Positive periodic solutions for a system of anisotropic parabolic equations

J. Math. Anal. Appl. 367 (2010) 204–228

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Positive periodic solutions for a system of anisotropic parabolic equations Genni Fragnelli 1 Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, via Roma 56, 53100, Italy

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 8 November 2009 Available online 7 January 2010 Submitted by V. Radulescu

We consider a system of two degenerate parabolic equations with nonlocal terms and Dirichlet boundary conditions. More precisely, the degeneracy in each equation of the system is of the type r (x)-Laplacian where r (x) is a function depending on x ∈ Ω , where Ω is a bounded smooth domain of Rn . The system models the diffusion and the interaction between two different biological species sharing the same territory Ω . The paper provides conditions on the parameters of the problem that guarantee the coexistence of a T -periodic non-negative solution (u , v ) with both non-trivial u , v. © 2009 Elsevier Inc. All rights reserved.

Keywords: Degenerate parabolic equations Periodic solutions Topological degree theory

1. Introduction In this paper we consider a system of { p (x), q(x)}-Laplacian parabolic equations with nonlocal terms and Dirichlet boundary conditions of the form

⎧     ut − div |∇ u | p (x)−2 ∇ u = f x, t , Φ1 (u ), Φ2 ( v ), a u , in Q T , ⎪ ⎪ ⎪     ⎨ v t − div |∇ v |q(x)−2 ∇ v = g x, t , Φ3 (u ), Φ4 ( v ), b v , in Q T , ⎪ u (x, t ) = v (x, t ) = 0, for (x, t ) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎩ u (·, 0) = u (·, T ) and v (·, 0) = v (·, T ).

(1)

Here Ω is an open bounded domain of Rn with smooth boundary ∂Ω , Q T := Ω × (0, T ), T > 0, p (x), q(x) ∈ C 1,α (Ω), p (x), q(x) > 2 for any x ∈ Ω , f , g and Φi (i = 1, 2, 3, 4) are continuous functions, a, b ∈ L ∞ ( Q T ) are extended to Ω × R by T -periodicity and we look for continuous and periodic weak solutions. Degenerate parabolic equations like those in system (1) model nonlinear diffusive phenomena and have been the subject of extensive study (see, for example, [8]). The interest in studying the existence of non-trivial non-negative periodic solutions for reaction–diffusion equations of this type relies in the consideration that the periodic behavior for certain biological quantities are the most desiderable one, see also [2,4,5,19,20,25,31]. In particular, system (1) is a possible model for the evolution of two biological species living in a common territory Ω (see, for example, [1] for a single elliptic equation with p (x) = p, [6] and [7] for a single elliptic equation involving nonlocal effects when p (x) = 2). Here u (x, t ) and v (x, t ) denote the respective densities of population at time t located at x ∈ Ω . Therefore, the term on the right-hand side of each equation in (1) denotes the actual increasing rate of the population at (x, t ) ∈ Q T . The nonlinear terms of the form { p (x), q(x)}-Laplacian, div(|∇ u | p (x)−2 ∇ u ) and div(|∇ v |q(x)−2 ∇ v ), replace the usual terms u and  v in order to describe the fact that the two species diffuse according to the position. In particular, they imply that the speed of the diffusion is rather slow. Observe that when the function p or q is a constant, we have the usual p- or q-Laplacian term. This seems to be a more realistic model when the region Ω ⊂ Rn presents non-homogeneous physical characteristic that affects locally

1

E-mail address: [email protected]. Research supported by the MIUR National Project Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari.

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.12.039

©

2009 Elsevier Inc. All rights reserved.

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

205

the growth in the state variable of u and v, giving rise to non-standard growth conditions depending on the position x ∈ Ω . For related results in the elliptic case we refer to [11,22] and the references therein. On the other hand such an anisotropic system is useful to describe the evolution of electrorheological fluids, which are special viscous liquids characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied [24]. This property is usefully employed in technological applications, such as actuators, clutches, shock absorbers, and rehabilitation equipment (for example, see [28]). The literature about systems of degenerate parabolic equations is poor and, to our best knowledge, there are a few papers dealing with the periodic problem (for systems of non-degenerate parabolic equations see, for example, [3,15,18]). On the other hand, there are a lot of results for the following single equation

  ⎧ p −2 ∇ u = f (x, t , u ), ⎨ ut − div |∇ u | u (x, t ) = 0, for (x, t ) ∈ ∂Ω × (0, T ), ⎩ u (x, 0) = u (x, T ), for x ∈ Ω.

(2)

In particular, if p > 2, f (x, t , u ) = f (x, t ) and it is periodic in t, we refer to [29]; if f (x, t , u ) = f (x, t )u α , where f is strictly positive and α > p − 1 or α < p − 1, we refer to [30] or [31]. If p = 2 and





f (x, t , u ) = f x, t , Φ(u ), u , a u ,

(3) belongs to L ∞ ( Q T )

we refer to [3]. Here Φ(u ) is a coercive and positive nonlocal term, i.e. Φ(u )  C u  L 1 (Ω) , a = a(x, t ) and f satisfies suitable growth and positivity conditions. The authors pointed out that monotonicity methods fail due to the presence of the nonlocal term and thus it is impossible to employ such methods to prove the existence of periodic solutions of





ut − u = f x, t , Φ(u ), u , a u , see [3, Theorem 0]. For this reason, they adopt a topological approach based on a priori bounds and the continuation property of the Leray–Schauder topological degree. Recently these results and the relative techniques have been extended in [33] to the case p > 2 and





f (x, t , u ) = a(x, t ) − Φ(u ) u .

.

Here Φ : L 2 (Ω)+ = {u ∈ L 2 (Ω): u  0, a.e. in Ω} → R+ is bounded, continuous, Φ(0) = 0 and is coercive. Moreover a(t , x) ∈ C ( Q T ) is periodic in T , may change sign, but



T x ∈ Ω:

a(x, t ) dt > 0 = ∅.

(4)

0

As stated before, the literature about systems of degenerate parabolic equations is scarce. Recently in [17] system (1) is considered when the degenerate nonlinear terms div(|∇ u | p (x)−2 ∇ u ) and div(|∇ v |q(x)−2 ∇ v ) are replaced by um and  v m , with m > 1. In particular in [17] the authors prove the existence of a continuous T -periodic weak solution (u , v ) for the following delayed nonlocal system

 ⎧ ∂u ⎪ m 2 2 ⎪ − u = a(x, t ) − K 1 (ξ, t )u (ξ, t − τ1 ) dξ + K 2 (ξ, t ) v (ξ, t − τ2 ) dξ u , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪  ⎨ ∂v −  v m = b(x, t ) + K 3 (ξ, t )u 2 (ξ, t − τ3 ) dξ − K 4 (ξ, t ) v 2 (ξ, t − τ4 ) dξ v , ⎪ ∂t ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪ ⎪ u (·, t )| = v (·, t )| = 0 , for t ∈ [ 0 , T ], ∂Ω ∂Ω ⎪ ⎩ u (·, 0) = u (·, T ) and v (·, 0) = v (·, T ),

(5)

assuming that the growth rates a, b and the functions K i belong to L ∞ ( Q T ) and satisfy conditions involving μ1 . Here μ1 denotes the least eigenvalue of − in Ω with Dirichlet boundary conditions. The delayed densities u , v at time t − τi , that appear in the nonlocal terms, take into account the time needed to an individual to become adult and, thus, to interact and compete. More precisely, a first existence result is obtained assuming the coercivity in L 2 (Ω) of the nonlocal terms corresponding to K 1 and K 4 and using suitable bounds on K 2 and K 3 . In particular, competitive systems (those with K 2  0 and K 3  0 a.e. in Q T ) and cooperative systems (those with K 2  0 and K 3  0 a.e. in Q T ) fall into this result. The existence theorem is then extended for competitive systems without assuming the coercivity in L 2 (Ω) of the nonlocal terms corresponding to K 1 and K 4 . An existence result is finally obtained without that coercivity and any sign condition on K 2 , K 3 by adding the technical assumption m > 3 in the degenerate terms um and  v m .

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G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

Systems of degenerate parabolic equations are treated also in [13], where only the coercive case is considered. In this paper a system similar to (5) is considered, but again um ,  v m , u 2 and v 2 (in the nonlocal terms) are replaced by div(|∇ u | p (x)−2 ∇ u ), div(|∇ v |q(x)−2 ∇ v ), u and v, respectively. In particular the authors prove that if

η := min

1

e 21 (x)a(x, t ) dx dt ;

T

1



e 21 (x)b(x, t ) dx dt

T

QT

− μ1 > 0

(6)

QT

and if there exist positive constants ki , ki such that

K i (x, t )  ki

for a.e. (x, t ) ∈ Q T and i = 1, 4,

K i (x, t )  ki

for a.e. (x, t ) ∈ Q T and i = 2, 3,

and

0 < k2 k3 < k1 k4 ,

(7)

then the system has a non-trivial non-negative periodic solution (u , v ). But in this case the pairs (0, v ) and (u , 0) are not excluded. As usual, in (6), μ1 is the first eigenvalue of the problem



− z = μ z, x ∈ Ω, z = 0, x ∈ ∂Ω

and e 1 is the associated positive eigenfunction such that e 1  L 2 (Ω) = 1. In the present paper, as in [17], we prove that the problem (1) has a T -periodic non-negative solution (u , v ) with both non-trivial u , v. Thus, in this case, the pairs (0, v ) and (u , 0) are excluded. Moreover, it is clear that the right-hand side of (1) generalizes the right-hand side of (5), since also in this case we can consider delays, as done in [12,14] or in [16]. The coexistence of a non-negative solution (u , v ) for (1) is proved assuming that, in general, there exist non-negative constants K j , j = 2, 3, and K i , i = 1, 2, 3, 4, such that for all u , v ∈ C ( Q T )

0  Φ1 (u )  K 1 u 2L 2 (Ω) ,

0  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  K 2  v 2L 2 (Ω) ,

− K 3 u 2L 2 (Ω)  Φ3 (u )  K 3 u 2L 2 (Ω) .

In particular, we say that the system is cooperative if K 2 = K 3 = 0 and it is competitive if K 2 = K 3 = 0, and, as in [17], we can face both the cooperative and the competitive cases. More precisely, a first existence result, Theorem 3.1, is obtained by assuming the coercivity in L 2 (Ω) of the nonlocal terms Φ1 (u ) and Φ4 ( v ) and using suitable bounds on K 2 , K 3 , K 1 and K 4 ; as already noted, for a single equation a coercivity assumption was considered in both [3] and [33]. In particular, competitive systems and cooperative systems fall into this existence result. Two further existence results are proved in Theorems 3.2 and 3.3 for competitive systems without assuming the coercivity in L 2 (Ω) of the nonlocal terms Φ1 (u ) and Φ4 ( v ). An other existence result is finally obtained in Theorem 3.4 without coercivity and any sign condition on Φ2 ( v ) and Φ3 (u ), by adding the technical assumption minΩ p (x), minΩ q(x) > 4 in the degenerate terms div(|∇ u | p (x)−2 ∇ u ) and div(|∇ v |q(x)−2 ∇ v ). The degeneracy of the equations is faced by perturbing the degenerate part of the equations (see for instance [5,13,21]). Specifically, we replace the two terms div(|∇ u | p (x)−2 ∇ u ) and div(|∇ v |q(x)−2 ∇ v ) by div((|∇ u |2 + ) p (x)−2

p (x)−2 2

∇ u )

and div((|∇ v |2 + ) 2 ∇ v ), with > 0 small enough. We thus obtain a family of regularized non-degenerate problems and we solve them by means of the topological degree theory. We finally get a solution of (1) passing to the limit as → 0. This procedure is carried out once for all in Section 2 and, in particular, in Theorem 2.1 where we show that the explicit knowledge of a priori bounds for the L 2 -norms of the solutions of the regularized problems with other information, is sufficient to prove the existence of a non-negative solution (u , v ) of (1) with non-trivial u , v (i.e. periodic coexistence). We devote Section 3 to finding the required a priori bounds and to state the definitive existence results. 2. Preliminary results and the coexistence theorem Throughout the paper we will make the following assumptions: Hypotheses 2.1. 1. The functions p , q are of class C 1,α (Ω), with p + := maxΩ p (x)  p − := minΩ p (x) > 2 and q+ := maxΩ q(x)  q− := minΩ q(x) > 2. 2. a and b belong to L ∞ ( Q T ).

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

207

3. Φi : C ( Q T ) → C ([0, T ]), i = 1, 2, 3, 4. 4. f , g : Q T × C ([0, T ])2 × L ∞ ( Q T ) → L 1 ( Q T ) are continuous functions, and for any given a, b ∈ L ∞ ( Q T ) there exist M i (x, t ), N i (x, t ) ∈ L ∞ ( Q T ), i = 1, 2, such that

M 1 (x, t ) − k1 α + k2 β  f (x, t , α , β, a)  M 2 (x, t ) − k1 α + k2 β and

N 1 (x, t ) − k4 β + k3 α  g (x, t , α , β, b)  N 2 (x, t ) − k4 β + k3 α for some non-negative constants ki , ki , i = 1, 2, 3, 4. Example of f , g and Φi are obtained considering the functions f (x, t , Φ1 (u ), Φ2 ( v ), a) := a −Φ1 (u )+Φ2 ( v ), g (x, t , Φ3 (u ),  Φ4 ( v ), b) := b − Φ4 ( v ) + Φ3 (u ) and Φi (u ) := Ω K i (x, t )u 2 dx, where a, b, K i ∈ L ∞ ( Q T ), i = 1, 2, 3, 4, and ki  K i (x, t )  ki , i = 1, 2, 3, 4, for some non-negative constants ki , ki , i = 1, 2, 3, 4. 2.1. Preliminary results In order to discuss system (1) and for the reader’s convenience, we recall some definitions and results concerning the spaces L p (x) and W 1, p (x) , which are called generalized Lebesgue–Sobolev spaces (for details see, for example, [9] and [10]). Here p is a continuous function on Ω such that p (x) > 1 for any x ∈ Ω and Ω is a bounded domain of Rn . The space L p (x) is defined as

 



L p (x) (Ω) := u  u is a measurable real-valued function with

   u (x) p (x) dx < ∞ .

Ω

Introducing in L p (x) (Ω) the following norm



   u (x)  p (x)  |u | L p(x) (Ω) := inf λ > 0:  dx  1 , λ  Ω

one has that ( L p (x) (Ω), | · | p (x) ) is a Banach space called a generalized Lebesgue space. An important role in manipulating the generalized Lebesgue space is played by the modular of the space L p (x) (Ω), which is the mapping ρ p (x) : L p (x) (Ω) → R defined by



ρ p(x) (u ) :=

  u (x) p (x) dx.

Ω

Proposition 2.1. The following relations hold true: 1. |u | L p(x) (Ω) < 1 ⇔ 2. |u | L p(x) (Ω) > 1 ⇔ 3. |u | L p(x) (Ω) = 1 ⇔

ρ p(x) (u ) < 1. ρ p(x) (u ) > 1. ρ p(x) (u ) = 1. p+ p− 4. |u | L p(x) (Ω) < 1 ⇒ |u | L p(x) (Ω)  ρ p (x) (u )  |u | L p(x) (Ω) . p−

p+

5. |u | L p(x) (Ω) > 1 ⇒ |u | L p(x) (Ω)  ρ p (x) (u )  |u | L p(x) (Ω) . 6. If {uk } ⊂ L p (x) (Ω) then the following statements are equivalent:

(i) (ii) (iii)

lim |uk − u | L p(x) (Ω) = 0,

k→+∞

lim

k→+∞

ρ p(x) (uk − u ) = 0,

uk → u in measure in Ω and

The following proposition holds.

lim

k→+∞

ρ p(x) (uk ) = ρ p(x) (u ).

208

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

Proposition 2.2. 1. The space ( L p (x) (Ω), | · | p (x) ) is a separable, uniformly convex Banach space and its conjugate space is L q(x) (Ω), where p (1x) + 1 = 1. Moreover, for any u ∈ L p (x) (Ω) and v ∈ L q(x) (Ω), we have q(x)

      uv dx  1 + 1 |u | p(x) | v | q(x) . L (Ω) L (Ω)   p− q− Ω

2. If p 1 and p 2 ∈ C (Ω) and 1  p 1 (x)  p 2 (x) for any x ∈ Ω , then L p 2 (x) (Ω) → L p 1 (x) (Ω) and the imbedding is continuous, so there exists a positive constant C p 1 (x), p 2 (x) so that

|u | L p1 (x) (Ω)  C p 1 (x), p 2 (x) |u | L p2 (x) (Ω) .

(8)

Now, define the generalized Sobolev space W 1, p (x) (Ω) as





u  := |u | L p(x) (Ω) + |∇ u | L p(x) (Ω) ,

for any u ∈ W 1, p (x) (Ω).

W 1, p (x) (Ω) := u ∈ L p (x) (Ω): |∇ u | ∈ L p (x) (Ω) , with the norm

1, p (x)

Moreover, we denote by W 0

(Ω) the closure of C 0∞ (Ω) in W 1, p (x) (Ω).

Proposition 2.3. 1, p (x)

1. W 1, p (x) (Ω) and W 0

(Ω) are separable and reflexive Banach spaces. 1, p (x) 2. If q ∈ C (Ω) is such that 1  q(x) < p ∗ (x) for any x ∈ Ω , then the imbedding of W 1, p (x) (Ω) or W 0 (Ω) into L q(x) (Ω) is compact and continuous. Here

∗

p (x) :=

+∞,

np (x) , n− p (x)

p (x)  n, p (x) < n.

3. There exists a constant C P > 0 such that

|u | L p(x) (Ω)  C P |∇ u | L p(x) (Ω) ,

1, p (x)

for any u ∈ W 0

4. |∇ u | L p(x) (Ω) and u  are equivalent norms in

(Ω) (Poincaré inequality).

(9)

1, p (x) W0 (Ω).

2.2. The main result −

We now give the definition of a solution for system (1). To this aim we introduce the space X p := {u ∈ L p (0, T ; 1, p (x)

W0

(Ω)):



QT

|∇ u | p (x) dx dt < +∞} equipped with the norm of L

and if p (x) = p for all x ∈ Ω then X p ≡

p−

1, p (x)

(0, T ; W 0

(Ω)). Obviously, X p is a Banach space

1, p L p (0, T ; W 0 (Ω)).

Definition 1. A pair of functions (u , v ) is said to be a weak solution of (1) if

u ∈ X p ∩ C ( Q T ),

v ∈ Xq ∩ C ( Q T )

and the couple (u , v ) satisfies



  ∂ϕ p (x)−2 + |∇ u | ∇ u · ∇ ϕ − f x, t , Φ1 (u ), Φ2 ( v ), a u ϕ dx dt = 0 −u ∂t

(10)

QT

and



  ∂ϕ q(x)−2 + |∇ v | ∇ v · ∇ ϕ − g x, t , Φ3 (u ), Φ4 ( v ), b v ϕ dx dt = 0, −v ∂t

QT

for any

ϕ ∈ C 1 ( Q T ), ϕ (x, T ) = ϕ (x, 0) for any x ∈ Ω and ϕ (x, t ) = 0 for any (x, t ) ∈ ∂Ω × [0, T ].

(11)

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

209

Here and in the following we assume that the functions t → u (·, t ) and t → v (·, t ) are extended from [0, T ] to R by T -periodicity so that (u , v ) is a solution for all t. Due to the degeneracy of the equation, we consider the following regularized (non-degenerate) problem:

⎧ ∂u   p(x)−2    ⎪ − div |∇ u |2 + 2 ∇ u = f x, t , Φ1 (u ), Φ2 ( v ), a u , ⎪ ⎪ ⎪ ∂t ⎪ ⎨ ∂v   q(x)−2    − div |∇ v |2 + 2 ∇ v = g x, t , Φ3 (u ), Φ4 ( v ), b v , ⎪ ∂ t ⎪ ⎪ u (x, t )| = v (x, t )| = 0, for t ∈ [0, T ], ⎪ ⎪ ∂Ω ∂Ω ⎩ u (·, 0) = u (·, T ) and v (·, 0) = v (·, T ), p + −2

(12)

q+ −2

where (x, t ) ∈ Q T and ∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }). A solution (u , v ) of (1) will be then obtained as the limit, for the following definition:

→ 0, of the solutions (u , v ) of (12), for which we give

Definition 2. A couple of functions (u , v ) is said to be a generalized solution of (12) if

u ∈ X p ∩ C ( Q T ),

v ∈ Xq ∩ C ( Q T )

and (u , v ) satisfies

  p(x)−2   ∂ϕ  + |∇ u |2 + 2 ∇ u · ∇ ϕ − f x, t , Φ1 (u ), Φ2 ( v ), a u ϕ dx dt = 0 −u

∂t

(13)

QT

and

  p(x)−2   ∂ϕ  + |∇ v |2 + 2 ∇ v · ∇ ϕ − g x, t , Φ3 (u ), Φ4 ( v ), b v ϕ dx dt = 0, −v

∂t

(14)

QT

for any

ϕ ∈ C 1 ( Q T ), ϕ (x, T ) = ϕ (x, 0) for any x ∈ Ω and ϕ (x, t ) = 0 for any (x, t ) ∈ ∂Ω × [0, T ].

Note that for any function r ∈ C 1,α (Ω) the space {ϕ ∈ C 1 ( Q T ): ϕ (x, T ) = ϕ (x, 0), x ∈ Ω} is dense in {ϕ ∈ X r ∩ C ( Q T ): ϕ (x, T ) = ϕ (x, 0), x ∈ Ω}, so that in the definition of solutions above u and v can be taken as test functions. To deal with the existence of T -periodic solutions (u , v ) of system (12), with u , v  0 in Q T , for any ∈ p + −2

q+ −2

(0, min{( 12 ) p− −2 , ( 12 ) q− −2 }), we introduce the map G : [0, 1] × L ∞ ( Q T ) × L ∞ ( Q T ) → L ∞ ( Q T ) × L ∞ ( Q T ) defined as follows:

(σ , f , g ) → (u , v ) = G (σ , f , g ) if and only if (u , v ) solves the following problem:

⎧   p(x2)−2  ⎪ u ,t − div σ |∇ u |2 +

∇ u = f , for a.e. (x, t ) ∈ Q T , ⎪ ⎪ ⎪ ⎨   q(x)−2  v ,t − div σ |∇ v |2 + 2 ∇ v = g , for a.e. (x, t ) ∈ Q T , ⎪ ⎪ ⎪ u (x, t )|∂Ω = v (x, t )|∂Ω = 0, for t ∈ [0, T ], ⎪ ⎩ u (·, 0) = u (·, T ) and v (·, 0) = v (·, T ). 1, p (x)

This map G is well defined since the operator A : W 0

1, p (x)

(Ω) → W 0

(15)

(Ω), ξ → − div[(|ξ |2 + )

coercive, hemicontinuous and strictly monotone, and so the equation u ,t − div((σ |∇ u | + ) periodic solution by [32, Theorem 32.D]. Consider now 2









p (x)−2 2

p (x)−2 2

ξ ] is bounded,

∇ u ) = f has a unique

f (λ, ) := f x, t , Φ1 (λ), Φ2 (), a λ, g (λ, ) := g x, t , Φ3 (λ), Φ4 (), b

,

belong to L ∞ ( Q T ). Clearly, if the positive functions u , v

where λ and  ∈ L ∞ ( Q T ) are such that (u , v ) = G (1, f (u , v ), g (u , v )), then (u , v ) is also a solution of (12) (with u and v  0) in Q T . Hence, the existence of a non-negative solution of (12) is equivalent to the existence of a fixed point (λ, ) for the map (λ, ) → G (1, f (λ, ), g (λ, )) with λ and   0.

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G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

Let T (σ , λ, ) := G (σ , f (λ, ), g (λ, )). Lemma 2.1. Let (λ, ) ∈ L ∞ ( Q T ) × L ∞ ( Q T ) and let

p + −2

q+ −2

∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }). Then (u , v ) = T (σ , λ, ) is a compact

continuous map from [0, 1] × L ∞ ( Q T ) × L ∞ ( Q T ) into L ∞ ( Q T ) × L ∞ ( Q T ). Moreover u , v ∈ C ( Q T ).

Proof. Let ( p , q) ∈ B, where B is a bounded set of L ∞ ( Q T )× L ∞ ( Q T ), then { f ( p , q): ( p , q) ∈ B } is a bounded set in L ∞ ( Q T ). The T -periodicity condition gives

  u (0, ·)

L 2 (Ω)

 c  f L∞ ( Q T ) ,

for some positive constant c. Since the initial data u (0, ·) is bounded in L 2 (Ω), it results

u  L 2 ( Q T )  c  f  L ∞ ( Q T ) . We may apply classical local estimates (see [23]) to bound u in L ∞ (Ω × [ T2 , 3T ]) and thus, by T -periodicity, for all t in 2 terms of  f  L ∞ ( Q T ) . Classical results (see, for example, [23, Theorem 1.1]) also show that u ∈ C α ,α /2 ( Q T ) for some α > 0. Moreover, due to its uniform continuity in Q T , u can be uniquely extended by continuity to Q T . The same arguments apply to v , thus the pair (u , v ) satisfies (15) and u , v ∈ C ( Q T ). Finally, by the Ascoli–Arzelà theorem, the compactness and continuity of T follow from the bound in the L ∞ -norm of u and v and the fact that they are equi-Hölder continuous in Q T [23, Theorem 1.1]. 2 Analogously to [13] and [17] and using [27] one can prove the next result. Proposition 2.4. If the non-trivial pair (u , v ) ∈ L ∞ ( Q T ) × L ∞ ( Q T ) solves

      (u , v ) = G σ , ρ f u + , v + + (1 − σ ), ρ g u + , v + + (1 − σ ) ,

(16)

for some σ ∈ [0, 1] and ρ ∈ [0, 1], then

u (x, t )  0 and

v (x, t )  0 for any (x, t ) ∈ Q T .

Moreover, if u = 0 or v = 0 then u > 0 or v > 0 in Q T , respectively. Observe that if ρ = 0, then (u , v ) = G (1, 0, 0) if and only if (u , v ) = (0, 0). The following result guarantees that the solutions (u , v ) of (12) we are going to find are not bifurcating from the p + −2

trivial solution (0, 0) as

1











e 21 (x) f x, t , Φ1 (u ), Φ2 ( v ), a dx dt > μ1

T

q+ −2

ranges in (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }). To this aim assume that for all u , v ∈ C ( Q T ) and

QT

1



e 21 (x) g x, t , Φ3 (u ), Φ4 ( v ), b dx dt > μ1 .

T

(17)

QT

Moreover, set r0 := min{ζ1 , ζ2 }, where

 

ζ1 := and

QT

e 21 (x) f (x, t , Φ1 (u ), Φ2 ( v ), a) dx dt − μ1 T

γ p+ (18)

K 1, p

  ζ2 :=

QT

e 21 (x) g (x, t , Φ3 (u ), Φ4 ( v ), b) dx dt − μ1 T

K 1,q := 2

p + −2 2

q + −2 2

 

1

γ p− 1 −

γq

+ +

2



(19)

.

K 1,q

Here

K 1, p := 2

γq+



2

1

1

+

−

C p + , p (x) max |∇ e 1 |2 |Ω| p+ max |Ω| γ p , |Ω| γ p p− Ω 2 q−



2



1

+

1

−

C q+ ,q(x) max |∇ e 1 |2 |Ω| q+ max |Ω| γq , |Ω| γq Ω

 

T  M 2  L ∞ ( Q T ) + k2 K 2 + 1

 

T  N 2  L ∞ ( Q T ) + k3 K 3 + 1 +

 γ1− p

 γ1− q

,

,

x) x) γ p− := minΩ { p(px()− }, γ p+ := maxΩ { p (px()− } and C p + , p (x) is the embedding constant of L p (Ω) into L p (x) (Ω). The constants 2 2 + − γq , γq , C q+ ,q(x) are defined analogously. We can prove the following result.

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

211

Proposition 2.5. Assume that 1. for all u , v ∈ C ( Q T )

0  Φ1 (u ),

0  Φ4 ( v )

and

Φ2 ( v )  K 2  v 2L 2 (Ω) ,

Φ3 (u )  K 3 u 2L 2 (Ω)

for some non-negative constants K i , i = 1, 2, 3, 4; 2. (17) is satisfied. If the non-trivial pair (u , v ) solves (u , v ) = G (σ , f (u + , v + ) + (1 − σ ), g (u + , v + ) + (1 − σ )) for some σ ∈ [0, 1], then





max u  L ∞ ( Q T ) ,  v  L ∞ ( Q T )  r0 . Moreover deg((u , v ) − T (1, u + , v + ), B r , 0) = 0 for all r ∈ (0, r0 ). Proof. By contradiction, assume that for some σ ∈ [0, 1] and r  min{1, r0 } there exists a pair (u , v ) = (0, 0) such that + + + (u , v ) = G (σ , f (u +

, v ) + (1 − σ ), g (u , v ) + (1 − σ )) with u  L ∞ ( Q T )  r and  v  L ∞ ( Q T )  r. Assume that u = 0 ∞ and take φ ∈ C 0 (Ω). Since by Proposition 2.4 we have that u > 0 in Q T , we can multiply the equation

  p(x)−2    ∂ u

− div |∇ u |2 + 2 ∇ u = f x, t , Φ1 (u ), Φ2 ( v ), a u + (1 − σ ) ∂t φ2

by u , integrate over Q T and pass to the limit in the Steklov averages (u )h ∈ H 1 ( Q T −δ ), δ, h > 0, in the standard way (see

[23, p. 85]), in order to obtain

− QT

φ2 u



div |∇ u |2 +

 p(x2)−2

 ∇ u dx dt =



   φ2 φ 2 (x) f x, t , Φ1 (u ), Φ2 ( v ), a + (1 − σ ) dx dt , u

QT

(20)

by the T -periodicity of u . Moreover a straightforward computation shows that



− QT

φ2 u



div |∇ u |2 +

 p(x2)−2

 ∇ u dx dt =





|∇ u |2 +

 p(x2)−2

|∇φ|2 dx dt

QT



−



|∇ u |2 +

 p(x2)−2

  2  φ  dx dt . u 

u 2 ∇



QT

Thus





|∇ u |2 +

 p(x2)−2

|∇φ|2 dx dt −

QT

QT

As in [13], the term



 2   φ (x) f x, t , Φ1 (u ), Φ2 ( v ), a dx dt  0.



 QT

|∇ u |2 +

QT

2

p + −2 2

(|∇ u |2 + )

 p(x2)−2

2

|∇ u | p (x)−2 |∇φ|2 dx dt + 2 

1

γ p−

2

+

 × max

p−

p + −2 2





p − −2 2



p − −2 2

|∇φ|2 dx dt QT





γ p+ −1

1

γ p+

T

γ p+

,

  ∇ u (x, t ) p (x) dx dt



γ p− −1 

1

γ p−

T

γ p−

QT

|∇φ|2 dx dt . QT

p + −2 2

  C p + , p (x) |∇φ|2  L p+ /2 (Ω)

  ∇ u (x, t ) p (x) dx dt

QT

+2

|∇φ|2 dx dt can be estimated in the following way:

|∇φ|2 dx dt

QT

p + −2 2

p (x)−2 2

(21)

212

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

+ + + Multiplying the first equation of (u , v ) = G (σ , f (u +

, v ) + (1 − σ ), g (u , v ) + (1 − σ )) by u , integrating over Q T and using the T -periodicity, one has



|∇ u | p (x) dx dt  QT



|∇ u |2 +

 p(x2)−2

|∇ u |2 dx dt

QT









f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt + (1 − σ )

 QT

u dx dt . QT

By Hypotheses 2.1 and the Hölder inequality, it results



|∇ u | p (x) dx dt 

T M 2 (x, t )u 2 dx dt + k2 K 2

QT

QT

   v (t )22

L (Ω)

  u (t )22

L (Ω)

dt + (1 − σ )u  L 1 ( Q T )

0

 M 2 L ∞ ( Q T ) u 2L 2 ( Q ) T

+ k2 K 2  v 2L 2 ( Q ) u 2L ∞ ( Q T ) + (1 − σ )u L 1 ( Q T ) T   2   M 2 L ∞ ( Q T ) + k2 K 2  v  L ∞ ( Q T ) |Ω| T u 2L ∞ ( Q T ) + |Ω| T u L ∞ ( Q T )     M 2 L ∞ ( Q T ) + k2 K 2 + 1 |Ω| T r . 

Thus, using (21),





|∇ u |2 +

QT

2

p + −2 2

 p(x2)−2

|∇φ|2 dx dt

|∇ u |

p (x)−2

QT

2

p + −2



1

2

−

γp

+2

p + −2 2



+

p − −2 2

2

2

|∇φ| dx dt + 2

p + −2 2



p − −2 2

|∇φ|2 dx dt QT









1

1

+

−

C p + , p (x) |∇φ|2  L p+ /2 (Ω) max |Ω| γ p , |Ω| γ p p−

 

T  M 2  L ∞ ( Q T ) + k2 K 2 + 1

 γ1− p

1

+

r γp



|∇φ|2 dx dt . QT

Taking φ(x) = e 1 (x), one has







e 21 (x) f x, t , Φ1 (u ), Φ2 ( v ), a dx dt QT

2

p + −2 2

· |Ω|



2 p+

p − −2 2

μ1 T + 2 

1

+

p + −2 2



1

γ p− 1

−

max |Ω| γ p , |Ω| γ p

+

2 p−

C p + , p (x) max |∇ e 1 |2 Ω

 

T  M 2  L ∞ ( Q T ) + k2 K 2 + 1

 γ1− p

1

+

r γp ,

(22)

since ∇ e 1 ∈ C 1 (Ω) (by regularity theory) and e 1  L 2 (Ω) = 1. Moreover, by the choice of , one has







e 21 (x) f x, t , Φ1 (u ), Φ2 ( v ), a dx dt − μ1 T QT

2

p + −2 2



1 −

γp

+

2 p−





1

+

1

−

C p + , p (x) max |∇ e 1 |2 · max |Ω| γ p , |Ω| γ p Ω

 

T  M 2  L ∞ ( Q T ) + k2 K 2 + 1

Thus

  r0 

QT

e 21 (x) f (x, t , Φ1 (u ), Φ2 ( v ), a) dx dt − μ1 T K 1, p

γ p+

 r,

which is, in any case, a contradiction with the choice of r. The same argument applies if v = 0.

 γ1− p

1

+

r γp .

(23)

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

213

Let us now fix any r ∈ (0, r0 ). We just proved that

      (u , v ) = G σ , f u + , v + + (1 − σ ), g u + , v + + (1 − σ ) , ∀(u , v ) ∈ ∂ B r and ∀σ ∈ [0, 1]. Hence the topological degree of (u , v ) − G (σ , f (u + , v + ) + (1 − σ ), g (u + , v + ) + (1 − σ )) is well defined in B r for all σ ∈ [0, 1]. From the homotopy invariance of the Leray–Schauder degree, we have























deg (u , v ) − T 1, u + , v + , B r , 0 = deg (u , v ) − G 0, f u + , v + + 1, g u + , v + + 1 , B r , 0



and the last degree is zero since the equation

      ( u , v ) = G 0, f u + , v + + 1 , g u + , v + + 1 admits neither trivial nor non-trivial solutions in B r .

2

The next proposition is crucial to prove the main theorem. Proposition 2.6. Let K > 0 and assume that u is a positive periodic continuous function such that

  p(x)−2  ∂u − div |∇ u |2 + 2 ∇ u  K u , ∂t

for a.e. (x, t ) ∈ Q T

and u (·, t )|∂Ω = 0, for t ∈ [0, T ]. Then there exists R > 0 and independent of such that

u  L ∞ ( Q T )  R . Proof. Multiplying

  p(x)−2  ∂ u

− div |∇ u |2 + 2 ∇ u  K u

∂t q+1

by u

, where q  0, and integrating over Ω , we have

d

1

q + 2 dt

 2 u (t )q+ q +2 L





+ (q + 1) (Ω)

|∇ u |2 +

 p(x2)−2

 q+2 q |∇ u |2 u dx  K u (t ) Lq+2 (Ω) .

+

−

Assuming, without loss of generality, that min{|∇ u | p , |∇ u | p } = |∇ u | p follows



p

p

(24)

Ω

q+p

  qp +1  p  p ∇ u

L





q

u |∇ u | p dx 

= (Ω) Ω

−

and writing p in place of p − for simplicity, it



q

u |∇ u | p (x) dx  Ω



|∇ u |2 +

 p(x2)−2

q

|∇ u |2 u dx.

Ω

Thus, by (24),

1

d

q + 2 dt

 2 u (t )q+ q +2 L

 + ( q + 1 ) (Ω)

p

p q+p

  qp +1  p  p ∇ u

L (Ω)

 q+2  K u (t ) Lq+2 (Ω) .

(25)

Setting

qk :=

p k +1 − p p−1

,

αk :=

p (qk + 2) qk + p

and

qk p

w k := u

+1

(k = 1, 2, . . .),

we obtain by (25)

d dt Since

 + ( q + 2 ) k k (Ω)

  w k (t )ααk L

1 qk + p

p

  ∇ w k (t ) pp

L (Ω)

 α  K (qk + 2) w k (t ) L αkk (Ω) .

(26)

αk < p, by the interpolation and the Sobolev inequalities, it results       θ 1−θk θ 1−θk  w k (t ) α   w k (t )Lk1 (Ω)  w k (t )Lq (Ω)  C  w k (t ) Lk1 (Ω) ∇ w k (t ) L p (Ω) . L k (Ω) q−α

Here θk = α (q−k1) , q > p is fixed (say q = p ∗ if p < n) and C is a positive constant. Using the fact that  w k (t ) L 1 (Ω) = k α  w k−1 (t ) L αk−k−11 (Ω) , then

214

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

 p   w k (t ) 1α−θk L

θk  pα     w k−1 (t ) α k−1 1−θk ∇ w k (t ) pp  C k (Ω) L (Ω) L k−1 (Ω) θ

k p αk−1 1−θ

 C xk−1

k

  ∇ w k (t ) pp

L (Ω)

,

where xk−1 := supt  w k−1 (t ) L αk−1 (Ω) . Thus by (26)

d dt

  −( q + 2 ) k k (Ω)

  w k (t )ααk L



 = K −C

p

1 qk + p

p

1 qk + p



 1−θp

θ

k p αk−1 θ − k 1

k C  w k (t ) L αk (Ω) xk−1

 α + K (qk + 2) w k (t ) L αkk (Ω)

θk   p −α p α    w k (t ) 1α−θk k x k−1 θk −1 (qk + 2) w k (t )ααk . k −1 L k (Ω) L k (Ω)

(27)

α

By the positiveness of w k it results that the function t →  w k (t ) L αkk (Ω) is increasing, hence (27) implies

   w k (t ) where

 L αk (Ω)



θ

k p αk−1 1−θ

K Mk

ηk

k

xk−1

(28)

,

1 p k ηk := p−α1k−θ (1−θk ) and M k := C ( qk + p ) . By definition of xk and (28) we get  ηk K k xk  xk− 1

Mk

with

k := p αk−1 p−αkθ(k1−θk ) .

qk−1 + p p

If xk−1  1 the thesis follows immediately, since xk−1 = supt u (t )q

k−1 +2

. Now, assume that xk−1 > 1. Since

αk < p, one

q− p

has that k < p and there exists k0 > 0 such that ηk  p1θ for all k  k0 , where θ := p (q−1) . Then, there exists a positive constant A such that p ηk  ηk  k +1 p K 2  x  p (k+1) p ηk x k  Ap θ x k

k −1

p−1

C

k −1

for all k  k0 . Thus

log xk  log A +

 log A

k+1

θ k−k0 −1 

log p + p log xk−1

pi +

k +1 log p 

i =0

 log A

1 − pk−k0 1− p

+

θ

ipk+1−i + pk−k0 log xk0

i =k0 +2

log p

θ

pk+1−(k−k0 )(k0 +2)



(k + 1)(k + 2) 2

−

(k0 + 1)(k0 + 2)



2

+ pk−k0 log xk0 .

(29)

It follows

xk  A

1− pk−k0 1− p

p

pk+1−(k−k0 )(k0 +2)

θ

( (k+1)(2 k+2) −

(k0 +1)(k0 +2) 2

k−k ) p 0

xk

0

.

qk + p p

Since xk = supt u (t )q +2 , we obtain k









supu (t ) L ∞ (Ω)  lim supu (t )q +2 k t k→∞

p  p 1− pk−k0  lim sup A qk + p 1− p p qk + p

pk+1−(k−k0 )(k0 +2)

θ

( (k+1)(2 k+2) −

(k0 +1)(k0 +2) 2

k−k ) p 0

k→∞

where R is a positive constant independent of

xk

0

p qk + p



= R,

as claimed. 2

The next result is our main tool to obtain coexistence results for (1). Roughly speaking, it says that the existence of non-negative non-trivial solutions u , v of (1) follows as soon as we can check the positivity of a value depending on the uniform a priori bounds of the L 2 -norm of any solution (u , v ) of the regularized problem (12) for any > 0 small enough. The next section will be devoted to the search of such explicit bounds when something more is known on the structure of the equations. As a matter of notation, throughout the paper we set









B R = (λ, ) ∈ L ∞ ( Q T ) × L ∞ ( Q T ): max λ L ∞ ( Q T ) ,  L ∞ ( Q T ) < R ,

R > 0.

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

215

Theorem 2.1. Assume that 1. for all u , v ∈ C ( Q T )

0  Φ1 (u )  K 1 u 2L 2 (Ω) ,

0  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  K 2  v 2L 2 (Ω) ,

− K 3 u 2L 2 (Ω)  Φ3 (u )  K 3 u 2L 2 (Ω)

for some non-negative constants K j , j = 2, 3, and K i , i = 1, 2, 3, 4; 2. there exist two constants C 1 , C 2 > 0 such that

u 2L 2 ( Q

T)

 C 1 and  v 2L 2 ( Q

T)

 C2

(30)

for all solutions (u , v ) of

     ( u , v ) = G 1, ρ f u + , v + , ρ g u + , v + , p + −2 p − −2

for all ∈ (0, min{( 12 )

, ( 12 )

q+ −2 q− −2

(31)

}) and ρ ∈ (0, 1].

Then there is a constant R > 0 such that

u  L ∞ ( Q T ) ,  v  L ∞ ( Q T ) < R p + −2

q+ −2

for all solution pairs (u , v ) of (31), for all ∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }) and ρ ∈ (0, 1]. In particular, one has that









deg (u , v ) − T 1, u + , v + , B R , 0 = 1. Moreover, if

η(C 1 , C 2 ) := min

1

e 21 (x) M 1 (x, t ) dx dt − μ1 −

T

k2 K 2 C 2 T

−

k1 K 1 C 1 T

QT

1



e 21 (x) N 1 (x, t ) dx dt − μ1 −

T

k3 K 3 C 1 T

−

k4 K 4 C 2

,



T

> 0,

(32)

QT

then problem (1) has a T -periodic non-negative solution (u , v ) with both non-trivial u , v. Proof. Assume that u = 0, thus u > 0 and v  0 in Q T by Proposition 2.4. Multiplying by u the first equation of (12), where f (x, t , Φ1 (u ), Φ2 ( v ), a) is replaced by ρ f (x, t , Φ1 (u ), Φ2 ( v ), a), integrating over Ω and using the Steklov averages (u )h , we obtain

1 d 2 dt



(u )h2 dx +

log

Ω (|∇(u )h |

2



Ω

+ )

p (x)−2 2

|∇(u )h |2 dx

2

Ω (u )h dx

      ρ  M 2 L ∞ ( Q T ) − k1 Φ1 (u )h + k2 Φ2 ( v )h .

(33)

Since t → u (t ) L 2 (Ω) is continuous in [0, T ], there exist t 1 and t 2 in [0, T ] such that





u 2 (x, t 1 ) dx = min

t ∈[0, T ]

Ω



u 2 (x, t ) dx Ω



u 2 (x, t 2 ) dx = max

and

u 2 (x, t ) dx.

t ∈[0, T ]

Ω

Ω

Integrating (33) between t 1 and t 2 and passing to the limit as h → 0, we obtain

1



2



Ω

and then

Ω



u 2 (x, t ) dx  C min

max

u 2 (x, t ) dx,

t ∈[0, T ]

t ∈[0, T ]

Ω

u 2 (x, t ) dx  T  M 2  L ∞ ( Q T ) + k2 K 2 C 2 ,

t ∈[0, T ]

t ∈[0, T ]





u 2 (x, t ) dx − log min

log max

Ω

(34)

216

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

and ρ . We claim that there is a constant γ > 0, independent of and ρ , such that

where C is independent of



u 2 (x, t ) dx  γ .

max

t ∈[0, T ]

Ω

Otherwise, inequality (34) would imply that the solutions u are unbounded in L 2 ( Q T ) as q+ −2

p + −2

ranges in (0, min{( 12 ) p− −2 ,

( 12 ) q− −2 }) and ρ in (0, 1], against our assumption (30). Of course, an analogous inequality holds for v . Now, we have

  2   p(x)−2   ∂ u

− div |∇ u |2 + 2 ∇ u   M 2 L ∞ ( Q T ) + k2 K 2 max  v (t )L 2 (Ω) u

t ∈[0, T ] ∂t     M 2  L ∞ ( Q T ) + k2 K 2 γ u .

Thus, by Proposition 2.6, there exists a positive constant R 1 independent of

(35)

ρ and such that

u  L ∞ ( Q T )  R 1 . Analogously,  v  L ∞ ( Q T )  R 2 for some constant R 2 > 0. Therefore it is enough to choose R > max{ R 1 , R 2 }. The homotopy invariance property of the Leray–Schauder degree implies that























deg (u , v ) − T 1, u + , v + , B R , 0 = deg (u , v ) − G 1, ρ f u + , v + , ρ g u + , v + , B R , 0 ,

ρ ∈ [0, 1]. If we take ρ = 0, using the fact that G at ρ = 0 is the zero map, it results       deg (u , v ) − T 1, u + , v + , B R , 0 = deg (u , v ), B R , 0 = 1.

for any

Now, let us assume that dent of , such that



η(C 1 , C 2 ) > 0 and observe that this ensures that (17) holds and that there are R > r > 0, indepen-













deg (u , v ) − G 1, f u + , v + , g u + , v + , B R \ B r , 0 = 1 p + −2

for any

q+ −2

∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }), by Proposition 2.5 and the excision property of the topological degree. p + −2

q+ −2

∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }). There is σ0 = σ0 ( ) ∈ (0, 1) such that still         deg (u , v ) − G σ , f u + , v + + (1 − σ ), g u + , v + + (1 − σ ) , B R \ B r , 0 = 1 for all σ ∈ [σ0 , 1]

Let us fix any

by the continuity of Leray–Schauder degree. This implies that the set of solution triples (σ , u , v ) ∈ [0, 1] × ( B R \ B r ) such that

      (u , v ) = G σ , f u + , v + + (1 − σ ), g u + , v + + (1 − σ )

(36)

contains a continuum S with the property that

  S ∩ {σ } × ( B R \ B r ) = ∅ for all σ ∈ [σ0 , 1].

Now, all the pairs (u , v ) such that (1, u , v ) ∈ S are T -periodic solutions of (12) with (u , v ) = (0, 0) and, hence, satisfy (30). Since the L 2 -norm is continuous with respect to the L ∞ -norm and S is a continuum, for every  > 0 there is σ ∈ [σ0 , 1) such that

u 2L 2 ( Q

T)

 C 1 +  and  v 2L 2 ( Q

T)

 C2 + 

for all (u , v ) with (σ , u , v ) ∈ S and σ ∈ [σ , 1]. Observe that, if (σ , u , v ) ∈ S for σ < 1, then u and v are positive solutions of (36). Moreover, if  is sufficiently small, then we still have η(C 1 + , C 2 + ) > 0. Now we can prove that, if  is sufficiently small, then

 γ +  γ +  T η (C 1 +  , C 2 +  ) p T η (C 1 +  , C 2 +  ) q u L ∞ ( Q T ) ,  v L ∞ ( Q T )  λ := min 1, , k1 K 1 |Ω| T + C k4 K 4 |Ω| T + C for all u , v such that (σ , u , v ) ∈ S and

K 2, p := 2

p + −2 2



γ p+ −1

·T

γ p+

1

γ p− 

+

2



(37)

σ ∈ [σ , 1), where C := max{ K 2, p , K 2,q }, 2



C p + , p (x) max |∇ e 1 |2 |Ω| p+ max  M 2  L ∞ ( Q T ) |Ω| T + k2 K 2 (C 2 + ) + |Ω| T p− Ω

,  M 2 L ∞ ( Q T ) |Ω| T + k2 K 2 (C 2 + ) + |Ω| T

 γ1− p

γ p− −1

T

γ p−



 γ1+ p

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

and

K 2,q := 2

q + −2 2



γq+ −1 γq+

·T

1

γq−

+

2



217



2

C q+ ,q(x) max |∇ e 1 |2 |Ω| q+ max  N 2  L ∞ ( Q T ) |Ω| T + k3 K 3 (C 1 + ) + |Ω| T q− Ω



,  N 2 L ∞ ( Q T ) |Ω| T + k3 K 3 (C 1 + ) + |Ω| T

γq− −1

 γ1−

T

q

γq−



 γ1+ q

.

Indeed, let (u , v ) be a solution of (36). Arguing by contradiction, assume that u  L ∞ ( Q T ) < λ and proceeding as in the proof of Proposition 2.5 (recall that u > 0 since (u , v ) solves (36) with σ < 1) we obtain the inequality

e 21 (x) f



1



γ p+

x, t , Φ1 (u ), Φ2 ( v ), a dx dt − μ1 T < C λ ,

QT

since



|∇ u |

p (x)

T 2

dx dt 

QT

M 2 (x, t )u dx dt + k2 K 2 QT

   v (t )22

L (Ω)

  u (t )22

dt + (1 − σ )u  L 1 ( Q T )

L (Ω)

0

  M 2 L ∞ ( Q T ) u 2L 2 ( Q ) + k2 K 2  v 2L 2 ( Q ) u 2L ∞ ( Q T ) + (1 − σ )u L 1 ( Q T ) T T     M 2 L ∞ ( Q T ) |Ω| T + k2 K 2 (C 2 + ) u 2L ∞ ( Q T ) + |Ω| T u L ∞ ( Q T )   <  M 2 L ∞ ( Q T ) |Ω| T + k2 K 2 (C 2 + ) + |Ω| T λ . Thus, by assumption,



e 21 (x) M 1 (x, t ) dx dt



k1 e 21 Φ1 (u ) − k2 e 21 Φ2 ( v )

− μ1 T <

QT



1

γ p+

dx dt + C λ

QT 1

γ+

 (k1 K 1 |Ω| T + C )λ p + k2 K 2 (C 2 + ).

η implies that 1  γ p+  2 T η (C 1 +  , C 2 +  )  e 1 (x) M 1 (x, t ) dx dt − μ1 T − k2 K 2 (C 2 + ) < k1 K 1 |Ω| T + C λ ,

The definition of

QT

which is a contradiction with the definition of λ . The same argument shows that  v  L ∞ ( Q T )  λ . Now, if we let σ → 1 and  → 0, then we obtain that (12) has at least a solution (u , v ) such that u  L ∞ ( Q T ) ,  v  L ∞ ( Q T )  λ0 , since S is a continuum and λ → λ0 as  → 0. Finally, we show that a solution (u , v ) of (1) with non-trivial u , v  0 is obtained as a limit of (u , v ) as → 0 since λ0 is independent of . Observe that since u , v are Hölder continuous in Q T , bounded in C ( Q T ) uniformly in > 0 and the structure conp + −2

ditions of [26] are satisfied for each equation of system (1), whenever Theorem 1.2] to conclude that the inequality

q+ −2

∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }), we can apply [26,

   β u (x1 , t 1 ) − u (x2 , t 2 )  Γ |x1 − x2 |β + |t 1 − t 2 | 2 holds for any (x1 , t 1 ), (x2 , t 2 ) ∈ Q T , where the constants Γ, β ∈ (0, 1) are independent of u  L ∞ ( Q T ) . The same inequality holds for v . Therefore, by the Ascoli–Arzelà theorem, a subsequence of (u , v ) converges uniformly in Q T to a pair (u , v ) satisfying

λ0   u  L ∞ ( Q T ) ,  v  L ∞ ( Q T )  R . Moreover, from (35), we have

  p(x)−2  ∂ u

− div |∇ u |2 + 2 ∇ u  C u , (38) ∂t where C is a positive constant independent of . Multiplying (38) by u , integrating over Q T and passing to the limit in the Steklov averages (u )h , one has





|∇ u | p (x) dx dt  QT

QT



|∇ u |2 +

 p(x2)−2

|∇ u |2 dx dt  C

u 2 dx dt . QT

(39)

218

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

Thus, if |∇ u | p (x) > 1, then

T

p− |∇ u | p (x) dt



|∇ u |

p (x)

QT

0

u 2 dx dt  M ,

dx dt  C QT

T

by the boundedness of u in L ∞ ( Q T ). Otherwise, if |∇ u | p (x)  1, then that there exists a positive constant M independent of such that

T

0

p−

|∇ u | p (x) dt  T . In any case one can conclude

p−

|∇ u | p (x) dt  M .

(40)

0

An analogous estimate holds for v . By estimate (40), u and v are uniformly bounded in X p and in X q , respectively. Thus, up to subsequence if necessary, (u , v ) weakly converges to (u , v ) ∈ X p × X q . We finally claim that the pair (u , v ) satisfies the identities



  ∂ϕ p (x)−2 + |∇ u | ∇ u · ∇ ϕ − f x, t , Φ1 (u ), Φ2 ( v ), a u ϕ dx dt = 0 −u ∂t

QT

and

 −v

  ∂ϕ + |∇ v |q(x)−2 ∇ v · ∇ ϕ − g x, t , Φ3 (u ), Φ4 ( v ), b v ϕ dx dt = 0, ∂t

QT

for any ϕ ∈ C 1 ( Q T ), ϕ (x, T ) = ϕ (x, 0) for any x ∈ Ω and ϕ (x, t ) = 0 for any (x, t ) ∈ ∂Ω × [0, T ], that is, (u , v ) is a generalized solution of (1). Indeed, by (39), there exists a positive constant C such that





|∇ u | p (x) dx dt  C and

|∇ u |2 dx dt  C .

QT

QT

Thus, by the Hölder inequality with q(x) :=



2( p (x)−1) , p (x)

  p(x)  p(x)−2  |∇ u |2 + 2 ∇ u  p(x)−1 dx dt  M p

one has



QT

|∇ u |

p (x)

dx dt +

QT

|∇ u |

p (x) p (x)−1

dx dt

 M,

QT p (x)

for some positive constants M and M p . This implies that there exists h ∈ ( L p(x)−1 ( Q T ))n such that (|∇ u |2 + ) weakly converges to h in ( L



∂ϕ −u dx dt + ∂t

QT

p (x) p (x)−1



p (x)−2 2

∇ u

( Q T )) as → 0. Now it is easy to prove that n



h · ∇ ϕ dx dt = QT





f x, t , Φ1 (u ), Φ2 ( v ), a u ϕ dx dt ,

(41)

QT

for any ϕ ∈ C 1 ( Q T ), ϕ (x, T ) = ϕ (x, 0) for any x ∈ Ω and ϕ (x, t ) = 0 for any (x, t ) ∈ ∂Ω × [0, T ] (and, by density, for any T -periodic ϕ ∈ X p ). It remains to prove that for every ϕ ∈ C 1 ( Q T )





|∇ u | p (x)−2 ∇ u · ∇ ϕ dx dt = QT

h · ∇ ϕ dx dt .

(42)

QT

To this aim consider the matrix function H (Y ) := (|Y |2 + )



H  ( Y ) = | Y |2 +



p (x)−2 2





I + p (x) − 2 |Y |2 +



p (x)−2 2

p (x)−4 2

Y . Then

YYT

is a positive definite matrix and taken v ∈ X p there exists a matrix Y such that









H (∇ u ) − H (∇ v ), ∇ u − ∇ v = H  (Y )(∇ u − ∇ v ), ∇ u − ∇ v  0.

The previous inequality is equivalent to



QT



|∇ u |2 +

 p(x2)−2

 p(x)−2   ∇ u − |∇ v |2 + 2 ∇ v · ∇(u − v )  0,

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

219

for all v ∈ X p . Multiplying by u the first equation of (12), integrating over Q T and using the periodicity of u , one has





|∇ u |2 +

 p(x2)−2



QT

Thus

QT



|∇ u |2 dx dt =



f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt . QT



|∇ u |2 +

 p(x2)−2

∇ u · ∇ v dx dt +



|∇ v |2 +

 p(x2)−2

∇ v · ∇(u − v ) dx dt

QT







f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt .

 QT

→ 0, we have   h · ∇ v dx dt + |∇ v | p (x)−2 ∇ v · ∇(u − v ) dx dt  f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt .

Letting

QT

QT

QT

On the other hand, take u = ϕ in (41) and obtain





QT





f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt .

h · ∇ u dx dt = QT

This implies







h − |∇ v | p (x)−2 ∇ v · ∇(u − v ) dx dt .

0

(43)

QT

Taking v := u − λϕ , with λ > 0 and



0





ϕ ∈ C 1 ( Q T ), we get

 p (x)−2

h − ∇(u − λϕ )

 ∇(u − λϕ ) · ∇ ϕ dx dt .

QT

Letting λ → 0 yields







h − |∇ u | p (x)−2 ∇ u · ∇ ϕ dx dt .

0 QT

If in (43) we take v := u + λϕ , with λ > 0,





ϕ ∈ C 1 ( Q T ) and letting again λ → 0, then



h − |∇ u | p (x)−2 ∇ u · ∇ ϕ dx dt  0.

QT

Thus (42) holds.

2

Remark. 1. Proposition 2.5, with assumption (17), does not guarantee that both components of a non-trivial solution (u , v ) of the regularized problem (12) are positive and, in fact, the proof of such a positivity is one of the main issues we had to handle in the proof of the preceding theorem with the help of the stronger assumption η(C 1 , C 2 ) > 0. However, if we consider the cooperative case, i.e., K 2 = K 3 = 0, then we have

η0 = η(C 1 , C 2 ) = min

1

e 21 (x) M 1 (x, t ) dx dt − μ1 −

T QT

k1 K 1 C 1 1 T

,

e 21 (x) N 1 (x, t ) dx dt − μ1 −

T

k4 K 4 C 2



T

QT

and the condition η0 > 0 implies (17). In particular, the coexistence in the cooperative case follows from a priori bounds on (u , v ).

η0 > 0 and

220

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

2. If the system is cooperative and k1 , k4 = 0, then

η0 = η(C 1 , C 2 ) = min

1



e 21 (x) M 1 (x, t ) dx dt − μ1 ,

T QT

1



e 21 (x) N 1 (x, t ) dx dt − μ1

T QT

and the coexistence follows as before from the condition η0 > 0 and a priori bounds of (u , v ), even if the constants C 1 , C 2 are not explicitly known. 3. The assumption η(C 1 , C 2 ) > 0 is used to show (37) and, therefore, guarantees the non-triviality of both the components of the non-negative T -periodic solution (u , v ) that is given by Theorem 2.1. However, when we proved the lower bounds (37), we used the estimate (23) for the first equation of the system in order to show that u  L ∞ ( Q T ) was not smaller than λ and, implicitly, we used for  v  L ∞ ( Q T ) the analogous estimate that holds for the second equation. On the other hand, if we use the second equation for u  L ∞ ( Q T ) (and the first equation for  v  L ∞ ( Q T ) ) we obtain a different choice for λ and, in particular, for η . But, in this case, the computations are more difficult and the expression of λ is more involved. 3. A priori bounds in L 2 ( Q T ) We apply Theorem 2.1 by looking for explicit a priori bounds in L 2 ( Q T ) for the solutions of the approximating problems (12) in different situations. We consider two main different cases. In the first one, which we call the “coercive case”, we assume that Φ1 (u )  K 1 u 2L 2 (Ω) > 0 and Φ4 ( v )  K 4  v 2L 2 (Ω) > 0. In the second one, the “non-coercive case”, we allow

Φ1 (u )  0 and Φ4 ( v )  0. Moreover, we distinguish also between cooperative and competitive situations by imposing sign conditions on Φ2 ( v ) and Φ3 (u ). 3.1. The coercive case Theorem 3.1. Assume p − , q− > 2 and that 1. there are some positive constants K i , K i , i = 1, 4, and non-negative constants K i , K i , i = 2, 3, such that

k2 K 2 k3 K 3 < k1 K 1 k4 K 4 , K 1 u 2L 2 (Ω)  Φ1 (u )  K 1 u 2L 2 (Ω) ,

K 4  v 2L 2 (Ω)  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  K 2  v 2L 2 (Ω) ,

− K 3 u 2L 2 (Ω)  Φ3 (u )  K 3 u 2L 2 (Ω)

for all u , v ∈ C ( Q T ); 2. the constants ki , i = 1, 4, of Hypotheses 2.1 are positive; 3. condition (32) of Theorem 2.1 is satisfied with

C1 =

T k4 K 4 k1 K 1 k4 K 4 − k2 K 2 k3 K 3

and

C2 =

T k1 K 1 k1 K 1 k4 K 4 − k2 K 2 k3 K 3

 k2 K 2  N 2 L∞ ( Q T )  M 2 L∞ ( Q T ) + k4 K 4

 k3 K 3  M 2 L∞ ( Q T ) .  N 2 L∞ ( Q T ) + k1 K 1

Then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. Proof. We just need to show that u 2L 2 ( Q )  C 1 and  v 2L 2 ( Q )  C 2 for any solution (u , v ) of (31) and then to invoke T T Theorem 2.1. Assume u = 0, thus u > 0 and v  0 in Q T by Proposition 2.4. Integrating Eq. (33) over [0, T ], and passing to the limit as h → 0, by the T -periodicity of u ,

T 0<



 p (x)−2 T  2 |∇ u |2 dx + ) 2 |∇ u |2 dx Ω (|∇ u |   Ω dt  dt

p + −2 2

2 Ω u dx

0

2 Ω u dx

0

T  T



M 2 L∞ ( Q T )

− k1

T Φ1 (u ) dt + k2

0

 T  M 2 L ∞ ( Q T ) − k1 K 1 u 2L 2 ( Q

Φ2 ( v ) dt 0

T)

+ k2 K 2  v 2L 2 ( Q ) . T

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

221

By Poincaré’s inequality we have



u 2 dx 

μ1 Ω

|∇ u |2 dx, Ω

hence p + −2 2



μ1 T  T  M 2 L ∞ ( Q T ) − k1 K 1 u 2L 2 ( Q T ) + k2 K 2  v 2L 2 ( Q T ) .

(44)

The same procedure, when applied to the second equation of (12), leads to p + −2 2



μ1 T  T  N 2 L ∞ ( Q T ) − k4 K 4  v 2L 2 ( Q T ) + k3 K 3 u 2L 2 ( Q T ) .

(45)

Hence from (44) and (45) we have

u 2L 2 ( Q ) T  v 2L 2 ( Q ) T

 

T  M 2 L∞ ( Q T ) k1 K 1 T  N 2 L∞ ( Q T ) k4 K 4

+ +

k2 K 2 k1 K 1 k3 K 3 k4 K 4

 v 2L 2 ( Q ) T u 2L 2 ( Q ) T

− −



p + −2 2

μ1 T

,

k1 K 1



p + −2 2

μ1 T

k4 K 4

.

These two inequalities imply that

 1−

 1−

k2 K 2 k3 K 3 k1 K 1 k4 K 4 k2 K 2 k3 K 3 k1 K 1 k4 K 4

u 2L 2 ( Q  v 2L 2 ( Q p + −2

for any

T

< )

T)

<



T k1 K 1

 M 2 L∞ ( Q T ) +

k2 K 2 k4 K 4

 N 2 L∞ ( Q T ) ,

 T k3 K 3 ∞ ∞  N 2 L ( Q T ) +  M 2 L ( Q T )

k4 K 4

k1 K 1

(46)

q+ −2

∈ (0, min{( 12 ) p− −2 , ( 12 ) q− −2 }) and the claim follows since k2 K 2k3 K 3 < k1 K 1k4 K 4 . 2 

Corollary 3.1. Let p − , q− > 2, Φi (u ) = Ω K i (x, t )u 2 dx, i = 1, 2, 3, 4, f (x, t , Φ1 (u ), Φ2 ( v ), a) = a(x, t ) − Φ1 (u ) + Φ2 ( v ) and g (x, t , Φ3 (u ), Φ4 ( v ), b) = b(x, t ) − Φ4 ( v ) + Φ3 (u ). Assume that a, b, K i ∈ L ∞ ( Q T ), i = 1, 2, 3, 4, and that there exist positive constants K i , K i , i = 1, 4, and non-negative constants K j , K j , j = 2, 3, such that K 2 K 3 < K 1 K 4 , K i  K i (x, t )  K i , i = 1, 4, and − K j  K j (x, t )  K j , j = 2, 3, for a.e. (x, t ) ∈ Q T . If the hypotheses 2 and 3 of Theorem 3.1 are satisfied, then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. Therefore, as immediate consequences of the previous theorem we obtain the following corollaries for the cooperative and the competitive cases. Corollary 3.2. Assume that the hypotheses of Theorem 3.1 are satisfied. If the system is cooperative, i.e. K 2 , K 3 = 0, then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. Corollary 3.3. Assume that the hypotheses of Theorem 3.1 are satisfied. If the system is competitive, i.e. K 2 , K 3 = 0, then problem (1) has a non-negative T -periodic solution (u , v ), with both non-trivial u , v. 3.2. The non-coercive case: weak competition and p − , q− > 2 Theorem 3.2. Assume p − , q− > 2 and that 1. for all u , v ∈ C ( Q T )

0  Φ1 (u )  K 1 u 2L 2 (Ω) ,

0  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  0,

− K 3 u 2L 2 (Ω)  Φ3 (u )  0

for some non-negative constants K j , j = 2, 3, and K i , i = 1, 4;

222

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

2. condition (32) of Theorem 2.1 is satisfied with



p+

p+



q+

q+

C 1 = T max C 2, p C P  M 2  L ∞ ( Q T )



2 p + −2

 2   p− p− , C 2 , p C P  M 2  L ∞ ( Q T ) p − −2

and

C 2 = T max C 2,q C P  N 2  L ∞ ( Q T )



2 q + −2

 q− q−  2  , C 2,q C P  N 2 L ∞ ( Q T ) q− −2 ,

where C 2, p , C 2,q and C P are the constants appearing in (8) and (9), respectively. Then problem (1) has a T -periodic non-negative solution (u , v ) with both non-trivial u , v. Proof. Multiplying the first equation of (31) by u , integrating in Q T and passing to the limit in the Steklov averages (u )h , we obtain by the T -periodicity of u





2

|∇ u | +

 p(x2)−2





2



f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt

|∇ u | dx dt = ρ

QT

QT

  M 2 L ∞ ( Q T ) u 2L 2 (Ω) .

(47)

By Proposition 2.2(2) and Poincaré’s inequality

u L 2 (Ω)  C 2, p |u | L p(x) (Ω)  C 2, p C P |∇ u | L p(x) (Ω)

 1+  1−  p p p (x) p (x) .  C 2, p C P max |∇ u | dx , |∇ u | dx Ω

(48)

Ω

Assuming that



|∇ u |

max

p (x)

dx



1 p+

|∇ u |

,

Ω

p+

T

dx

1 p−





T )

p + −2 2

T

p+ 2

T

p (x)

dx

1 p+

,

Ω

and by (47) and (48), we obtain

  + u (t ) p2 L

0 p + −2 2

|∇ u |

=

Ω

by Hölder’s inequality with r =

u  L 2 ( Q

p (x)

( C 2, p C P )

p+

dt  T (Ω)



p + −2 2

( C 2, p C P ) p

+

|∇ u | p (x) dx dt QT



|∇ u |2 +

 p(x2)−2

|∇ u |2 dx dt

QT

T

p + −2 2

( C 2, p C P )

p+

 M 2 L ∞ ( Q T ) u 2L 2 ( Q ) . T

Thus p + −2

u  L 2 ( Q

T)

T

p + −2 2

+

( C 2, p C P ) p  M 2  L ∞ ( Q T ) .

If



|∇ u |

max

p (x)

Ω

dx





1 p+

|∇ u |

,

p (x)

dx

Ω

1 p−



 =

|∇ u |

p (x)

dx

1 p−

,

Ω

then p − −2

u  L 2 ( Q

T)

T

p − −2 2

−

( C 2, p C P ) p  M 2  L ∞ ( Q T )

and the estimate for u 2L 2 ( Q ) readily follows. The same arguments apply to v . T An immediate consequence of the previous theorem is the following corollary.

2

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

223



Corollary 3.4. Let p − , q− > 2, Φi (u ) = Ω K i (x, t )u 2 dx, i = 1, 2, 3, 4, f (x, t , Φ1 (u ), Φ2 ( v ), a) = a(x, t ) − Φ1 (u ) + Φ2 ( v ) and g (x, t , Φ3 (u ), Φ4 ( v ), b) = b(x, t ) − Φ4 ( v ) + Φ3 (u ). Assume that a, b, K i ∈ L ∞ ( Q T ), i = 1, 2, 3, 4, and that there exist non-negative constants K j , j = 2, 3, K i , i = 1, 4, such that − K j  K j (x, t )  0, j = 2, 3, and 0  K i (x, t )  K i , i = 1, 4, for a.e. (x, t ) ∈ Q T . If the hypothesis 2 of Theorem 3.2 is satisfied, then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. 3.3. The non-coercive case: strong competition and p − , q− > 2 Theorem 3.3. Assume p − , q− > 2 and that 1. for all u , v ∈ C ( Q T )

0  Φ1 (u )  K 1 u 2L 2 (Ω) ,

0  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  − K 2  v 2L 2 (Ω) < 0,

− K 3 u 2L 2 (Ω)  Φ3 (u )  − K 3 u 2L 2 (Ω) < 0

for some positive constants K i , K i , i = 2, 3, and non-negative constants K i , i = 1, 4; 2. the constants ki , i = 2, 3, of Hypotheses 2.1 are positive; 3. condition (32) of Theorem 2.1 is satisfied with

C 1 = T max

C 2 = T max

 

p+

p+

q+

q+

C 2, p C P  M 2  L ∞ ( Q T ) C 2,q C P  N 2  L ∞ ( Q T )





2 p + −2

  p− p−  2  N 2 L∞ ( Q T ) , C 2 , p C P  M 2  L ∞ ( Q T ) p − −2 , , k3 K 3

2

q + −2



q−

q−

, C 2,q C P  N 2 L ∞ ( Q T )



2

q − −2

,

 M 2 L∞ ( Q T ) k2 K 2

 .

Then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. Proof. If (u , v ) is a solution of (31) with u ≡ 0, or alternatively v ≡ 0, one can argue as in the proof of Theorem 3.2 to obtain that

 v L2 ( Q T ) 

√



q+

q+



p+

p+

T max C 2,q C P  N 2  L ∞ ( Q T )



1 q + −2

 1   q− q− , C 2,q C P  N 2 L ∞ ( Q T ) q− −2 ,

or, alternatively

u  L 2 ( Q T ) 

√

T max C 2, p C P  M 2  L ∞ ( Q T )



1 p + −2

 1   p− p− , C 2 , p C P  M 2  L ∞ ( Q T ) p − −2 .

If u = 0, then u > 0 and v  0 in Q T by Proposition 2.4. Moreover u ∈ C ( Q T ) and, hence, there exists t 1 ∈ [0, T ] such that





u 2 (x, t 1 ) dx = min

u 2 (x, t ) dx.

t ∈[0, T ]

Ω

Ω

Multiplying the first equation of (31) by u , integrating over Ω and using the Steklov averages (u )h , we obtain

1 d



(u )h2 dx 

2 dt Ω





d

(u )h2 dx +

dt Ω



|∇ u |2 +

  p(x2)−2  ∇(u )h 2 dx

Ω

 2    M 2 L ∞ ( Q T ) − k2 K 2 ( v )h L 2 (Ω) 

(u )h2 dx. Ω

Hence, we have

d dt



 t  exp 2

( v )h2 (ξ, s) dξ

k2 K 2 t1







−  M 2 L ∞ ( Q T ) ds

Ω

 (u )h2 (x, t ) dx

 0,

for t  t 1 ,

Ω

which implies that

 t  exp 2

( v )h2 (ξ, s) dξ

k2 K 2 t1



Ω



(u )h2 (x, t ) dx 

−  M 2 L ∞ ( Q T ) ds Ω

(u )h2 (x, t 1 ) dx Ω

224

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

for t  t 1 . Passing to the limit as h → 0 and taking t = t 1 + T ,





t 1 + T 2

t1

Ω

u 2 (x, t 1 + T ) dx 

v (ξ, t ) dξ ds − 2T  M 2  L ∞ ( Q T )

exp 2k2 K 2

Ω

u 2 (x, t 1 ) dx Ω



u 2 (x, t 1 + T ) dx.

 Ω

Therefore we have that t 1 + T

2

v 2 (x, t ) dx dt 

v (x, t ) dx dt = t1

QT

Ω

T  M 2 L∞ ( Q T ) k2 K 2

by the T -periodicity of v . Finally, if v = 0, we can prove that u 2L 2 ( Q )  T  N 2  L ∞ ( Q T ) /k3 K 3 in a similar way. T

2

The following corollary is immediate.



Corollary 3.5. Let p − , q− > 2, Φi (u ) = Ω K i (x, t )u 2 dx, i = 1, 2, 3, 4, f (x, t , Φ1 (u ), Φ2 ( v ), a) = a(x, t ) − Φ1 (u ) + Φ2 ( v ) and g (x, t , Φ3 (u ), Φ4 ( v ), b) = b(x, t ) − Φ4 ( v ) + Φ3 (u ). Assume that a, b, K i ∈ L ∞ ( Q T ), i = 1, 2, 3, 4, and that there exist positive constants K j , K j , j = 2, 3, and non-negative constants K i , i = 1, 4, such that − K j  K j (x, t )  − K j , j = 2, 3, and 0  K i (x, t )  K i , i = 1, 4, for a.e. (x, t ) ∈ Q T . If the hypotheses 2 and 3 of Theorem 3.3 are satisfied, then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. 3.4. The non-coercive case: p − , q− > 4 If p − , q− > 4, we are able to find explicit bounds (although complicated) without any assumption on the sign of the functionals Φ2 , Φ3 , as shown in the next result. Theorem 3.4. Assume p − , q− > 4 and that 1. for all u , v ∈ C ( Q T )

0  Φ1 (u )  K 1 u 2L 2 (Ω) ,

0  Φ4 ( v )  K 4  v 2L 2 (Ω)

and

− K 2  v 2L 2 (Ω)  Φ2 ( v )  K 2  v 2L 2 (Ω) ,

− K 3 u 2L 2 (Ω)  Φ3 (u )  K 3 u 2L 2 (Ω)

for some non-negative constants K j , j = 2, 3, and K i , i = 1, 2, 3, 4; 2. condition (32) of Theorem 2.1 is satisfied with

C1 = T

1 2

 max

  

C2 = T

1 2

 max



( p − −2)(q− −2)

( p − − 2)(q− − 2) − − − − α p− ,q− + β pp− ,qq−−2p −2q p − q− − 2p − − 2q− ( p + −2)(q+ −2)

( p + − 2)(q+ − 2) + + + + α p+ ,q+ + β pp+ ,qq+−2p −2q p + q+ − 2p + − 2q+ ( p + −2)(q− −2)

( p + − 2)(q− − 2) + − + − α p+ ,q− + β pp+ ,qq−−2p −2q p + q− − 2p + − 2q− ( p − −2)(q+ −2)

( p − − 2)(q+ − 2) − + − + α p− ,q+ + β pp− ,qq+−2p −2q p − q+ − 2p − − 2q+ ( p − −2)(q− −2)

( p − − 2)(q− − 2) − − − − αq− , p− + βqp− ,qp−−2p −2q p − q− − 2p − − 2q− ( p + −2)(q+ −2)

( p + − 2)(q+ − 2) + + + + αq+ , p+ + βqp+ ,qp+−2p −2q + + + + p q − 2p − 2q

1/2 , 1/2 , 1/2 , 1/2  , 1/2 , 1/2 ,

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228

 

( p + −2)(q− −2)

( p + − 2)(q− − 2) + − + − αq− , p+ + βqp− ,qp+−2p −2q p + q− − 2p + − 2q− ( p − −2)(q+ −2)

( p − − 2)(q+ − 2) − + − + αq+ , p− + βqp+ ,qp−−2p −2q − + − + p q − 2p − 2q

225

1/2 , 1/2  (49)

.

Here



α p− ,q− := T 2K 2p  M 2 2L ∞ ( Q T ) 4− p −

β p − ,q− := 2 p− −2 T



4− p −    2  2 + 2 p− −2 T 2( K p k2 K 2 )2 2K q2  N 2 2L ∞ ( Q T ) q− −2 p− −2 ,

2 p − −2

( p − −4)(q− −2)+2(q− −4) ( p − −2)(q− −2)



αq− , p− := T 2K q2  N 2 2L ∞ ( Q T )



2 q − −2





2( K p k2 K 2 )2 2( K q k3 K 3 )2

+2

4−q− q − −2





2 q − −2



2 p − −2

,



T 2( K q k3 K 3 )2 2K 2p  M 2 2L ∞ ( Q T )



2 p − −2



2 q − −2

and 4−q−

(q− −4)( p − −2)+2( p − −4) ( p − −2)(q− −2)

βq− , p − := 2 q− −2 T





2( K q k3 K 3 )2 2( K p k2 K 2 )2



2 p − −2



2 q − −2

.

Analogously for α p + ,q+ , α p − ,q+ , α p + ,q− , β p + ,q+ , β p − ,q+ , β p + ,q− , αq+ , p + , αq− , p + , αq+ , p − , βq+ , p + , βq− , p + and βq+ , p − . Then problem (1) has a non-negative T -periodic solution (u , v ) with both non-trivial u , v. Proof. Let (u , v ) be a solution of (31). By Poincaré’s inequality

 |u | L p(x) (Ω)  C P |∇ u | L p(x) (Ω)  C P max

1+  1−  p p |∇ u | p (x) dx , |∇ u | p (x) dx ,

Ω

where C P is the constant appearing in (9). p+

p+

Thus either |u | L p(x) (Ω)  C P



min



Ω

Ω

p−

p−

|∇ u | p (x) dx or |u | L p(x) (Ω)  C P

 p+ p− |u | L p(x) (Ω) , |u | L p(x) (Ω)



 Kp

|∇ u |

p (x)

 Ω



dx  K p

Ω p+

(50)



|∇ u | p (x) dx. In any case

|∇ u |2 +

 p(x2)−2

|∇ u |2 dx,

(51)

Ω

p−

where K p := max{C P , C P }. Multiplying the first equation of (31) by u , integrating in Q T and passing to the limit in the Steklov averages (u )h , we obtain by the T -periodicity of u





|∇ u |2 +

 p(x2)−2





|∇ u |2 dx dt = ρ



f x, t , Φ1 (u ), Φ2 ( v ), a u 2 dx dt

QT

QT

T 



 M 2 L ∞ ( Q T ) + k2 K 2  v 2L 2 (Ω)

 

0

u 2 dx dt .

(52)

Ω

From (51) and (52) and Hölder’s inequality it follows that

 T min

  + u (t ) p2

L (Ω)

T dt ,

0

  − u (t ) p2

L (Ω)

dt

0

 T  KP



2  M 2 L ∞ ( Q T ) + k2 K 2  v 2L 2 (Ω) dt

0

 12  T

  u (t )42

L (Ω)

 12 dt

(53)

,

0 p+

p−

where K p := K p max{C 2, p (x) , C 2, p (x) } and C 2, p (x) is the constant appearing in (8). Assume, without loss of generality, that

u  L 2 (Ω)  1. Then, by (53),

T

  − u (t ) p2

L (Ω)

0

 T dt  K P 0



2  M 2 L ∞ ( Q T ) + k2 K 2  v 2L 2 (Ω) dt

 12  T

  u (t )42

L (Ω)

0

 12 dt

.

226

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228 p− 4

On the other hand, by Hölder’s inequality with r =

T

  u (t )42

L (Ω)

dt  T

p − −4 p−

 T



  − u (t ) p2

L (Ω)

0

> 1, we obtain 4 p−

dt

(54)

.

0

Thus (53) and (54) imply

T

  u (t )42

L (Ω)

dt  T

p − −4 p − −2

4 p − −2

 T

Kp

0



 M 2 L ∞ ( Q T ) + k2 K 2  v 2L 2 (Ω)

2



2 p − −2

dt

0

T

p − −4 p − −2

4 p − −2



Kp



T 2T  M 2 2L ∞ ( Q T )

+ 2k22 K 22

 v 4L 2 (Ω) dt

2 p − −2

0

T



2K 2p  M 2 2L ∞ ( Q T )



2

p − −2

+T

p − −4 p − −2



2K 2p k22 K 22



 T

2



p − −2

 v 4L 2 (Ω) dt

2 p − −2

.

0 p+ 4

On the other hand, if u  L 2 (Ω) < 1, proceeding as before and applying the Hölder inequality with r =

T

  u (t )42

L (Ω)

dt  T



2K 2p  M 2 2L ∞ ( Q T )



2 p + −2

+T

p + −4 p + −2



2K 2p k22 K 22



2 p + −2

 T

  v 4L 2 (Ω) dt

0

> 1, we obtain

2 p + −2

.

0

Thus one can conclude that

T



  u (t )42

L (Ω)

dt  max T



2K 2p  M 2 2L ∞ ( Q T )



2 p + −2

p + −4 p + −2

+T



2K 2p k22 K 22



2 p + −2

 T

  v 4L 2 (Ω) dt

0

2 p + −2

,

0

T



2K 2p  M 2 2L ∞ ( Q T )



2

p − −2

p − −4 p − −2

+T



2K 2p k22 K 22



 T

2



p − −2

 v 4L 2 (Ω) dt

2 p − −2

.

0

In an analogous way we can show that

T



   v (t )42

L (Ω)





dt  max T 2K q2  N 2 2L ∞ ( Q T )

2

q + −2

q + −4 q + −2

+T



 2

2K q2 k23 K 3

2

 T

q + −2

 u 4L 2 (Ω) dt

0



2K q2  N 2 2L ∞ ( Q T )



2 q − −2

+T

q − −4 q − −2



2K q2 k23 K 23



2 q − −2

 T

 u 4L 2 (Ω) dt

0 q+

q−

q+

q−

where K q := max{C P , C P } · max{C 2,q(x) , C 2,q(x) }. Without loss of generality assume that



2K 2p  M 2 2L ∞ ( Q T )



2 p − −2

p − −4 p − −2

+T



2K 2p k22 K 22



2 p − −2

 T

  v 4L 2 (Ω) dt

2 p − −2

0

 = max T



2K 2p  M 2 2L ∞ ( Q T )



2

p + −2

+T

p + −4 p + −2



2K 2p k22 K 22



2

 T

p + −2

  v 4L 2 (Ω) dt

2 p + −2

,

0

T



2K 2p  M 2 2L ∞ ( Q T )



2

p − −2

+T

p − −4 p − −2



2K 2p k22 K 22



2

 T

p − −2

  v 4L 2 (Ω) dt

0

and

,

0

T

T

2 q + −2

2 p − −2



2 q − −2

,

G. Fragnelli / J. Math. Anal. Appl. 367 (2010) 204–228



T 2K q2  N 2 2L ∞ ( Q T )





2

q − −2

+T

q − −4 q − −2



 2

2K q2 k23 K 3

2

 T



q − −2

u 4L 2 (Ω) dt 0



= max T 2K q2  N 2 2L ∞ ( Q T )



2

q + −2

+T

q + −4 q + −2



 2

2K q2 k23 K 3

2 q − −2

 T

2

q + −2

 u 4L 2 (Ω) dt

0

T



2K q2  N 2 2L ∞ ( Q T )



2 q − −2

+T

q − −4 q − −2



2K q2 k23 K 23



227

2 q − −2

 T

,

 u 4L 2 (Ω) dt

2 q + −2

2 q − −2

,

0

otherwise replace p − with p + or q− with q+ . Hence,



U  T 2K 2p  M 2 2L ∞ ( Q T )

+T

p − −4 p − −2





2 p − −2

2( K p k2 K 2 )2



2 p − −2

 

T 2K q2  N 2 2L ∞ ( Q T )



2 q − −2

q − −4  2  2  2 + T q− −2 2( K q k3 K 3 )2 q− −2 U q− −2 p− −2

4− p −   2    2  2  T 2K 2p  M 2 2L ∞ ( Q T ) p− −2 + 2 p− −2 T 2( K p k2 K 2 )2 2K q2  N 2 2∞ q− −2 p− −2 4− p −

+ 2 p − −2 T with U =

( p − −4)(q− −2)+2(q− −4) ( p − −2)(q− −2)





2( K p k2 K 2 )2 2( K q k3 K 3 )2



2 q − −2



2 p − −2

4

U ( p− −2)(q− −2)

T  2 2 0 ( Ω u dx) dt. The last inequality has the form: 4

U  α + β U ( p− −2)(q− −2) , 4

with

α , β > 0. Since p − > 4, the function f (U ) := α + β U ( p− −2)(q− −2) is concave, and then U  f (U )  f (U 0 ) + f  (U 0 )(U − U 0 ),

where U 0 := β

U

( p − −2)(q− −2) p − q− −2p − −2q−

( p−

(55)

. Using the fact that f (U 0 ) = α + U 0 and (55), one has

− 2)(q−

( p − −2)(q− −2) − 2) p − q− −2p − −2q− . α + β p − q− − 2p − − 2q−

A final application of Hölder’s inequality shows that u 2L 2 ( Q )  T 1/2 U 1/2 . The argument for v proceeds in a similar T way. 2 Again one can prove the following immediate consequence.



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