Ψ on Σ r for some constant λ ∈ R and for every r ∈ ( r , r ). Solving the differential equation, we find the Ψ is ofthe form Ψ( x, y ) = ψ ( y ) e λx . This together with the equation ∆Ψ = 0 yields, ψ ′′ + λ ψ = 0 ⇒ Ψ( x, y ) = [ a cos( λy ) + b sin( λy )] e λx ∀ ( x, y ) ∈ ( r , r ) × R , and Ψ can be uniquely extended to R by the unique continuation principle for harmonic functions. SinceΨ satisfies the condition (2.19) and is nontrivial, it follows that λ > 0. The proof is complete, recallingthe periodicity in y of the function Ψ and computing its Almgren quotient. (cid:3) Proof of Theorem 1.1 In this section we construct a solution to (1.1) modeled on the harmonic function Φ( x, y ) = cosh x sin y .3.1. Existence in bounded cylinders. For every R > u R , v R ) to(3.1a) − ∆ u = − uv in C R − ∆ v = − u v in C R u, v > − R, R ) × (0 , π ) with periodic boundary condition on thesides [ − R, R ] × { , π } ) with Dirichlet boundary condition(3.1b) u = Φ + , v = Φ − on Σ R ∪ Σ − R , and exhibiting the same symmetries of (Φ + , Φ − ). To be precise: Proposition 3.1. There exists a solution ( u R , v R ) to problem (3.1a) with the prescribed boundary con-ditions (3.1b) , such that u R ( − x, y ) = u R ( x, y ) and v R ( − x, y ) = v R ( x, y ) , the symmetries v R ( x, y ) = u R ( x, y − π ) u R ( π − x, y ) = v R ( π + x, y ) u R (cid:16) x, π y (cid:17) = u R (cid:16) x, π − y (cid:17) v R (cid:18) x, π + y (cid:19) = v R (cid:18) x, π − y (cid:19) hold, u R − v R ≥ in { Φ > } and v R − u R ≥ in { Φ < } , u R > Φ + and v R > Φ − . Remark 3.2. In light of the eveness of ( u R , v R ) in x , it results ∂ x u = 0 = ∂ x v on Σ . As a consequence, the monotonicity formulae proved in subsection 2.1 hold true for ( u R , v R ) in thesemi-cylinder C (0 ,R ) .In order to keep the notation as simple as possible, in what follows we will refer to a solution of(3.1a)-(3.1b) as to a solution of (3.1). Proof. Let U R := ( u, v ) ∈ ( H ( C R )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = Φ + , v = Φ − on Σ R ∪ Σ − R , u ≥ ,u − v ≥ { Φ ≥ } , v ( x, y ) = u ( x, y − π ) , u ( − x, y ) = u ( x, y ) ,u ( x, π − y ) = v ( x, π + y ) , u (cid:0) x, π + y (cid:1) = u (cid:0) x, π − y (cid:1) . Note that if ( u, v ) ∈ U R then v is nonnegative, even in x and symmetric in y with respect to π ; moreover, u − v ≤ { Φ < } . It is immediate to check that U R is weakly closed with respect to the H topology.We seek solutions of (3.1) as minimizers of the energy functional J ( u, v ) := Z C R |∇ u | + |∇ v | + u v in U R . The existence of at least one minimizer is given by the direct method of the calculus of variations;for the coercivity of the functional J , we use the following Poincar´e inequality:(3.2) Z C R u ≤ C Z Σ − R u + Z C R |∇ u | ! ∀ u ∈ H ( C R ) , where C depends only on R . To show that a minimizer satisfies equation (3.1), we consider the parabolicproblem(3.3) U t − ∆ U = − U V in (0 , + ∞ ) × C R V t − ∆ V = − U V in (0 , + ∞ ) × C R U = Φ + , V = Φ − on (0 , + ∞ ) × (Σ R ∪ Σ − R )with initial condition in U R . There exists a unique local solution ( U, V ); by Lemma A.1 if follows U, V ≥ ≤ U ≤ sup C R Φ + and 0 ≤ V ≤ sup C R Φ − . This control reveals that ( U, V ) can be uniquely extended in the whole (0 , + ∞ ). Since(3.4) dd t J ( U ( t, · ) , V ( t, · )) = − Z C R (cid:0) U t + V t (cid:1) ≤ , that is, the energy is a Lyapunov functional, from the parabolic theory it follows that for every sequence t i → + ∞ there exists a subsequence ( t j ) such that ( U ( t j · ) , V ( t j , · )) converges to a solution ( u, v ) of (3.1).Therefore, in order to prove that ( u R , v R ) solves (3.1), it is sufficient to show that there exists an initialcondition in U R such that the limiting profile ( u, v ) coincides with ( u R , v R ). We use the fact that(3.5) U R is positively invariant under the parabolic flow . To prove this claim, we firstly note that by the symmetry of initial and boundary conditions and by theuniqueness of the solution to problem (3.3), we have V ( t, x, y ) = U ( t, x, y − π ) , U ( t, − x, y ) = U ( t, x, y ) ,V ( t, x, π + y ) = U ( t, x, π − y ) , U (cid:16) t, x, π y (cid:17) = U (cid:16) t, x, π − y (cid:17) . (3.6) XPONENTIAL ENTIRE SOLUTIONS 19 This implies U ( t, x, π ) − V ( t, x, π ) = 0 ∀ ( t, x ) ∈ (0 , + ∞ ) × [ − R, R ] . Furthermore, using the (3.6) and the periodicity of ( U, V ) U ( t, x, − V ( t, x, 0) = U ( t, x, − V ( t, x, π ) = 0 ∀ ( t, x ) ∈ (0 , + ∞ ) × [ − R, R ] U ( t, x, π ) − V ( t, x, π ) = U ( t, x, π ) − V ( t, x, 0) = 0 ∀ ( t, x ) ∈ (0 , + ∞ ) × [ − R, R ] . This means that U − V = 0 on { Φ = 0 } . Let us introduce D R := { Φ > } ∩ C R . For every ( u , v ) ∈ U R ,we have(3.7) ( U − V ) t − ∆( U − V ) = U V ( U − V ) in (0 , + ∞ ) × D R U − V ≥ { } × D R U − V ≥ , + ∞ ) × ∂D R . Lemma A.1 implies U − V ≥ , + ∞ ) × D R . This completes the proof of the claim.Let us consider the equation (3.3) with the initial conditions U (0 , x, y ) = u R ( x, y ), V (0 , x, y ) = v R ( x, y ); let us denote ( U R , V R ) the corresponding solution. On one side, by minimality, J ( u R , v R ) ≤ J ( U R ( t, · ) , V R ( t, · )) ∀ t ∈ (0 , + ∞ );we point out that this comparison is possible because of (3.5). On the other side, by (3.4), J ( U R ( t, · ) , V R ( t, · )) ≤ J ( u R , v R ) ∀ t ∈ (0 , + ∞ ) . We deduce that J ( U R , V R ) is constant, which in turns implies (use again (3.4)), U Rt ( t, x, y ) = V Rt ( t, x, y ) ≡ ⇒ U R ( t, x, y ) = u R ( x, y ) , V R ( t, x, y ) = v R ( x, y ) . By the above argument, as ( u R , v R ) coincides with the asymptotic profile of a solution of the parabolicproblem (3.3), it solves (3.1). Points 1)-3) of the thesis are satisfied due to the positive invariance of U R .The strong maximum principle yields u R > v R > 0. Moreover, ( − ∆( u R − v R − Φ) = u R v R ( u R − v R ) ≥ D R u R − v R − Φ = 0 on ∂D R ⇒ u R − v R − Φ ≥ D R , so that by the strong maximum principle and the fact that u R , v R > u R > Φ + . Analogously, v R > Φ − . (cid:3) Remark 3.3. The existence of a positive solution satisfying the conditions 1)-2) of the Proposition canbe proved by means of the celebrated Palais’ Principle of Symmetric Criticality. To do this, it is sufficientto minimize the functional J in the weakly closed set V R := ( u, v ) ∈ ( H ( C R )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = Φ + , v = Φ − on Σ R ∪ Σ − R ,v ( x, y ) = u ( x, y − π ) , u ( − x, y ) = u ( x, y ) ,u ( x, π − y ) = v ( x, π + y ) , u (cid:0) x, π + y (cid:1) = u (cid:0) x, π − y (cid:1) , and apply the maximum principle. We choose a more complicated proof since we will strongly use thepointwise estimates given by point 4).3.2. Compactness of the family { ( u R , v R ) } . In this section we aim at proving that, up to a subse-quence, the family { ( u R , v R ) : R > } obtained in Proposition 3.1 converges, as R → + ∞ , to a solution( u, v ) of (1.1) defined in the whole C ∞ . Then, by looking at ( u, v ) as defined in R (this is possiblethanks to the periodicity), we obtain a solution of (1.1) satisfying the conditions 1)-5) of Theorem 1.1.At a later stage, we will also obtain the estimates of points 6) and 7).We denote E R , E R , H R , N R and N R the functions E sym , H, E sym , N sym and N sym (which have beendefined in subsection 2.1) when referred to ( u R , v R ). As observed in Remark 3.2, for these quantities theresults of subsection 2.1 apply.We will obtain compactness of the sequence ( u R , v R ) using some uniform-in- R control on N R and H R .We start with a uniform (in both r and R ) upper bound for the Almgren quotients N R ( r ). Lemma 3.4. There holds N R ( r ) ≤ , for every R > and r ∈ (0 , R ) . Proof. It is an easy consequence of the monotonicity of N R and of the minimality of ( u R , v R ) for thefunctional J in U R : noting that J ( u R , v R ) = E R ( R ), we compute N R ( r ) ≤ N R ( R ) ≤ E R ( R ) H R ( R ) ≤ R Σ R Φ Z C (0 ,R ) |∇ Φ | = 2 tanh R. We used the fact that the restriction of (Φ + , Φ − ) in C R is an element of U R for every R , and the boundarycondition of ( u R , v R ) on Σ R . (cid:3) In the proof of the following Lemma we will exploited the compactness of the local trace operator T Σ : u ∈ H ( C (0 , ) u | Σ ∈ L (Σ ), see Corollary A.4. Lemma 3.5. There exists C > such that H R (1) ≤ C for every R > .Proof. By contradiction, assume that H R n (1) → + ∞ for a sequence R n → + ∞ . Let us introduce thesequence of scaled functions(ˆ u n ( x, y ) , ˆ v n ( x, y )) := 1 p H R n (1) ( u R n ( x, y ) , v R n ( x, y )) . We wish to prove a convergence result for such a sequence, in order to obtain a uniform lower bound for N R n (1). In a natural way, the scaling leads us to consider, for r ∈ (0 , E n ( r ) := Z C (0 ,r ) |∇ ˆ u n | + |∇ ˆ v n | + 2 H R n (1)ˆ u n ˆ v n , ˆ H n ( r ) := Z Σ r ˆ u n + ˆ v n , ˆ N n ( r ) := ˆ E n ( r )ˆ H n ( r ) . By construction, it holds ˆ H n (1) = 1 and ˆ N n ( r ) = N R n ( r ) ≤ 2; therefore, thanks to Lemma 3.4(3.8) Z C (0 , |∇ ˆ u n | + |∇ ˆ v n | ≤ ˆ E n (1) = ˆ N n (1) ˆ H n (1) ≤ ∀ r ∈ (0 , , which gives a uniform bound in the H ( C (0 , ) norm of the sequence (ˆ u n , ˆ v n ) (we can use a Poincar´einequality of type (3.2)). Then, we can extract a subsequence which converges weakly in H ( C (0 , ) tosome limiting profile (ˆ u, ˆ v ), which is nontrivial in light of the compactness of the local trace operator T Σ and of the fact that ˆ H n (1) = 1. Since the set of the restrictions to C (0 , of functions of U R is closed inthe weak H ( C (0 , ) topology, ˆ u and ˆ v are nonnegative functions with the same symmetries of ( u R , v R );moreover we can show that (ˆ u, ˆ v ) satisfies the segregation condition ˆ u ˆ v = 0 a.e. in C (0 , . Indeed, by thecompactness of the Sobolev embedding H ( C (0 , ) ֒ → L ( C (0 , ) we deduce that the interaction term I ( u, v ) := Z C (0 , u v is continuous in the weak topology of ( H ( C (0 , )) . From the estimate (3.8), we infer2 H R n (1) I (ˆ u n , ˆ v n ) ≤ ˆ E n (1) ≤ n → + ∞ , we conclude I (ˆ u, ˆ v ) = lim n →∞ I (ˆ u n , ˆ v n ) = 0 ⇒ ˆ u ˆ v = 0 a.e. in C (0 , . Moreover, from the compactness of the local trace operator T Σ , we also deduce R Σ ˆ u + ˆ v = 1. Let usconsider the functional J ∞ ( u, v ) := Z C (0 , |∇ u | + |∇ v | , defined in the set M := (cid:26) ( u, v ) ∈ ( H ( C (0 , )) (cid:12)(cid:12)(cid:12)(cid:12) R Σ u + v = 1 ,v ( x, y ) = u ( x, y − π ) , uv = 0 a.e. in C (cid:27) . XPONENTIAL ENTIRE SOLUTIONS 21 Due to the compactness of the trace operator, one can check that M is closed in the weak ( H ( C (0 , )) topology. It is clear that (ˆ u, ˆ v ) ∈ M . We claim thatinf ( u,v ) ∈M J ∞ ( u, v ) =: m > . Indeed, let us assume by contradiction that the infimum is 0: since the set M is weakly closed and J ∞ is weakly lower semi-continuous and coercive, there exists (¯ u, ¯ v ) such that J ∞ (¯ u, ¯ v ) = 0. It followsthat (¯ u, ¯ v ) is a vector of constant functions; the symmetry and the segregation condition imply that(¯ u, ¯ v ) ≡ (0 , u, ¯ v ) ∈ M . Thus, the weak convergence of thesequence (ˆ u n , ˆ v n ) entailslim inf n →∞ ˆ N n (1) ≥ lim inf n →∞ Z C (0 , |∇ ˆ u n | + |∇ ˆ v n | ≥ m > , so that whenever n is sufficiently large(3.9) N R n (1) = ˆ N n (1) ≥ m Thanks to Lemma 3.4 we know that m ≤ N R n (1) ≤ 2, and from the assumption H R n (1) → + ∞ wededuce that (recall the (2.4)) ϕ R n ( r ; 1) : = Z r d sH R n ( s ) / ≤ e N Rn (1) H R n (1) N R n (1) h e − N Rn (1) − e − N Rn (1) r i → n → ∞ , for every r > 1. In particular, there exists C > ϕ R n ( r ; 1) ≤ C ∀ ≤ r ≤ R n , ∀ n. This implies that the sequence ( E R n (1)) n is bounded. To see this, we firstly note that ( u R n , v R n ) satisfiesthe symmetry condition (2.6) which is necessary to apply Lemma 2.8; consequently, the variationalcharacterization of ( u R n , v R n ) (see also the proof of Lemma 3.4 and the (3.10)) implies that E R n (1) e ≤ e Cϕ Rn ( R n ;1) E R n ( R n ) e R n ≤ C E R n ( R n ) e R n ≤ C R C (0 ,Rn ) |∇ Φ | e R n = C sinh R n cosh R n e R n ≤ C, where C does not depend on n . Since ( E R n (1)) n is bounded and ( H R n (1)) n tends to infinity, we obtainlim n →∞ N R n (1) = lim n →∞ E R n (1) H R n (1) = 0 , in contradiction with the (3.9) (cid:3) Proposition 3.6. There exists a subsequence of ( u R , v R ) which converges in C loc ( C ∞ ) , as R → + ∞ , toa solution ( u, v ) of (1.1) in the whole C ∞ . This solution satisfies point 2)-5) of Theorem 1.1, and itsAlmgren quotient N is such that N ( r ) ≤ ∀ r > and lim r → + ∞ N ( r ) ≥ . Proof. As H R (1) is bounded in R and N R (1) ≤ 2, also E R (1) is bounded in R . By means of a Poincar´einequality of type (3.2), this induces a uniform-in- R bound for the H ( C (0 , ) norm of ( u R , v R ), which inturns, by the compactness of the trace operator, gives a uniform-in- R bound for the L ( ∂C (0 , ) norm.Due to the subharmonicity of ( u R , v R ), the L ( ∂C (0 , ) bound provides a uniform-in- R bound for the L ∞ norm of ( u R , v R ) in every compact subset of C (0 , ; the regularity theory for elliptic equations (see [8])ensures that, up to a subsequence, ( u R , v R ) converges in C loc ( C (0 , ), as R → + ∞ , to a solution ( u , v )of (1.1) in C (0 , . As each ( u R , v R ) is even in x , this solution can be extended by even symmetry in x to C , and here satisfies the conditions 1)-4) of Proposition 3.1 (hence both u and v are nontrivial). Theprevious argument can be iterated: indeed, by Corollary 2.5 and Lemma 3.4, we deduce H R ( r ) ≤ H R (1) e e r ≤ Ce r ∀ r > R bound for H R (1) implies a uniform-in- R bound for H R ( r ) for every r > 1. As aconsequence we obtain, for every r > 1, a solution ( u r , v r ) to equation (1.1) in C r . A diagonal selectiongives the existence of a solution ( u, v ) to (1.1) in the whole C ∞ . This solution inherits by ( u r , v r ) theconditions 1)-4) of Proposition 3.1, and thanks to the C loc ( C ∞ ) convergence and Lemma 3.4 there holds N ( r ) = R C (0 ,r ) |∇ u | + |∇ v | + 2 u v R Σ r u + v ≤ ∀ r > . From Lemma 2.9, which we can apply in light of the symmetries of ( u, v ), we concludelim r → + ∞ N ( r ) ≥ . (cid:3) The following Lemma completes the proof of point 6) of Theorem 1.1. After that, by means of thepointwise estimates u > Φ + and v > Φ − and Corollary 2.5, it is straightforward to obtain also point 7). Lemma 3.7. There holds l := lim r →∞ N ( r ) = 1 . Proof. In light of the fact that l ≥ 1, it is sufficient to show that l ≤ 1. Let ( u R n , v R n ) be the convergentsubsequence found in Proposition 3.6, which we will simply denote { ( u n , v n ) } . For r > f n ( r ) := R C (0 ,r ) u n v n H R n ( r ) , g n ( r ) := R Σ r u n v n H R n ( r ) . With f and g we identify the same quantities computed for the limiting profile ( u, v ). Observe that f n , g n , f and g are continuous and nonnegative. By definition,(3.11) f n ( r ) ≤ N R n ( r ) ≤ ∀ r > u n , v n ) implies that f n → f and g n → g uniformly on compact intervals, while by Theorem 2.4 we have Z r g n ( s ) d s ≤ N R n ( r ) and Z r g ( s ) d s ≤ N ( r ) , so that in particular g n ∈ L (0 , R ) and g ∈ L ( R + ). By means of the monotonicity formula for theAlmgren quotient N , Proposition 2.4, it is possible to refine the computation in Lemma 3.4: N R n ( r ) = N R n ( r ) + f R n ( r ) ≤ N R n ( R n ) + f R n ( r ) ≤ f n ( r ) . In light of the strong H loc ( C ∞ ) convergence of ( u n , v n ) to ( u, v ), we deduce N ( r ) ≤ n → + ∞ f n ( r ) = 1 + f ( r ) . We have to show that f ( r ) → r → + ∞ . To prove this, we begin by computing the logarithmicderivative of f n : f ′ n ( r ) f n ( r ) = R Σ r u n v n R C (0 ,r ) u n v n − E R n ( r ) H R n ( r ) = g n ( r ) f n ( r ) − N R n ( r ) , where we used the fact that H ′ R n ( r ) = 2 E R n ( r ), see the (2.3). Exploiting the strong H convergence ofthe sequence { ( u n , v n ) } and the fact that lim r → + ∞ N ( r ) ≥ 1, we deduce that there exist r , δ > N R n ( r ) > δ for every n sufficiently large. Consequently, f n satisfies the inequality f ′ n ( r ) + 2 δf n ( r ) ≤ g n ( r ) for r ∈ ( r , R n ) . XPONENTIAL ENTIRE SOLUTIONS 23 Multiplying for e δr and integrating in ( r , r ) for r < r < r < R n , we obtain f n ( r ) ≤ e δ ( r − r ) f n ( r ) + Z r r g n ( s ) e δ ( s − r ) d s ≤ e δ ( r − r ) + Z r r g n ( s ) d s, where we used the estimate (3.11). This implies f ( r ) ≤ e δ ( r − r ) + Z r r g ( s ) d s for r < r < r . Since g ∈ L ( R + ) and f ≥ 0, choosing r = r we findlim sup r → + ∞ f ( r ) = 0 = lim r → + ∞ f ( r ) . (cid:3) Proof of Theorem 1.5 In this section we construct a solution to (1.1) modeled on the harmonic function Γ( x, y ) = e x sin y .Our construction is based on the trivial observation thatΦ R ( x, y ) := 2 cosh( x + R ) e − R sin y → Γ( x, y ) as R → + ∞ . Existence in bounded cylinders. As a first step, using the same line of reasoning developed inProposition 3.1, it is possible to show the existence of solution to the system(4.1a) − ∆ u = − uv in C ( − R,R ) − ∆ v = − u v in C ( − R,R ) u, v > − R, R ) × (0 , π ) with periodic boundarycondition on the sides [ − R, R ] × { , π } ) and such that(4.1b) u R = Φ + R , v R = Φ − R on Σ R ∪ Σ − R More precisely: Proposition 4.1. There exists a solution ( u R , v R ) to problem (4.1a) with the prescribed boundary con-ditions (4.1b) , such that u R ( − R − x, y ) = u R ( − R + x, y ) and v R ( − R − x, y ) = v R ( − R + x, y ) , the symmetries v R ( x, y ) = u R ( x, y − π ) u R ( x, π − y ) = v R ( x, π + y ) u R (cid:16) x, π y (cid:17) = u R (cid:16) x, π − y (cid:17) v R (cid:18) x, π + y (cid:19) = v R (cid:18) x, π − y (cid:19) hold, u R − v R ≥ in { Φ R > } and v R − u R ≥ in { Φ R < } , u R > (Φ R ) + and v R > (Φ R ) − .Sketch of proof. One can recast the proof of Proposition 3.1 in this setting. (cid:3) Remark 4.2. In light of point 1) of the Proposition, it results ∂ x u R = 0 = ∂ x v R on Σ − R . Therefore, the monotonicty formulae proved in subsection 2.1 hold true for ( u R , v R ) in the semi-cylinder C R . Compactness of the family { ( u R , v R ) } . As in the previous section, we denote as E R , E R , N R and N R the functions E sym , E sym , N sym and N sym defined in subsection 2.1 when referred to ( u R , v R ). Wefollow here the same line of reasoning adopted in subsection 3.2. Firstly, it is not difficult to modify theproof of Lemmas 3.4 and 3.5 obtaining the following estimates: Lemma 4.3. There holds N R ( r ) ≤ , for every R > and r ∈ ( − R, R ) . Lemma 4.4. There exists C > such that H R (1) ≤ C for every R > . We are in position to show that the family { ( u R , v R ) } is compact, in the following sense. Proposition 4.5. There exists a subsequence of { ( u R , v R ) } which converges in C loc ( C ∞ ) , as R → + ∞ ,to a solution ( u, v ) of (1.1) in the whole C ∞ . This solution has the properties 2)-4) of Proposition 4.1.Proof. As H R (1) is bounded in R and N R (1) ≤ 2, also E R (1) is bounded in R , and a fortiori Z C |∇ u R | + |∇ v R | ≤ C ∀ R > . This estimate, the boundedness of H R (1) and a Poincar`e inequality of type (3.2) imply that { ( u R , v R ) } is bounded in H ( C ). Consequently, it is possible to argue as in the proof of Proposition 3.6 and obtainthe existence of a subsequence of { ( u R , v R ) } which converges in C loc ( C ) to a solution ( u , v ) of (1.1)in C , which inherits by { ( u R , v R ) } the properties 2)-4) of Proposition 4.1. In light of Corollary 2.5 andLemma 4.3, this procedure can be iterated: indeed H R ( r ) ≤ H R (1) e e r ≤ Ce r ∀ r > , so that applying the previous argument we obtain a subsequence of { ( u R , v R ) } which converges in C loc ( C r )to a solution ( u r , v r ) of (1.1) in C r , and inherits by { ( u R , v R ) } the properties 2)-4) of Proposition 4.1.A diagonal selection gives the existence of a solution ( u, v ) of (1.1) in the whole C ∞ , and this solutionenjoys the properties 2)-4) of Proposition 4.1. (cid:3) Remark 4.6. The monotonicity formulae proved in subsection 2.1 do not apply on ( u, v ), because passingto the limit we lose the Neumann condition ∂ x u R = 0 = ∂ x v R on Σ − R .In the next Lemma, we show that ( u, v ) is a solution with finite energy, so that the achievementsproved in subsection 2.2 applies. Lemma 4.7. Let ( u, v ) be the solution found in Proposition 4.5. It results (4.2) E unb ( r ) := Z C ( −∞ ,r ) |∇ u | + |∇ v | + u v < + ∞ ∀ r ∈ R and lim r →−∞ H ( r ) = lim r →−∞ Z Σ r u + v = 0 . Recall that E unb has been defined in subsection 2.2.Proof. Let { ( u R n , v R n ) } be the converging subsequence found in Proposition 4.5, which we will simplydenote { ( u n , v n ) } . Since { ( u n , v n ) } converges to ( u, v ) in C loc ( C ∞ ), it follows thatlim n →∞ (cid:0) |∇ u n | + |∇ v n | + u n v n (cid:1) χ C ( − Rn,r ) = (cid:0) |∇ u | + |∇ v | + u v (cid:1) χ C ( −∞ ,r ) a.e. in C ( −∞ ,r ) , for every r > 1. Therefore, applying Corollary 2.5 on ( u n , v n ), Lemma 4.4 and the Fatou lemma, wededuce E unb ( r ) ≤ lim inf n →∞ Z C ( −∞ ,r ) (cid:0) |∇ u n | + |∇ v n | + u n v n (cid:1) χ C ( − Rn,r ) ≤ lim inf n →∞ E R n ( r )= lim inf n →∞ N R n ( r ) H R n ( r ) ≤ lim inf n →∞ H R n (1) e e r ≤ Ce r , XPONENTIAL ENTIRE SOLUTIONS 25 which proves the (4.2). To complete the proof, we firstly note that necessarily E unb ( r ) → r → −∞ ,and hence the same holds for E unb (which has been defined in subsection 2.2). Assume by contradictionthat for a sequence r n → −∞ it results H ( r n ) ≥ C > 0. We define(ˆ u n ( x, y ) , ˆ v n ( x, y )) := 1 p H ( r n ) ( u ( x + r n , y ) , v ( x + r n , y )) . A direct computation shows that Z C ( −∞ , |∇ ˆ u n | + |∇ ˆ v n | ≤ Z C ( −∞ , |∇ ˆ u n | + |∇ ˆ v n | + 2 H ( r n )ˆ u n ˆ v n = 1 H ( r n ) E unb ( r n ) → n → ∞ . Consequently, (ˆ u n , ˆ v n ) tend to be a pair of constant functions of type (ˆ u, ˆ v ) with ˆ u = ˆ v (thisfollows from the symmetries of ( u, v )). As C Z C ( −∞ , ˆ u n ˆ v n ≤ H ( r n ) Z C ( −∞ , ˆ u n ˆ v n → , necessarily (ˆ u n , ˆ v n ) → (0 , 0) almost everywhere in C ( −∞ , . This is in contradiction with the fact that R Σ ˆ u n + ˆ v n = H ( r n ) ≥ C . (cid:3) So far we proved that the solution ( u, v ), found in Proposition 4.5, enjoys properties 1)-5) of Theorem1.5, and is such that H ( r ) → r → −∞ . The previous Lemma enables us to apply the achievementsof subsection 2.2 for E unb , H, N unb and N unb (which we consider referred to the solution ( u, v ) found inProposition 4.5), and permits to complete the description of the growth of ( u, v ), points 6)-7) of Theorem1.5. Lemma 4.8. Let ( u, v ) be the solution found in Proposition 4.5. It results lim r → + ∞ N unb ( r ) = 1 . Proof. Let { ( u R n , v R n ) } be the converging subsequence found in Proposition 4.5, , which we will simplydenote { ( u n , v n ) } . Firstly, arguing as in the proof of the previous Lemma, we note that by the C loc ( C ∞ )convergence of ( u n , v n ) to ( u, v ) it follows that N unb ( r ) ≤ lim inf n →∞ N R n ( r ) ≤ ∀ r ∈ R , thanks to the Fatou lemma. This, together with the symmetries of ( u, v ), permits to use Lemma 2.17,which gives lim r → + ∞ N unb ( r ) ≥ 1. To complete the proof, it is sufficient to show that lim r → + ∞ N unb ( r ) ≤ 1. For any r > 0, let f n ( r ) := R C r u n v n H R n ( r ) , g n ( r ) := R Σ r ∪ Σ − r u n v n H R n ( r ) , and let f and g the same quantities referred to the solution ( u, v ). Observe that f n , g n , f and g arecontinuous and nonnegative. The uniform convergence of ( u n , v n ) to ( u, v ) implies that f n → f and g n → g , as n → ∞ , uniformly on compact intervals. By definition,(4.3) f n ( r ) ≤ N R n ( r ) ≤ ∀ r > . whenever R n ≥ r . We claim that g ∈ L ( R + ). Indeed, by the monotonicity of H and Proposition 2.14,it follows that Z r g ( s ) d s = Z r R Σ s u v H ( s ) d s + Z − r R Σ s u v H ( − s ) d s ≤ Z r − r R Σ s u v H ( s ) d s ≤ Z r −∞ R Σ s u v H ( s ) d s ≤ N unb ( r ) , for every r > 0. Let r > 0; it is possible to refine the computation on Lemma 3.4 to obtain N R n ( r ) ≤ f n ( r ) + R C ( − Rn, − r ) u n v n H R n ( r ) ≤ f n ( r ) + E R n ( − r ) H R n ( r ) Therefore, using again the Fatou lemma we deduce N unb ( r ) ≤ lim inf n →∞ N R n ( r ) ≤ f ( r ) + lim inf n →∞ E R n ( − r ) H R n ( r ) , and to complete the proof we will show that(4.4) lim r →∞ (cid:18) f ( r ) + lim inf n →∞ E R n ( − r ) H R n ( r ) (cid:19) = 0 . Firstly, we note thatlim inf n →∞ E R n ( − r ) H R n ( r ) = lim inf n →∞ N R n ( − r ) H R n ( − r ) H R n ( r ) ≤ n →∞ H R n ( − r ) H R n ( r ) . From the C loc ( C ∞ ) convergence of ( u n , v n ) to ( u, v ) it follows2 lim inf n →∞ H R n ( − r ) H R n ( r ) = 2 H ( − r ) H ( r ) → r → + ∞ where we used Lemma 4.7 and the fact that H ( r ) > H (0) > r > 0. For the (4.4) it remainsto prove that f ( r ) → r → + ∞ . Having observed that lim r → + ∞ N ( r ) ≥ g ∈ L ( R + ), it isnot difficult to adapt the conclusion of the proof of Lemma 3.7. (cid:3) Systems with many components In this section we are going to prove the existence of entire solutions with exponential growth for the k component system (1.4). Our construction is based on the elementary limitlim d → + ∞ ℑ (cid:20)(cid:16) zd (cid:17) d (cid:21) = e x sin y, which shows that the harmonic function e x sin y can be obtained as limit of homogeneous harmonicpolynomial. We wish to prove that the same idea applies to solutions of the system (1.4): there existsan entire solution to (1.4) having exponential growth which can be obtained as limit of entire solutionshaving algebraic growth.5.1. Preliminary results. We recall some results contained in [2]. For d ∈ N , let G d be the rotation ofangle πd . Theorem 5.1 (Theorem 1.6 of [2]) . Let k ≥ be a positive integer, let d ∈ N be such that d = hk for some h ∈ N . There exists a solution ( u d , . . . , u dk ) to the system (1.4) which enjoys the following symmetries u di ( x, y ) = u di ( G kd ( x, y )) u di ( x, y ) = u di +1 ( G d ( x, y )) u dk +1 − i ( x, y ) = u di ( x, − y )(5.1) where we recall that indexes are meant mod k . Moreover lim r → + ∞ r d Z ∂B r k X i =1 (cid:0) u di (cid:1) = b ∈ (0 , + ∞ ) , and (5.2) lim r → + ∞ r R B r P ki =1 |∇ u di | + P ≤ i In the figure we represent some of the solutions obtained in Theorem 5.1.Here the number of components is set as k = 3: each component is drawn with adifferent color. On the other hand the periodicity (that is, how many times the patchof 3-components is replicated in the circle) is given by h = 1 (up left), h = 2 (up right), h = 3 (down left) and h = 4 (down right), respectively. As a consequence, the growthrate d varies as d = , , , 6, following the same order.The solution ( u d , . . . , u dk ) is modeled on the harmonic polynomial ℑ ( z d ), as specified by the symmetries(5.1). In the quoted statement, the authors modeled their construction on the functions ℜ ( z d ): it isstraightforward to obtain an analogous result replacing the real part with the imaginary one. Remark 5.2. We point out that the symmetries (5.1) implies that u d is symmetric with respect to thereflection with the axis y = tan (cid:0) π d (cid:1) x .For a solution ( u , . . . , u k ) of system (1.4) in R , we introduce the functionals E alg ( r ; Λ) := Z B r k X i =1 |∇ u i | + Λ X ≤ i In [2] the authors consider the case Λ = 1.We work in the plane R , so that it is possible to choose Λ = 2 in Proposition 5.3. We denote E d ( · ; Λ)and H d the quantities defined in (5.3) when referred to the functions ( u d , . . . , u dk ) defined in Theorem 5.1;also, we denote N d ( · ; Λ) := E d ( · ; Λ) H d . In case Λ = 2, we will simply write E d and N d to ease the notation. Lemma 5.5. Let ( u d , . . . , u dk ) be defined in Theorem 5.1. There holds lim r → + ∞ N d ( r ) = d .Proof. It is an easy consequence of the (5.2) and of Corollary 5.8 in [2], where it is proved that for thesolution ( u d , . . . , u dk ) there holds lim r → + ∞ E d ( r ; 2) r d = lim r → + ∞ E d ( r ; 1) r d . XPONENTIAL ENTIRE SOLUTIONS 29 Therefore, lim r → + ∞ N d ( r ) = lim r → + ∞ E d ( r ; 2) H d ( r ) = lim r → + ∞ E d ( r ; 2) r d · lim r → + ∞ r d H d ( r )= lim r → + ∞ E d ( r ; 1) r d · lim r → + ∞ r d H d ( r ) = lim r → + ∞ N d ( r ; 1) = d. (cid:3) As a consequence, the following doubling property holds true: Proposition 5.6 (See Proposition 5.3 of [2]) . For any < r < r it holds H d ( r ) r d ≤ H d ( r ) r d . Proof. A direct computation shows thatdd r log H d ( r ) r d = 2 N d ( r ) r − dr ≤ (cid:3) Let us consider the scaling(5.5) ( u d ,R , . . . , u dk,R ) := (cid:18) dkH d ( R ) (cid:19) (cid:0) u d ( Rx, Ry ) , . . . , u dk ( Rx, Ry ) (cid:1) , where R will be determined later as a function of d . We see that(5.6) − ∆ u di,R = − β dR u di,R X j = i (cid:0) u dj,R (cid:1) in R Z ∂B k X i =1 (cid:0) u di,R (cid:1) = 2 dk where β dR := k d H d ( R ) R . Remark 5.7. As a function of R , β dR is continuous and such that β dR → R → β dR → ∞ if R → ∞ .Accordingly with our scaling, we introduce the new Almgren quotient N d,R ( r ) := E d,R ( r ) H R ( r ) = r Z B r k X i =1 |∇ u di,R | + 2 β dR X ≤ i In order to understand the behavior of (ˆ u d ,R , . . . , ˆ u dk,R ) when d → ∞ , we fix R = R ( d ) to get anon-degeneracy condition. Lemma 5.8. For every d ∈ N there exists R d > such that ˆ H d,R d (1) = Z ∂ r ˆ S d X i (cid:0) ˆ u di,R d (cid:1) = 1 . Proof. By (5.10) we know that ˆ H d (1) = β dR k , so that we have to find R d such that β dR = k . As observedin Remark 5.7, this choice is possible. (cid:3) We denote (ˆ u d , . . . , ˆ u dk ) := (ˆ u d ,R d , . . . , ˆ u dk,R d ), ˆ H d := ˆ H d,R d , ˆ E d := ˆ E d,R d , ˆ N d := ˆ N d,R d and β d := β dR d .We aim at proving that , up to a subsequence, the family (cid:8) (ˆ u d , . . . , ˆ u dk ) : d ∈ N (cid:9) converges, as d → + ∞ ,to a solution of (1.4). To this aim, major difficulties arise from the fact that ˆ S dr and ˆ S d depend on d ; inthe next Lemma we show that this problem can be overcome thanks to a convergence property of thesedomains. Lemma 5.9. For any r > , the sets ˆ S dr converge to R × (0 , kπ ) as k → + ∞ , in the sense that R × (0 , kπ ) = Int \ n ∈ N [ d>n ˆ S dr , where for A ⊂ R we mean that Int( A ) denotes the inner part A . Analogously, R × (0 , kπ ) = Int \ n ∈ N [ d>n ˆ S d and ( −∞ , × (0 , kπ ) = Int \ n ∈ N [ d>n ˆ S d , and for every ¯ x ∈ R ( −∞ , ¯ x ) × (0 , kπ ) = Int \ n ∈ N [ d>n ˆ S d ¯ xd . Proof. We prove only the first claim. Let r > Step 1) . R × (0 , kπ ) ⊂ \ n ∈ N [ d>n ˆ S dr .Let ( x, y ) ∈ R × (0 , kπ ). We show that for every d ∈ N sufficiently large ( x, y ) ∈ ˆ S dr , that is, (cid:0) xd , yd (cid:1) ∈ S dr , which means r(cid:16) xd (cid:17) + (cid:16) yd (cid:17) < r and arctan (cid:18) yx + d (cid:19) ∈ (cid:18) , kπd (cid:19) . For the first condition it is possible to choose d sufficiently large, as r > 1. To prove the second condition,we start by considering d > − x , so that arctan (cid:16) yx + d (cid:17) > 0. Now, provided d is sufficiently largearctan (cid:18) yx + d (cid:19) < kπd ⇔ y < ( x + d ) tan (cid:18) kπd (cid:19) . Since y < kπ , there exists ε > y ≤ k (1 − ε ) π . Let ¯ d be sufficiently large so that x + d > (cid:16) − ε (cid:17) d and dkπ tan (cid:18) kπd (cid:19) > − ε d > ¯ d . Then ( x + d ) tan (cid:18) kπd (cid:19) > (cid:16) − ε (cid:17) kπ > (1 − ε ) kπ ≥ y whenever d > ¯ d . Figure 2. Visualization of the construction in Lemma 5.9. In red the limiting set R × (0 , kπ ). In blue some of the scaled domains ˆ S dr , for r > Step 2) . \ n ∈ N [ d>n ˆ S dr ⊂ R × [0 , kπ ].We show that ( R × [0 , kπ ]) c ⊂ (cid:16)T n ∈ N S d>n ˆ S dr (cid:17) c . If ( x, y ) R × [0 , kπ ], then y > kπ or y < 0. Weconsider only the case y > kπ ; in such a situation y > kπ = lim d →∞ ( x + d ) tan (cid:18) kπd (cid:19) , so that ( x, y ) ˆ S dr for every d sufficiently large. (cid:3) Remark 5.10. As a consequence of the previous result, we see that ∂ r ˆ S d → { } × [0 , kπ ] and ∂ r ˆ S d ¯ xd → { ¯ x } × [0 , kπ ]for every ¯ x ∈ R . Remark 5.11. Recall the expression of r in the new variable, given by (5.11). For every r > d ∈ N there exists ξ ( r, d ) such that r = 1 + ξ ( r, d ) d ⇔ ξ ( r, d ) = d ( r − . Note that for every ( x, y ) ∈ ∂ r ˆ S dr it results x < ξ ( r, d ). On the contrary, fixing ( x, y ) ∈ ∂ r ˆ S dr there exists ζ ( d, x, y ) such that r = r(cid:16) xd (cid:17) + (cid:16) yd (cid:17) = 1 + xd + ζ ( d, x, y ) . In particular, if y = 0 we have ζ ( d, x, 0) = 0, while if y > ζ ( d, x, y ) ∼ d − .We are ready to prove the convergence of { (ˆ u d , . . . , ˆ u dk ) } as d → ∞ . Lemma 5.12. Up to a subsequence, { (ˆ u d , . . . , ˆ u dk ) } converges in C loc ( C ∞ ) , as d → ∞ , to a nontrivialsolution (ˆ u , . . . , ˆ u k ) of (1.4) . This solution, which is kπ -periodic in y , enjoys the symmetries ˆ u i +1 ( x, y ) = ˆ u i ( x, y − π ) and ˆ u (cid:16) x, y + π (cid:17) = ˆ u (cid:16) x, y − π (cid:17) Proof. From Proposition 5.6 and Lemma 5.8, we deduce that for any r ≥ d the inequalityˆ H d ( r ) r d = β d H d ( r ) d r d ≤ β d d H d (1) = ˆ H d (1) = 1 XPONENTIAL ENTIRE SOLUTIONS 33 holds. For every x > 0, let r = 1 + xd ; for every d sufficiently large, we have(5.13) ˆ H d (cid:16) xd (cid:17) ≤ (cid:16) xd (cid:17) d ≤ e x Recalling the (5.12) (which we apply for R = R d ), we deduce(5.14) ˆ E d (cid:16) xd (cid:17) = ˆ N d (cid:16) xd (cid:17) ˆ H d (cid:16) xd (cid:17) ≤ e x for every d sufficiently large. Recall that (ˆ u d , . . . , ˆ u dk ) can be extended by angular periodicity in the wholeplane R . Let us introduce T dr := n ( ρ, θ ) : ρ < r, θ ∈ (cid:16) − πd , ( k + 1) πd (cid:17)o ⊃ S dr , and let ˆ T dr := d (cid:0) T dr − (1 , (cid:1) ⊃ ˆ S dr . Suitably modifying the argument in Lemma 5.9, it is not difficult tosee that Int \ n ∈ N [ d>n ˆ T d ¯ xd = ( −∞ , ¯ x ) × ( − π, ( k + 1) π )for every ¯ x ∈ R . Hence, let B an open ball contained in R × ( − π, ( k + 1) π ), and let x B := sup { x : ( x, y ) ∈ B } , so that B ⊂ ( −∞ , x B + 1) × ( − π, ( k + 1) π ). Using the same argument in the proof of Lemma 5.9, itis possible to show that B ⊂ ˆ T dr , for every d sufficiently large, and by the (5.14) and the periodicity of (ˆ u , . . . , ˆ u k ) we deduce Z B X i |∇ u i | ≤ E d (cid:18) x B + 1 d (cid:19) ≤ e x B +1) whenever d is sufficiently large. This, together with (5.13), implies that { (ˆ u d , . . . , ˆ u dk ) } is uniformlybounded in H ( B ), for every B ⊂ R × ( − π, ( k + 1) π ). By the compactness of the trace operator, thisbound provides a uniform-in- d bound on the L ( ∂K ) norm for every compact K ⊂⊂ R × ( − π, ( k + 1) π ),which in turns, due to the subharmonicity of u di , gives a uniform-in- d bound on the L ∞ ( K ) norm of { (ˆ u d , . . . , ˆ u dk ) } , for every compact set K ⊂⊂ R × ( − π, ( k + 1) π ). The standard regularity theory forelliptic equations guarantees that when d → ∞ then { (ˆ u d , . . . , ˆ u dk ) } converges in C loc ( R × ( − π, ( k + 1) π )),up to a subsequence, to a function (ˆ u , . . . , ˆ u k ) which is a solution to (1.4). By the convergence and bythe normalization required in Lemma 5.8, we deduce that (recall also the convergence of the boundaries ∂ ˆ S d , Remark 5.10) Z kπ X i ˆ u i (0 , y ) d y = 1;in particular, (ˆ u , . . . , ˆ u k ) is nontrivial. The kπ -periodicity in y follows directly form the convergenceof the domains, Lemma 5.9. By the pointwise convergence of (ˆ u d , . . . , ˆ u dk ) to (ˆ u , . . . , ˆ u k ) and by thesymmetries of each function (ˆ u d , . . . , ˆ u dk ) (see equation (5.1) and Remark 5.2) we deduce also thatˆ u i +1 ( x, y ) = ˆ u i ( x, y − π ) and ˆ u (cid:16) x, y + π (cid:17) = ˆ u (cid:16) x, y − π (cid:17) . (cid:3) Characterization of the growth of (ˆ u , . . . , ˆ u k ) . So far we proved the existence of a solution(ˆ u , . . . , ˆ u k ) of (1.4) which enjoys the properties 1) and 2) of Theorem 1.8. In this subsection, we aregoing to complete the proof of the quoted statement, showing that (ˆ u , . . . , ˆ u k ) enjoys also the properties3)-5). We denote as ˆ E , ˆ E, ˆ H and ˆ N the quantities E unb , E unb , H and N unb introduced in subsection 2.2when referred to the function (ˆ u , . . . , ˆ u k ). Firstly, we show that (ˆ u , . . . , ˆ u k ) has finite energy, point 3)of Theorem 1.8, and that ˆ H ( x ) → −∞ as x → −∞ . Lemma 5.13. For every x ∈ R there holds ˆ E ( x ) < + ∞ . In particular ˆ E ( x ) ≤ lim inf d →∞ ˆ E d (cid:16) xd (cid:17) and ˆ E ( x ) ≤ lim inf d →∞ ˆ E d (cid:16) xd (cid:17) . Furthermore, lim x →−∞ ˆ H ( x ) = 0 .Proof. By the C loc ( R ) convergence of (ˆ u d , . . . , ˆ u dk ) to (ˆ u , . . . , ˆ u k ) and by the convergence properties ofthe domains ˆ S d xd , Lemma 5.9, we deducelim d →∞ X i |∇ ˆ u di | + X i There holds lim x → + ∞ ˆ N ( x ) = 1 . Proof. By Proposition 2.14, we know that ˆ N is nondecreasing in x , and thanks to the symmetries of(ˆ u , . . . , ˆ u k ), see Lemma 5.12, Lemma 2.17 implies that lim x → + ∞ ˆ N ( x ) ≥ 1. It remains to show that thislimit is smaller then 1. This follows from the estimates of Lemma 5.13 and from the strong convergenceof (ˆ u d , . . . , ˆ u dk ) → (ˆ u , . . . , ˆ u k ), which implies that ˆ H d (cid:0) xd (cid:1) → ˆ H ( x ) as d → ∞ : therefore, for every x ∈ R ˆ N ( x ) = ˆ E ( x )ˆ H ( x ) ≤ lim inf d →∞ ˆ E d ( x )lim d →∞ ˆ H d ( x ) = lim inf d →∞ ˆ N d ( x ) ≤ , where we used the (5.12). (cid:3) In light of this achievement, we can apply Corollary 2.15 to complete the proof of point 5) of Theorem1.8. The fact that γ > r → + ∞ ˆ H ( r ) e r = lim r → + ∞ ˆ E ( r ) e r · lim r → + ∞ N ( r ) > . Remark 5.15. With a similar construction, it is possible to obtain the existence of solutions to (1.4) in R modeled on cosh x sin y . To do this, we can first construct solutions of (1.4) having algebraic growthdefined outside the ball of radius 1, with homogeneous Neumann boundary conditions on ∂B . Thiscan be done suitably modifying the proof of Theorem 1.6 in [2]. Then, performing a new blow-up ina neighborhood of (1 , R , with homogeneous Neumanncondition on { x = 0 } ; this solution can be extended by even-symmetry in x in the whole R .6. Asymptotics of solutions which are periodic in one variable In this section we prove Theorem 1.9. Proof of Theorem 1.9. Let us start with case ( i ). First of all, let us recall that, since the solution ( u, v )is non trivial, N (0) > 0: in particular, from point ( i ) of Corollary 2.15 it follows that H ( r ) → + ∞ as r → + ∞ . Let us consider the shifted functions( u R ( x, y ) , v R ( x, y )) := 1 p H ( R ) ( u ( x + R, y ) , v ( x + R, y )) XPONENTIAL ENTIRE SOLUTIONS 35 which solve the system − ∆ u R = − H ( R ) u R v R in C ∞ − ∆ v R = − H ( R ) u R v R in C ∞ Z Σ u R + v R = 1and share the same periodicity of ( u, v ). We introduce E R ( r ) := Z C ( −∞ ,r ) |∇ u R | + |∇ R | + 2 H ( R ) u R v R ,H R ( r ) := Z Σ r u R + v R and N R ( r ) := E R ( r ) H R ( r ) . It is easy to see that E R ( r ) = 1 H ( R ) E unb ( r + R ) H R ( r ) = 1 H ( R ) H ( r + R ) ⇒ N R ( r ) = N unb ( r + R )for any r (recall that E unb and N unb have been defined in subsection 2.2). We point out that, bydefinition, N R ( r ) ≤ N R ( r ) for every R < R . Furthermore, N R ( r ) ≤ N (+ ∞ ) for every r, R and N R ( r ) → N (+ ∞ ) as R → ∞ for every r ∈ R . Therefore, N R tends to the constant function N (+ ∞ ) in L ( R ).Thanks to the normalization condition H R (0) = 1 and the uniform bound N R ( r ) < N (+ ∞ ), applyingCorollary 2.15 (see also Remark 2.18) we deduce that H R ( r ) is uniformly bounded in R for every r > E R ( r ) is uniformly bounded in R for every r > 0, and this reveals that the sequence( u R , v R ) is uniformly bounded in H ( C ∞ ) and, by standard elliptic estimates, in L ∞ loc ( C ∞ ). FromTheorem 2.6 of [11] (it is a local versione of Theorem 1.1 of [9]), we evince that the sequence ( u R , v R )is uniformly bounded also in C ,α loc ( C ∞ ) for any α ∈ (0 , u R , v R )converges in C loc ( C ∞ ) and in H loc ( C ∞ ) to a pair (Ψ + , Ψ − ), where Ψ is a nontrivial harmonic function(this is a combination of the main results in [9] and [5]). By the convergence, Ψ has to be 2 π -periodic in y . Firstly, we prove that H ( r ; Ψ) → r → −∞ , so that the results of subsection 2.3 hold true forΨ. As already observed, N R ( r ) ≥ N ¯ R ( r ) for every r ∈ R , for every R > ¯ R . By the expression of thelogarithmic derivative of H R , see Corollary 2.15 (see also Remark 2.18) we havedd r log H R ( r ) = 2 N R ( r ) ≥ N ¯ R ( r ) = dd r log H ¯ R ( r ) ∀ r. As a consequence, taking into account that H R (0) = 1 for every R , for every r < H R (0) H R ( r ) ≥ H ¯ R (0) H ¯ R ( r ) ⇔ H ¯ R ( r ) ≥ H R ( r ) ∀ R > ¯ R. Passing to the limit as R → + ∞ , by the C ( R ) convergence of ( u R , v R ) to (Ψ + , Ψ − ) it follows that H ¯ R ( r ) ≥ H ( r ; Ψ), which gives H ( r ; Ψ) → r → −∞ in light of our assumption on ( u, v ).Using again the expression of the logarithmic derivative of H R and H ( · ; Ψ), we deducelog H R ( r ) H R ( r ) = 2 Z r r N R ( s )d s and log H ( r ; Ψ) H ( r ; Ψ) = 2 Z r r N ( s ; Ψ)d s, where r < r . The left hand side of the first identity converges to the left hand side of the secondidentity; recalling that N R ( r ) → N (+ ∞ ) for every r , we deduce Z r r N ( s ; Ψ)d s = lim R → + ∞ Z r r N R ( s )d s = N (+ ∞ )( r − r ) ⇒ r − r Z r r N ( s ; Ψ) d s = N (+ ∞ ) . for every r < r . It is well known that, being N ( · ; Ψ) ∈ L ( R ), the limit as r → r of the left handside converges to N ( r ; Ψ) for almost every r ∈ R . Hence, N ( r ; Ψ) = N (+ ∞ ) for every r ∈ R . We arethen in position to apply Proposition 2.21:lim R → + ∞ N ( R ) = lim R → + ∞ N R (0) = N (0; Ψ) = d ∈ N \ { } , and Ψ( x, y ) = [ C cos( dy ) + C sin( dy )] e dx for some constant C , C ∈ R .As far as case ( ii ) is concerned, for the sake of simplicity we assume a = 0. One can repeat the proofwith minor changes replacing E unb and N unb with E sym and N sym (which have been defined in subsection2.1). The unique nontrivial step consists in proving that in this setting H ( r ; Ψ) → r → −∞ . To thisaim, we note that, as before, H R ( r ) ≤ H ¯ R ( r ) ∀ R > ¯ R, for every r > − ¯ R . In particular, if r > − ¯ R , by Proposition 2.4 and Corollary 2.5 we deduce H R ( r ) ≤ H ¯ R ( r ) = H ( r + ¯ R ) H ( ¯ R ) ≤ e N (1)( r + ¯ R ) e N (1) ¯ R = e N (1) r ∀ R > ¯ R, for every r > − ¯ R . Passing to the limit as R → + ∞ , by C ( R ) convergence we obtain H ( r ; Ψ) ≤ e N (1) r ∀ r ∈ R , which yields H ( r ; Ψ) → r → −∞ . (cid:3) Appendix A.We start with the following version of the parabolic minimum principle, which we used in the proof ofProposition 3.1. Lemma A.1. Let N ≥ , let Ω = ( a, b ) × Ω ′ ⊂ R N be open and connected, let c ∈ L ∞ (Ω) and let w ∈ H (Ω) be such that w t − ∆ w ≥ c ( x ) w in [0 , T ] × Ω w ≥ on { } × Ω w ≥ on (0 , T ) × ( a, b ) × ∂ Ω ′ , and w has ( b − a ) -periodic boundary condition on { a, b } × Ω ′ . Then w ≥ .Proof. Let J ( t ) := R Ω ( w − ) . A direct computation shows that J ′ ( t ) ≤ k c k L ∞ (Ω) J ( t ), where we usedthe boundary conditions. Consequently, J ( t ) ≤ J (0) e k c k L ∞ (Ω) t = 0 ∀ t ∈ [0 , T ]where the last identity follows by the initial condition. (cid:3) Remark A.2. Note that we do not require anything about the sign of c .In sections 3 and 4, we exploited many times the following properties of the trace operators. Theorem A.3. For a < b real numbers, let C ( a,b ) = ( a, b ) × S k be a bounded cylinder. The trace operator T r : u ∈ H ( C ( a,b ) ) u | Σ a ∪ Σ b ∈ L (Σ a ∪ Σ b ) is compact.Proof. Let ( u n ) ⊂ H ( C ( a,b ) ) be such that u n ⇀ 0. We show that u n | Σ a ∪ Σ b → L (Σ a ∪ Σ b ). For thesake of simplicity we consider the case a = 0 and b = 1. Let w ( x, y ) := x ( x − ∂ ν w = 1on Σ ∪ Σ . Let F ( x, y ) = ∇ w ( x, y ) = (2 x − , 0) and g ( x, y ) = ∆ w ( x, y ) = 2 . By the divergence theorem2 Z C (0 , u n = Z C (0 , (div F ) u n = − Z C (0 , u n F · ∇ u n + Z Σ a ∪ Σ b u n , XPONENTIAL ENTIRE SOLUTIONS 37 so that Z Σ a ∪ Σ b u n ≤ k u n k L ( C (0 , + 2 k u n k L ( C (0 , k∇ u n k L ( C (0 , → n → ∞ , by the compactness of the Sobolev embedding H ( C (0 , ֒ → L ( C (0 , ). (cid:3) Corollary A.4. For a < b real numbers, let C ( a,b ) = ( a, b ) × S k be a bounded cylinder. The local traceoperator T Σ b : u ∈ H ( C ( a,b ) ) u | Σ b ∈ L (Σ b ) is compact.Proof. It is an easy consequence of Theorem A.3 and of the fact that the linear operator L f : ϕ ∈ L (Σ a ∪ Σ b ) f ϕ ∈ L (Σ a ∪ Σ b ) is continuous for every f ∈ L ∞ (Σ a ∪ Σ b ). As T Σ b = L χ Σ b ◦ T r C ( a,b ) ,where χ Σ b is the characteristic function of Σ b , T Σ b is compact. (cid:3) Acknowledgments: the authors thank Prof. Alberto Farina, Prof. Susanna Terracini and Prof. Gi-anmaria Verzini for many valuable discussions related to this problem. The first author is partiallysupported by PRIN 2009 grant ”Critical Point Theory and Perturbative Methods for Nonlinear Differ-ential Equations”. References [1] H. Berestycki, T.-C. Lin, J. Wei, and C. Zhao. On Phase-Separation Models: Asymptotics and Qualitative Properties. Arch. Ration. Mech. Anal. , 208(1):163–200, 2013.[2] H. Berestycki, S. Terracini, K. Wang, and J. Wei. On entire solutions of an elliptic system modeling phase-separation. to appear on Adv. Math. , 2013.[3] L. A. Caffarelli and F.-H. Lin. 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