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Periodic p -Laplacian with nonlocal terms
Nonlinear Analysis 66 (2007) 442–453 www.elsevier.com/locate/na
Periodic p-Laplacian with nonlocal terms Qian Zhou a , Yuanyuan Ke a,∗ , Yifu Wang a,b , Jingxue Yin a a Department of Mathematics, Jilin University, Changchun, Jilin 130012, People’s Republic of China b Beijing Institute of Technology, People’s Republic of China
Received 20 January 2005; accepted 22 November 2005
Abstract In this paper we study the existence of non-trivial periodic solutions for a periodic p-Laplacian with nonlocal terms based on the theory of Leray–Schauder degree. The key step is dealing with the degeneracy of the p-Laplacian and the logistic-type terms arising in the right hand side of the equation. c 2005 Elsevier Ltd. All rights reserved. Keywords: p-Laplacian; Periodic solutions; Nonlocal terms
1. Introduction In this paper we consider the periodic problem for a periodic p-Laplacian with nonlocal terms of the following form, ∂u − div(|∇u| p−2 ∇u) = (m − Φ[u])u, ∂t u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u(x, T ),
(x, t) ∈ Q T ,
x ∈ Ω,
(1.1) (1.2) (1.3)
where p > 2, Ω is a bounded domain in Rn with smooth boundary, Q T = Ω × (0, T ). This problem is motivated by models which have been proposed for some problems in mathematical biology and fisheries management, where m = m(x, t) represents the maximal rate of natural increase at location x and time t, u = u(x, t) represents the density of the species at position x and time t, the nonlocal term Φ[u] : L 2 → R is a bounded continuous functional, and m − Φ[u] ∗ Corresponding author. Tel.: +86 431 516 7051.
E-mail address: [email protected] (Y. Ke). c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.11.038
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
443
denotes the actual growth rate with self-limitation, which means that the growth rate is not influenced by the density of the species at some local point, but, perhaps more important in the real world, by the total amount of the species. Problems with nonlocal terms have been considered by many authors, such as [6–8]. We remark that the periodic aspects of some related nonlinear diffusion equations were recently considered by [2–5,9–12]. Our consideration is motivated by the work of Allegretto and Nistri [1], where they studied the equation ∂u − u = f (x, t, Φ[u], u, m)u, (1.4) ∂t which reflects that the speed of the diffusion is relatively fast, and the typical case of f (x, t, Φ[u], u, m) is m − Φ[u]. In this paper, we consider another type of equation, namely the p-Laplacian equation, which reflects that the speed of the diffusion is rather slow. However, due to the relevant connections to gas or fluid flow media, this case is closer to reality. We should remark that it is more difficult to address than the case where the operator is linear. This is mainly due to the weaker regularization of the p-Laplacian ( p = 2) as compared to that of the Laplace operator ( p = 2). Also we can note that the nonlinear source in [2] is a strong source, so the key step in [2] is using the blow-up method to find the upper bound estimates of the related solutions. The nonlinear source in [9–12] emphasized the perturbations, so the authors used a variational process to deal with the problems, and did not consider the non-trivial solutions. In our paper, due to the nonlinear term being of logistic type, we mainly emphasize finding the uniform lower bound of maximum modulus of the related solutions, which is much more difficult than finding the upper bound estimates in some circumstances. In fact, the technique employed in [2] is not applicable to our case, and here the result of [4] on the sup-bounds estimate of |∇u| plays an important role in getting the uniform lower bound. 2. Main results and its proof First, we present the assumptions and the definitions of solutions. (H1) Let Φ[·] : L 2 (Ω )+ → R+ be a bounded continuous functional, Φ[0] = 0, and C1 ω2L 2 (Ω ) ≤ Φ[ω] ≤ C2 ω2L 2 (Ω ) , where 0 < C1 ≤ C2 are constants independent of ω, R+ = [0, +∞), L 2 (Ω )+ = {u ∈ L 2 (Ω ) | u ≥ 0, a.e. Ω }. (H2) m ∈ C T (Q T ) may change sign, but 1 T m(x, t)dt > 0 = ∅, x ∈Ω : T 0 where C T (Q T ) is a class of functions which are continuous in Ω¯ ×R and T -periodic with respect to t. By (H2) and the continuity of function m(x, t), there exist x 0 ∈ Ω , r0 > 0 and constant T m 0 > 0 such that T1 0 m(x, t)dt ≥ m 0 for all x ∈ B(x 0 , r0 ) ⊂ Ω . Let μ1 be the first eigenvalue of the following eigenvalue problem 1 −v = μv, in B x 0 , r0 , 2
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Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
1 on ∂ B x 0 , r0 . 2
v = 0,
Our main efforts centers on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem (1.1)–(1.3). Definition 2.1. A function u is called a generalized solution of the problem (1.1)–(1.3), if 1, p u ∈ L p (0, T ; W0 (Ω )) ∩ C T (Q T ), and u satisfies ∂ϕ p−2 + |∇u| ∇u · ∇ϕ − (m − Φ[u])uϕ dxdt = 0, (2.1) −u ∂t QT for any ϕ ∈ C 1 (Q T ) with ϕ(x, 0) = ϕ(x, T ) on ∂Ω × (0, T ). Due to the degeneracy of the equation, we should consider the following regularized problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = (m − Φ[u σ ])u + σ, ∂t (x, t) ∈ ∂Ω × (0, T ), u σ (x, t) = 0,
u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
x ∈ Ω,
(2.2) (2.3) (2.4)
where p > 2, σ is a positive constant. The desired solution of the problem (1.1)–(1.3) will be obtained as a limit point of the nonnegative solutions u σ of the problem (2.2)–(2.4). For fixed σ > 0, Eq. (2.2) is uniformly parabolic. However, though u σ is smooth enough, we cannot ensure Φ[u σ ] is smoother than C 0 . So we would not expect the right side of (2.2) to have C α smoothness. Furthermore, we do not expect the problem (2.2)–(2.4) to have a classical solution. Now we should define the strong generalized solution of the problem (2.2)–(2.4). Definition 2.2. A function u σ is called a strong generalized solution of problem (2.2)–(2.4), if •
u σ ∈ Wq2,1 (Q T ) ∩ C T (Q T ) and u satisfies the Eq. (2.2) almost everywhere. Now we introduce a map and deduce some useful lemma. Consider the problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = f, ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), x ∈ Ω. u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
Utilizing the results in [2], we know that for all given functions f ∈ C T (Q T ), there exists a unique generalized solution u σ ∈ C T (Q T ) ∩ C 1 (Q T ), which satisfies ∂u∂tσ ∈ L 2 (Q T ). We define a map u σ = G f with G : C T (Q T ) → C T (Q T ). Using methods similar to those in [2], we can get that the map is completely continuous. Let f (v) = (m − Φ[v])v + , where v + = max{v, 0}, by the condition (H1), we can see when v is continuous with respect to t, Φ[v] is continuous with respect to t. So we can study the existence of the fixed point of the complete continuous map u σ = G((m − Φ[v])v + ) instead of obtaining the existence of the solution of (2.2)–(2.4). First, we prove the nonnegativity of the solutions of the regularized problem. Lemma 2.1. If u σ ∈ C T (Q T ) is a non-trivial periodic solution of problem (2.2)–(2.4), then u σ (x, t) > 0,
x ∈ Ω , t ≥ 0.
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Proof. We first prove u σ ≥ 0. Multiplying u − σ by Eq. (2.2), and integrating over Q T , we can see T T p−2 ∂u σ − u σ dtdx + (|∇u σ |2 + σ ) 2 ∇u σ ∇u − σ dtdx ∂t Ω 0 Ω 0 T − = (m − Φ[u σ ])u + σ u σ dtdx, Ω
0
u− σ
= min{u σ , 0}. Considering the above terms respectively, T T ∂u σ − 1 − 2 u σ dtdx = − (u σ ) dx = 0, ∂t 2 Ω Ω 0 0 T T p−2 p−2 2 (|∇u σ |2 + σ ) 2 ∇u σ ∇u − dtdx = (|∇u σ |2 + σ ) 2 |∇u − σ σ | dtdx,
where
Ω
Then
0
Ω
0
Ω
Ω
T
T 0
(m
− − Φ[u σ ])u + σ u σ dtdx
(|∇u σ |2 + σ )
p−2 2
0
= 0.
2 |∇u − σ | dtdx = 0.
p−2
Notice that (|∇u σ |2 + σ ) 2 > 0, then T 2 |∇u − σ | dtdx = 0. Ω
(2.5)
0
Utilizing Poincar`e’s inequality, we have − 2 2 |u σ | dx ≤ C |∇u − σ | dx. Ω
Ω
This combined with (2.5) implies u σ ≥ 0. Next we prove u σ > 0. By the fact that u σ ∈ C T (Q T ) is non-trivial, there exist ξ ∈ (0, T ] and x ∈ Ω , such that u σ (x, ξ ) ≡ 0. Let 0 ≤ ψ(x) ∈ C0∞ (Ω ) be non-trivial with ψ(x) < u σ (x, ξ ). For a constant D > 0, let v solve the problem p−2 ∂v − div((|v|2 + σ ) 2 ∇v) + Dv = 0, ∂t v(x, t) = 0, (x, t) ∈ ∂Ω × [ξ, T ],
v(x, ξ ) = ψ(x),
x ∈ Ω , t > ξ,
x ∈ Ω.
Noticing that u σ ∈ C T (Q T ), and by the condition (H1), we can obtain Φ[u σ ] ∈ C T (Q T ). Combining this with m ∈ C T (Q T ), we know m − Φ[u σ ] ∈ C T (Q T ). By the comparison theorem, we can see that when D is large enough, we have u σ (x, t) ≥ v(x, t). And by using the maximum principle, we have for all x ∈ Ω and t > ξ , v(x, t) > 0. Finally, utilizing the periodicity of u σ , we can find that u σ (x, t) ≥ v(x, t) > 0, that is for all x ∈ Ω and t > 0, u σ (x, t) > 0. The proof is completed. Lemma 2.2. There exists a constant r > 0, such that no solutions u of the problem (2.2)–(2.4) satisfy 0 < u σ L ∞ (Q T ) ≤ r.
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Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
Proof. Let u σ be the solution of the problem (2.2)–(2.4), and 0 < u σ L ∞ (Q T ) ≤ r . By Lemma 2.1, we know that for all (x, t) ∈ Q T , u σ (x, t) > 0. Now for all φ(x) ∈ C0∞ (Ω ),
we choose uφσ as the test function. Then multiplying uφσ by Eq. (2.2), and then integrating over Q T , we have 2 p−2 ∂u σ φ 2 φ 2 2 (|∇u σ | + σ ) ∇u σ ∇ dtdx + dtdx uσ uσ Q T ∂t QT = φ 2 (m − Φ[u σ ])dtdx. 2
2
QT
By the periodicity of u σ , the first term of the equation can be written as QT
∂u σ ∂t
φ2 uσ
dtdx =
Ω
T
φ2 0
∂(ln u σ ) dtdx = 0, ∂t
and the second term can be considered as 2 p−2 φ 2 2 dtdx (|∇u σ | + σ ) ∇u σ ∇ uσ QT p−2 φ 2 dtdx (|∇u σ | + σ ) 2 ∇u σ ∇ φ = uσ QT p−2 ∇φ = (|∇u σ |2 + σ ) 2 φ∇u σ dtdx uσ QT p−2 φ (|∇u σ |2 + σ ) 2 φ∇u σ ∇ dtdx + uσ QT p−2 ∇φ φ u σ ∇φ − u 2σ ∇ dtdx (|∇u σ |2 + σ ) 2 = u u σ σ QT p−2 φ dtdx (|∇u σ |2 + σ ) 2 φ∇u σ ∇ + u σ QT p−2 = (|∇u σ |2 + σ ) 2 |∇φ|2 dtdx QT
− QT
(|∇u σ |2 + σ )
p−2 2
2 φ u 2σ ∇ dtdx. u σ
So the inequality p−2 2 2 (|∇u σ | + σ ) 2 |∇φ| dtdx − QT
=
(|∇u σ | + σ ) 2
QT
p−2 2
u 2σ
∇ φ u
σ
QT 2
φ 2 (m − Φ[u σ ])dtdx
dtdx ≥ 0
follows. By Theorem 5.1 and some remarks in [4] (page 238, page 243), it follows that there exists a constant γ = γ (N, p) such that
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
sup
[(x 0 ,t0 )+Q(τ θ,τρ)]
447
|∇u|
1 1 2 p−2 2 ρ γ θ/ρ 2 p |∇u| dxdt ∧ , ≤ −− θ (1 − τ )(N+2)/2 [(x 0 ,t0 )+Q(θ,ρ)]
(2.6)
for any (x 0 , t0 ) ∈ Q (T ,3T ) = Ω × (T, 3T ), [(x 0 , t0 ) + Q(θ, ρ)] ⊂ Q (T ,3T ) , and any τ ∈ (0, 1). Substituting θ = r0 , ρ = min{T,
√ m 0 r0 2
p+6 2
}, τ =
1 2
in (2.6) gives
|∇u|
sup [(x 0 ,t0 )+Q( 21 r0 , 12 ρ)]
≤ C(N, p, r0 , m 0 , μ1 ) − −
1 [(x 0 ,t0 )+Q(r0 ,ρ)]
On the other hand, by (2.2)–(2.4), we have |∇u σ | p dtdx ≤ max |m(x, t)| QT
QT
QT
|∇u| p dxdt
2
∧
1 2
m0 4μ1
1 p−2
.
|u σ |2 dtdx.
So sup [(x 0 ,t0 )+Q( 21 r0 , 12 ρ)]
|∇u σ | ≤ C(N, p, r0 , m 0 , μ1 )
1 |u σ | dtdx 2
QT
2
1 ∧ 2
m0 4μ1
1 p−2
,
which implies ∇u σ L ∞ (B(x0, 1 r0 )×(0,T )) 2
1 ≤ Cu σ L ∞ (Q T ) ∧ 2
m0 4μ1
1 p−2
,
where C is a constant independent of σ . p−2
p−2
p−2
Since (|∇u σ |2 + σ ) 2 ≤ 2 2 (|∇u σ | p−2 + σ 2 ), we have p−2 p−2 2 2 (|∇u σ | p−2 + σ 2 )|∇φ|2 dtdx − φ 2 (m − Φ[u σ ])dtdx QT QT p−2 ≥ (|∇u σ |2 + σ ) 2 |∇φ|2 dtdx − φ 2 (m − Φ[u σ ])dtdx ≥ 0. QT
QT
By the approximating process, taking φ = φ1 , where φ1 be the eigenfunction of the first eigenvalue μ1 , we get φ12 (m − Φ[u σ ])dtdx B(x 0 , 12 r0 )×(0,T )
≤
B(x 0 , 21 r0 )×(0,T )
2
p−2 2
(|∇u σ | p−2 + σ
)|∇φ1 |2 dtdx
p−2 p−2 m0 + 2 2 σ 2 |∇φ1 |2 dtdx 4μ1 B(x 0 , 21 r0 )×(0,T )
p−2 p−2 m0 = T Cμ1r p−2 ∧ + 2 2 μ1 σ 2 φ12 dx. 1 4 B(x 0, 2 r0 )
≤
p−2 2
Cr p−2 ∧
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On the other hand, B(x 0 , 21 r0 )×(0,T )
φ12 (m − Φ[u σ ])dtdx
≥
B(x 0 , 12 r0 )
φ12 (x)
≥ (T m 0 − T C2r 2 )
(0,T )
m(x, t)dt − T C2r 2 dx
B(x 0 , 12 r0 )
φ12 (x)dx.
Therefore we obtain m 0 ≤ C2r 2 + Cμ1r p−2 ∧
p−2 m0 + 2 2 μ1 σ 4
p−2 2
.
Obviously this inequality does not hold if 2 1 1 p−2 1 m 0 p−2 m0 2 m0 σ ≤ , r ≤ min , . 2 4μ1 4C2 4Cμ1 Therefore there exists one positive constant r , such that no solutions u σ of the problem (2.2)–(2.4) satisfy 0 < u σ L ∞ (Q T ) ≤ r. The proof is completed.
In this paper, we should use the theory of topological degree to find the solution of the problem (2.2)–(2.4). We will consider the problem in the set Σ = {u σ ∈ C T (Q T ) | u σ ∈ B R , u σ ∈ Br }, where B R is a ball centered at the origin with radius R in C T (Q T ), 0 < r < R, where r, R are all constants independent of σ . Now we will discuss the topological degree in B R and Br separately. In order to apply the homotopy invariant of Leray–Schauder degree, let λ ∈ [0, 1], and consider the map u σ = G(λ(m − Φ[v])v + ). First we verify that u σ = G(λ(m − Φ[v])v + ) is the homotopic map of the map u σ = G((m − Φ[v])v + ), that is the value of the topological degree deg(u σ − G(λ(m − Φ[u σ ])u + σ ), B R , 0) is identical with respect to the parameter λ ∈ [0, 1]. Lemma 2.3. Let λ ∈ [0, 1]. Then there exists a positive constant R independent of σ , such that no solutions u σ ∈ C T (Q T ) satisfy both the following problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = λ(m − Φ[u σ ])u σ , ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × [0, T ],
u σ (x, 0) = u σ (x, T ),
x ∈ Ω,
(x, t) ∈ Q T ,
(2.7) (2.8) (2.9)
and the condition u σ L ∞ (Q T ) = R. Proof. Like for the problem (2.2)–(2.4), we can also define the strong generalized solution of the problem (2.7)–(2.9). Let u σ be the strong generalized solution of the problem (2.7)–(2.9). Multiplying u σ by Eq. (2.7), and then integrating over Q T , we have
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449
p−2 ∂u σ dtdx − uσ div((|∇u σ |2 + σ ) 2 ∇u σ )u σ dtdx ∂t QT QT p−2 ∂u σ uσ (|∇u σ |2 + σ ) 2 |∇u σ |2 dtdx = dtdx + ∂t Q QT T 2 = λ(m − Φ[u σ ])u σ dtdx.
QT
By the periodicity of u σ , we obtain ∂u σ dtdx = 0. uσ ∂t QT Since
QT
then
(|∇u σ |2 + σ )
p−2 2
|∇u σ |2 dtdx ≥ 0,
QT
λ(m − Φ[u σ ])u 2σ dtdx ≥ 0.
Setting M = supx∈Ω ,t ∈[0,T ] m(x, t), and combining with the assumption (H1), we can see that λ(m − Φ[u σ ])u 2σ dtdx 0≤ Q T ≤ (m − Φ[u σ ])u 2σ dtdx Q T ≤ Mu 2σ dtdx − (u 2σ Φ[u σ ])dxdt QT
QT
≤ M QT
that is
T
u 2σ dtdx − C 2
Ω
0
u 2σ dx
T
Ω
0
T
dt ≤ C
2 u 2σ dx
Ω
0
dt,
u 2σ dx dt,
(2.10)
where C is a constant independent of λ and σ . Furthermore, utilizing Cauchy’s inequality, we have 2 1 u 2σ dx ≤ 2 + ε2 u 2σ dx , 4ε Ω Ω that is
T 0
Ω
u 2σ dxdt ≤
1 + ε2 4ε2
T
0
2
Ω
u 2σ dx
Combining (2.10) with (2.11), we have u σ L 2 (Q T ) ≤ C, where C is a constant independent of λ and σ .
dt.
(2.11)
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By (2.2)–(2.4), we have p |u σ | dtdx ≤ QT
p
QT
|∇u σ | dtdx ≤ max |m(x, t)| QT
QT
|u σ |2 dtdx.
The assumption (H1) implies that there exists a constant K > 0, such that u σ satisfies p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) ≤ K u σ . ∂t
Therefore by the Young inequality and Theorem 3.2 in [4] (page 121) with δ = p, q = and κ = Np , we have the following estimate u σ (x, t) ≤ C
3T 2 T 2
12
Ω
p(N+2) N
u 2σ dxdt
,
for t ∈ [ T2 , 3T 2 ]. Then by the periodicity of u σ , we have u σ L ∞ (Q T ) ≤ C, where C is independent of λ and σ . So the proof is completed if R > C.
Utilizing the above result, and the result in [3] with the case λ = 0, we can apply the homotopy invariant of Leray–Schauder degree and discuss the topological degree of the problem (2.2)–(2.4) in B R . Lemma 2.4. There exists a constant R, such that deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0) = 1. Proof. Using Lemma 2.3, and the existence and uniqueness of the solution of the problem (2.2)–(2.4) in the case λ = 0, we apply the homotopy invariant of Leray–Schauder degree and have 1 = deg(u σ , B R , 0) = deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0).
Now we discuss the topological degree of the problem (2.2)–(2.4) in Br . Due to the fact that the problem (2.2)–(2.4) has the solution u σ = 0, we cannot deal with the problem directly. So for all given constants γ ≥ 0, and a smooth function θ = θ (x) > 0 in Ω , we introduce a map u σ = G((m − Φ[v])v + ) + γ G(θ ). By the theory of Leray–Schauder degree, we can see that when γ ≥ 0 and θ > 0, the map u σ = G((m − Φ[v])v + ) + γ G(θ ) is the homotopic map of the map u σ = G((m − Φ[v])v + ), that is for all θ > 0, the topological degree deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) is identical with respect to γ ≥ 0. Lemma 2.5. For a given smooth function θ = θ (x) > 0 in Ω , and a constant γ > 0, all the solutions u σ ∈ C T (Q T ) of the following problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = (m − Φ[u σ ])u σ + γ θ, ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × [0, T ], x ∈ Ω, u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
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451
satisfy u σ L ∞ (Q T ) > r > 0, where r is a constant independent of σ, γ and θ . Proof. Like for Lemma 2.2, we can find a constant r independent of σ, γ and θ , such that when 0 ≤ u σ L ∞ (Q T ) ≤ r , it would contradict the condition (H2). Noticing the fact that after introducing the map u σ = G((m − Φ[v])v + ) + γ G(θ ), u σ = 0 is not the solution of the problem, we can get the following lemma. Lemma 2.6. There exists a constant r , such that deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) = 0. Proof. Using Lemma 2.5, and the homotopy invariant of Leray–Schauder degree, we have 0 = deg(u σ − G((m − Φ[u σ ])u + σ ) − γ G(θ ), Br , 0) = deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0). The proof is completed.
Summing up the results above, we can see when Φ[u σ ] and m satisfy the conditions (H1) and (H2), there exist constants r, R > 0, such that deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0) = 1, deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) = 0, that is deg(u σ − G((m − Φ[u σ ])u + σ ), Σ , 0) = 1. By the theory of Leray–Schauder degree, we can conclude that the problem (2.2)–(2.4) has a non-trivial periodic solution in Σ . Then by Lemma 2.1, we can see that the problem (2.2)–(2.4) has a nonnegative non-trivial periodic solution in Q T . Finally, we can get our main result Theorem 2.1. The problem (1.1)–(1.3) has a nonnegative non-trivial periodic solution u. Proof. Using Lemma 2.2 and the proof of Lemma 2.3, we know that all the solutions u σ ∈ Σ of the problem (2.2)–(2.4) satisfy r < u σ L ∞ (Q T ) < R, where r and R are all constants which are independent of σ . In the proof of Lemma 2.2, we have ∇u σ L ∞ (Q T ) ≤ C, where C is independent of σ . Combining this with the conditions (2.3) and (2.4), it follows that 1, p
u σ ∈ L p (0, T ; W0 (Ω )) ∩ C T (Q T ).
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σ In order to get the uniform estimate of ∂u ∂t , we multiply (u σ )t by Eq. (2.2), and integrate it over Q T . By the periodicity of u σ , the condition (H1) and Cauchy’s inequality, (u σ )2t dtdx QT = (m − Φ[u σ ])u σ (u σ )t dtdx − A(∇u σ )∇u σ ∇(u σ )t dtdx
QT
QT
holds. Noticing the periodicity of u σ , we have A(∇u σ )∇u σ ∇(u σ )t dtdx QT
T
= 0
∂ ∂t s
Ω
B(∇u)dx dt = 0,
where B(s) = 0 A(ξ )ξ dξ . So (u σ )2t dtdx QT = (m − Φ[u σ ])u σ (u σ )t dtdx − A(∇u σ )∇u σ ∇(u σ )t dtdx QT QT = (m − Φ[u σ ])u σ (u σ )t dtdx QT 1 1 ≤ ((m − Φ[u σ ])u σ )2 dtdx + (u σ )2t dtdx. 2 2 QT QT By the boundedness of Φ[ω] and u σ , we have (u σ )2t dtdx ≤ C, QT
with constant C independent of σ . So we can get the uniform estimate of ∂u σ ≤ C. ∂t 2 L (Q T )
∂u σ ∂t
,
By the above estimates of u σ , and noticing that the constant C is independent of σ , we can conclude there exist a subsequence {u σi } in Σ and u ∈ Σ , which satisfy weak in L p (Q T ), |∇u σi | p−2 ∇u σi |∇u| p−2 ∇u, ∂u σi ∂u , weak in L 2 (Q T ), ∂t ∂t in C(Q T ). u σi → u, It is easy to see that u is the nonnegative non-trivial periodic solution of the problem (1.1)–(1.3). The proof is completed. Acknowledgments This work was partially supported by the National Science Foundation of China (Grant Nos 10401006 and 10426018), partially supported by a Specific Foundation for Ph.D. Specialities of
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the Educational Department of China and partially supported by the 985 project. The authors would like to thank the referees for their valuable suggestions for the revision of the manuscript. References [1] W. Allegretto, P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal term, IMA J. Math. Control Inform. 16 (1) (1999) 43–58. [2] Y. Wang, J. Yin, Z. Wu, Periodic solutions of evolution p-Laplacian equations with nonlinear sources, J. Math. Anal. Appl. 219 (1) (1998) 76–96. [3] Z.Q. Wu, Q.N. Zhao, J.X. Yin, H.L. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001. [4] E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [5] Y. Wang, Z. Wu, J. Yin, Periodic solutions of evolution p-Laplacian equations with weakly nonlinear sources, Int. J. Math. Game Theory Algebra 10 (1) (2000) 67–77. [6] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [7] A. Calsina, C. Perello, Equations for biological evolution, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 939–958. [8] A. Calsina, C. Perello, J. Saldana, Non-local reaction–diffusion equations modelling predator–prey coevolution, Publ. Mat. 32 (1994) 315–325. [9] J. Crema, J.L. Boldrini, More on forced periodic solutions of quasi-parabolic equations, Cadernos de Matem´atica 01 (2000) 71–88. [10] J.L. Boldrini, J. Crema, On forced periodic solutions of superlinear quasi-parabolic problems, Electron. J. Differential Equations 1998 (14) (1998) 1–18. [11] J. Crema, J.L. Boldrini, Periodic solutions of quasilinear equations with discontinuous perturbations, Cadernos de Matem´atica 01 (2000) 53–69. [12] J. H´uska, Periodic solutions in superlinear parabolic problems, Acta Math. Univ. Comenianae LXXI (1) (2002) 19–26.
Periodic p-Laplacian with nonlocal terms Qian Zhou a , Yuanyuan Ke a,∗ , Yifu Wang a,b , Jingxue Yin a a Department of Mathematics, Jilin University, Changchun, Jilin 130012, People’s Republic of China b Beijing Institute of Technology, People’s Republic of China
Received 20 January 2005; accepted 22 November 2005
Abstract In this paper we study the existence of non-trivial periodic solutions for a periodic p-Laplacian with nonlocal terms based on the theory of Leray–Schauder degree. The key step is dealing with the degeneracy of the p-Laplacian and the logistic-type terms arising in the right hand side of the equation. c 2005 Elsevier Ltd. All rights reserved. Keywords: p-Laplacian; Periodic solutions; Nonlocal terms
1. Introduction In this paper we consider the periodic problem for a periodic p-Laplacian with nonlocal terms of the following form, ∂u − div(|∇u| p−2 ∇u) = (m − Φ[u])u, ∂t u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u(x, T ),
(x, t) ∈ Q T ,
x ∈ Ω,
(1.1) (1.2) (1.3)
where p > 2, Ω is a bounded domain in Rn with smooth boundary, Q T = Ω × (0, T ). This problem is motivated by models which have been proposed for some problems in mathematical biology and fisheries management, where m = m(x, t) represents the maximal rate of natural increase at location x and time t, u = u(x, t) represents the density of the species at position x and time t, the nonlocal term Φ[u] : L 2 → R is a bounded continuous functional, and m − Φ[u] ∗ Corresponding author. Tel.: +86 431 516 7051.
E-mail address: [email protected] (Y. Ke). c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.11.038
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
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denotes the actual growth rate with self-limitation, which means that the growth rate is not influenced by the density of the species at some local point, but, perhaps more important in the real world, by the total amount of the species. Problems with nonlocal terms have been considered by many authors, such as [6–8]. We remark that the periodic aspects of some related nonlinear diffusion equations were recently considered by [2–5,9–12]. Our consideration is motivated by the work of Allegretto and Nistri [1], where they studied the equation ∂u − u = f (x, t, Φ[u], u, m)u, (1.4) ∂t which reflects that the speed of the diffusion is relatively fast, and the typical case of f (x, t, Φ[u], u, m) is m − Φ[u]. In this paper, we consider another type of equation, namely the p-Laplacian equation, which reflects that the speed of the diffusion is rather slow. However, due to the relevant connections to gas or fluid flow media, this case is closer to reality. We should remark that it is more difficult to address than the case where the operator is linear. This is mainly due to the weaker regularization of the p-Laplacian ( p = 2) as compared to that of the Laplace operator ( p = 2). Also we can note that the nonlinear source in [2] is a strong source, so the key step in [2] is using the blow-up method to find the upper bound estimates of the related solutions. The nonlinear source in [9–12] emphasized the perturbations, so the authors used a variational process to deal with the problems, and did not consider the non-trivial solutions. In our paper, due to the nonlinear term being of logistic type, we mainly emphasize finding the uniform lower bound of maximum modulus of the related solutions, which is much more difficult than finding the upper bound estimates in some circumstances. In fact, the technique employed in [2] is not applicable to our case, and here the result of [4] on the sup-bounds estimate of |∇u| plays an important role in getting the uniform lower bound. 2. Main results and its proof First, we present the assumptions and the definitions of solutions. (H1) Let Φ[·] : L 2 (Ω )+ → R+ be a bounded continuous functional, Φ[0] = 0, and C1 ω2L 2 (Ω ) ≤ Φ[ω] ≤ C2 ω2L 2 (Ω ) , where 0 < C1 ≤ C2 are constants independent of ω, R+ = [0, +∞), L 2 (Ω )+ = {u ∈ L 2 (Ω ) | u ≥ 0, a.e. Ω }. (H2) m ∈ C T (Q T ) may change sign, but 1 T m(x, t)dt > 0 = ∅, x ∈Ω : T 0 where C T (Q T ) is a class of functions which are continuous in Ω¯ ×R and T -periodic with respect to t. By (H2) and the continuity of function m(x, t), there exist x 0 ∈ Ω , r0 > 0 and constant T m 0 > 0 such that T1 0 m(x, t)dt ≥ m 0 for all x ∈ B(x 0 , r0 ) ⊂ Ω . Let μ1 be the first eigenvalue of the following eigenvalue problem 1 −v = μv, in B x 0 , r0 , 2
444
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
1 on ∂ B x 0 , r0 . 2
v = 0,
Our main efforts centers on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem (1.1)–(1.3). Definition 2.1. A function u is called a generalized solution of the problem (1.1)–(1.3), if 1, p u ∈ L p (0, T ; W0 (Ω )) ∩ C T (Q T ), and u satisfies ∂ϕ p−2 + |∇u| ∇u · ∇ϕ − (m − Φ[u])uϕ dxdt = 0, (2.1) −u ∂t QT for any ϕ ∈ C 1 (Q T ) with ϕ(x, 0) = ϕ(x, T ) on ∂Ω × (0, T ). Due to the degeneracy of the equation, we should consider the following regularized problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = (m − Φ[u σ ])u + σ, ∂t (x, t) ∈ ∂Ω × (0, T ), u σ (x, t) = 0,
u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
x ∈ Ω,
(2.2) (2.3) (2.4)
where p > 2, σ is a positive constant. The desired solution of the problem (1.1)–(1.3) will be obtained as a limit point of the nonnegative solutions u σ of the problem (2.2)–(2.4). For fixed σ > 0, Eq. (2.2) is uniformly parabolic. However, though u σ is smooth enough, we cannot ensure Φ[u σ ] is smoother than C 0 . So we would not expect the right side of (2.2) to have C α smoothness. Furthermore, we do not expect the problem (2.2)–(2.4) to have a classical solution. Now we should define the strong generalized solution of the problem (2.2)–(2.4). Definition 2.2. A function u σ is called a strong generalized solution of problem (2.2)–(2.4), if •
u σ ∈ Wq2,1 (Q T ) ∩ C T (Q T ) and u satisfies the Eq. (2.2) almost everywhere. Now we introduce a map and deduce some useful lemma. Consider the problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = f, ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), x ∈ Ω. u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
Utilizing the results in [2], we know that for all given functions f ∈ C T (Q T ), there exists a unique generalized solution u σ ∈ C T (Q T ) ∩ C 1 (Q T ), which satisfies ∂u∂tσ ∈ L 2 (Q T ). We define a map u σ = G f with G : C T (Q T ) → C T (Q T ). Using methods similar to those in [2], we can get that the map is completely continuous. Let f (v) = (m − Φ[v])v + , where v + = max{v, 0}, by the condition (H1), we can see when v is continuous with respect to t, Φ[v] is continuous with respect to t. So we can study the existence of the fixed point of the complete continuous map u σ = G((m − Φ[v])v + ) instead of obtaining the existence of the solution of (2.2)–(2.4). First, we prove the nonnegativity of the solutions of the regularized problem. Lemma 2.1. If u σ ∈ C T (Q T ) is a non-trivial periodic solution of problem (2.2)–(2.4), then u σ (x, t) > 0,
x ∈ Ω , t ≥ 0.
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445
Proof. We first prove u σ ≥ 0. Multiplying u − σ by Eq. (2.2), and integrating over Q T , we can see T T p−2 ∂u σ − u σ dtdx + (|∇u σ |2 + σ ) 2 ∇u σ ∇u − σ dtdx ∂t Ω 0 Ω 0 T − = (m − Φ[u σ ])u + σ u σ dtdx, Ω
0
u− σ
= min{u σ , 0}. Considering the above terms respectively, T T ∂u σ − 1 − 2 u σ dtdx = − (u σ ) dx = 0, ∂t 2 Ω Ω 0 0 T T p−2 p−2 2 (|∇u σ |2 + σ ) 2 ∇u σ ∇u − dtdx = (|∇u σ |2 + σ ) 2 |∇u − σ σ | dtdx,
where
Ω
Then
0
Ω
0
Ω
Ω
T
T 0
(m
− − Φ[u σ ])u + σ u σ dtdx
(|∇u σ |2 + σ )
p−2 2
0
= 0.
2 |∇u − σ | dtdx = 0.
p−2
Notice that (|∇u σ |2 + σ ) 2 > 0, then T 2 |∇u − σ | dtdx = 0. Ω
(2.5)
0
Utilizing Poincar`e’s inequality, we have − 2 2 |u σ | dx ≤ C |∇u − σ | dx. Ω
Ω
This combined with (2.5) implies u σ ≥ 0. Next we prove u σ > 0. By the fact that u σ ∈ C T (Q T ) is non-trivial, there exist ξ ∈ (0, T ] and x ∈ Ω , such that u σ (x, ξ ) ≡ 0. Let 0 ≤ ψ(x) ∈ C0∞ (Ω ) be non-trivial with ψ(x) < u σ (x, ξ ). For a constant D > 0, let v solve the problem p−2 ∂v − div((|v|2 + σ ) 2 ∇v) + Dv = 0, ∂t v(x, t) = 0, (x, t) ∈ ∂Ω × [ξ, T ],
v(x, ξ ) = ψ(x),
x ∈ Ω , t > ξ,
x ∈ Ω.
Noticing that u σ ∈ C T (Q T ), and by the condition (H1), we can obtain Φ[u σ ] ∈ C T (Q T ). Combining this with m ∈ C T (Q T ), we know m − Φ[u σ ] ∈ C T (Q T ). By the comparison theorem, we can see that when D is large enough, we have u σ (x, t) ≥ v(x, t). And by using the maximum principle, we have for all x ∈ Ω and t > ξ , v(x, t) > 0. Finally, utilizing the periodicity of u σ , we can find that u σ (x, t) ≥ v(x, t) > 0, that is for all x ∈ Ω and t > 0, u σ (x, t) > 0. The proof is completed. Lemma 2.2. There exists a constant r > 0, such that no solutions u of the problem (2.2)–(2.4) satisfy 0 < u σ L ∞ (Q T ) ≤ r.
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Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
Proof. Let u σ be the solution of the problem (2.2)–(2.4), and 0 < u σ L ∞ (Q T ) ≤ r . By Lemma 2.1, we know that for all (x, t) ∈ Q T , u σ (x, t) > 0. Now for all φ(x) ∈ C0∞ (Ω ),
we choose uφσ as the test function. Then multiplying uφσ by Eq. (2.2), and then integrating over Q T , we have 2 p−2 ∂u σ φ 2 φ 2 2 (|∇u σ | + σ ) ∇u σ ∇ dtdx + dtdx uσ uσ Q T ∂t QT = φ 2 (m − Φ[u σ ])dtdx. 2
2
QT
By the periodicity of u σ , the first term of the equation can be written as QT
∂u σ ∂t
φ2 uσ
dtdx =
Ω
T
φ2 0
∂(ln u σ ) dtdx = 0, ∂t
and the second term can be considered as 2 p−2 φ 2 2 dtdx (|∇u σ | + σ ) ∇u σ ∇ uσ QT p−2 φ 2 dtdx (|∇u σ | + σ ) 2 ∇u σ ∇ φ = uσ QT p−2 ∇φ = (|∇u σ |2 + σ ) 2 φ∇u σ dtdx uσ QT p−2 φ (|∇u σ |2 + σ ) 2 φ∇u σ ∇ dtdx + uσ QT p−2 ∇φ φ u σ ∇φ − u 2σ ∇ dtdx (|∇u σ |2 + σ ) 2 = u u σ σ QT p−2 φ dtdx (|∇u σ |2 + σ ) 2 φ∇u σ ∇ + u σ QT p−2 = (|∇u σ |2 + σ ) 2 |∇φ|2 dtdx QT
− QT
(|∇u σ |2 + σ )
p−2 2
2 φ u 2σ ∇ dtdx. u σ
So the inequality p−2 2 2 (|∇u σ | + σ ) 2 |∇φ| dtdx − QT
=
(|∇u σ | + σ ) 2
QT
p−2 2
u 2σ
∇ φ u
σ
QT 2
φ 2 (m − Φ[u σ ])dtdx
dtdx ≥ 0
follows. By Theorem 5.1 and some remarks in [4] (page 238, page 243), it follows that there exists a constant γ = γ (N, p) such that
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
sup
[(x 0 ,t0 )+Q(τ θ,τρ)]
447
|∇u|
1 1 2 p−2 2 ρ γ θ/ρ 2 p |∇u| dxdt ∧ , ≤ −− θ (1 − τ )(N+2)/2 [(x 0 ,t0 )+Q(θ,ρ)]
(2.6)
for any (x 0 , t0 ) ∈ Q (T ,3T ) = Ω × (T, 3T ), [(x 0 , t0 ) + Q(θ, ρ)] ⊂ Q (T ,3T ) , and any τ ∈ (0, 1). Substituting θ = r0 , ρ = min{T,
√ m 0 r0 2
p+6 2
}, τ =
1 2
in (2.6) gives
|∇u|
sup [(x 0 ,t0 )+Q( 21 r0 , 12 ρ)]
≤ C(N, p, r0 , m 0 , μ1 ) − −
1 [(x 0 ,t0 )+Q(r0 ,ρ)]
On the other hand, by (2.2)–(2.4), we have |∇u σ | p dtdx ≤ max |m(x, t)| QT
QT
QT
|∇u| p dxdt
2
∧
1 2
m0 4μ1
1 p−2
.
|u σ |2 dtdx.
So sup [(x 0 ,t0 )+Q( 21 r0 , 12 ρ)]
|∇u σ | ≤ C(N, p, r0 , m 0 , μ1 )
1 |u σ | dtdx 2
QT
2
1 ∧ 2
m0 4μ1
1 p−2
,
which implies ∇u σ L ∞ (B(x0, 1 r0 )×(0,T )) 2
1 ≤ Cu σ L ∞ (Q T ) ∧ 2
m0 4μ1
1 p−2
,
where C is a constant independent of σ . p−2
p−2
p−2
Since (|∇u σ |2 + σ ) 2 ≤ 2 2 (|∇u σ | p−2 + σ 2 ), we have p−2 p−2 2 2 (|∇u σ | p−2 + σ 2 )|∇φ|2 dtdx − φ 2 (m − Φ[u σ ])dtdx QT QT p−2 ≥ (|∇u σ |2 + σ ) 2 |∇φ|2 dtdx − φ 2 (m − Φ[u σ ])dtdx ≥ 0. QT
QT
By the approximating process, taking φ = φ1 , where φ1 be the eigenfunction of the first eigenvalue μ1 , we get φ12 (m − Φ[u σ ])dtdx B(x 0 , 12 r0 )×(0,T )
≤
B(x 0 , 21 r0 )×(0,T )
2
p−2 2
(|∇u σ | p−2 + σ
)|∇φ1 |2 dtdx
p−2 p−2 m0 + 2 2 σ 2 |∇φ1 |2 dtdx 4μ1 B(x 0 , 21 r0 )×(0,T )
p−2 p−2 m0 = T Cμ1r p−2 ∧ + 2 2 μ1 σ 2 φ12 dx. 1 4 B(x 0, 2 r0 )
≤
p−2 2
Cr p−2 ∧
448
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
On the other hand, B(x 0 , 21 r0 )×(0,T )
φ12 (m − Φ[u σ ])dtdx
≥
B(x 0 , 12 r0 )
φ12 (x)
≥ (T m 0 − T C2r 2 )
(0,T )
m(x, t)dt − T C2r 2 dx
B(x 0 , 12 r0 )
φ12 (x)dx.
Therefore we obtain m 0 ≤ C2r 2 + Cμ1r p−2 ∧
p−2 m0 + 2 2 μ1 σ 4
p−2 2
.
Obviously this inequality does not hold if 2 1 1 p−2 1 m 0 p−2 m0 2 m0 σ ≤ , r ≤ min , . 2 4μ1 4C2 4Cμ1 Therefore there exists one positive constant r , such that no solutions u σ of the problem (2.2)–(2.4) satisfy 0 < u σ L ∞ (Q T ) ≤ r. The proof is completed.
In this paper, we should use the theory of topological degree to find the solution of the problem (2.2)–(2.4). We will consider the problem in the set Σ = {u σ ∈ C T (Q T ) | u σ ∈ B R , u σ ∈ Br }, where B R is a ball centered at the origin with radius R in C T (Q T ), 0 < r < R, where r, R are all constants independent of σ . Now we will discuss the topological degree in B R and Br separately. In order to apply the homotopy invariant of Leray–Schauder degree, let λ ∈ [0, 1], and consider the map u σ = G(λ(m − Φ[v])v + ). First we verify that u σ = G(λ(m − Φ[v])v + ) is the homotopic map of the map u σ = G((m − Φ[v])v + ), that is the value of the topological degree deg(u σ − G(λ(m − Φ[u σ ])u + σ ), B R , 0) is identical with respect to the parameter λ ∈ [0, 1]. Lemma 2.3. Let λ ∈ [0, 1]. Then there exists a positive constant R independent of σ , such that no solutions u σ ∈ C T (Q T ) satisfy both the following problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = λ(m − Φ[u σ ])u σ , ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × [0, T ],
u σ (x, 0) = u σ (x, T ),
x ∈ Ω,
(x, t) ∈ Q T ,
(2.7) (2.8) (2.9)
and the condition u σ L ∞ (Q T ) = R. Proof. Like for the problem (2.2)–(2.4), we can also define the strong generalized solution of the problem (2.7)–(2.9). Let u σ be the strong generalized solution of the problem (2.7)–(2.9). Multiplying u σ by Eq. (2.7), and then integrating over Q T , we have
Q. Zhou et al. / Nonlinear Analysis 66 (2007) 442–453
449
p−2 ∂u σ dtdx − uσ div((|∇u σ |2 + σ ) 2 ∇u σ )u σ dtdx ∂t QT QT p−2 ∂u σ uσ (|∇u σ |2 + σ ) 2 |∇u σ |2 dtdx = dtdx + ∂t Q QT T 2 = λ(m − Φ[u σ ])u σ dtdx.
QT
By the periodicity of u σ , we obtain ∂u σ dtdx = 0. uσ ∂t QT Since
QT
then
(|∇u σ |2 + σ )
p−2 2
|∇u σ |2 dtdx ≥ 0,
QT
λ(m − Φ[u σ ])u 2σ dtdx ≥ 0.
Setting M = supx∈Ω ,t ∈[0,T ] m(x, t), and combining with the assumption (H1), we can see that λ(m − Φ[u σ ])u 2σ dtdx 0≤ Q T ≤ (m − Φ[u σ ])u 2σ dtdx Q T ≤ Mu 2σ dtdx − (u 2σ Φ[u σ ])dxdt QT
QT
≤ M QT
that is
T
u 2σ dtdx − C 2
Ω
0
u 2σ dx
T
Ω
0
T
dt ≤ C
2 u 2σ dx
Ω
0
dt,
u 2σ dx dt,
(2.10)
where C is a constant independent of λ and σ . Furthermore, utilizing Cauchy’s inequality, we have 2 1 u 2σ dx ≤ 2 + ε2 u 2σ dx , 4ε Ω Ω that is
T 0
Ω
u 2σ dxdt ≤
1 + ε2 4ε2
T
0
2
Ω
u 2σ dx
Combining (2.10) with (2.11), we have u σ L 2 (Q T ) ≤ C, where C is a constant independent of λ and σ .
dt.
(2.11)
450
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By (2.2)–(2.4), we have p |u σ | dtdx ≤ QT
p
QT
|∇u σ | dtdx ≤ max |m(x, t)| QT
QT
|u σ |2 dtdx.
The assumption (H1) implies that there exists a constant K > 0, such that u σ satisfies p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) ≤ K u σ . ∂t
Therefore by the Young inequality and Theorem 3.2 in [4] (page 121) with δ = p, q = and κ = Np , we have the following estimate u σ (x, t) ≤ C
3T 2 T 2
12
Ω
p(N+2) N
u 2σ dxdt
,
for t ∈ [ T2 , 3T 2 ]. Then by the periodicity of u σ , we have u σ L ∞ (Q T ) ≤ C, where C is independent of λ and σ . So the proof is completed if R > C.
Utilizing the above result, and the result in [3] with the case λ = 0, we can apply the homotopy invariant of Leray–Schauder degree and discuss the topological degree of the problem (2.2)–(2.4) in B R . Lemma 2.4. There exists a constant R, such that deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0) = 1. Proof. Using Lemma 2.3, and the existence and uniqueness of the solution of the problem (2.2)–(2.4) in the case λ = 0, we apply the homotopy invariant of Leray–Schauder degree and have 1 = deg(u σ , B R , 0) = deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0).
Now we discuss the topological degree of the problem (2.2)–(2.4) in Br . Due to the fact that the problem (2.2)–(2.4) has the solution u σ = 0, we cannot deal with the problem directly. So for all given constants γ ≥ 0, and a smooth function θ = θ (x) > 0 in Ω , we introduce a map u σ = G((m − Φ[v])v + ) + γ G(θ ). By the theory of Leray–Schauder degree, we can see that when γ ≥ 0 and θ > 0, the map u σ = G((m − Φ[v])v + ) + γ G(θ ) is the homotopic map of the map u σ = G((m − Φ[v])v + ), that is for all θ > 0, the topological degree deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) is identical with respect to γ ≥ 0. Lemma 2.5. For a given smooth function θ = θ (x) > 0 in Ω , and a constant γ > 0, all the solutions u σ ∈ C T (Q T ) of the following problem p−2 ∂u σ − div((|∇u σ |2 + σ ) 2 ∇u σ ) = (m − Φ[u σ ])u σ + γ θ, ∂t u σ (x, t) = 0, (x, t) ∈ ∂Ω × [0, T ], x ∈ Ω, u σ (x, 0) = u σ (x, T ),
(x, t) ∈ Q T ,
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satisfy u σ L ∞ (Q T ) > r > 0, where r is a constant independent of σ, γ and θ . Proof. Like for Lemma 2.2, we can find a constant r independent of σ, γ and θ , such that when 0 ≤ u σ L ∞ (Q T ) ≤ r , it would contradict the condition (H2). Noticing the fact that after introducing the map u σ = G((m − Φ[v])v + ) + γ G(θ ), u σ = 0 is not the solution of the problem, we can get the following lemma. Lemma 2.6. There exists a constant r , such that deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) = 0. Proof. Using Lemma 2.5, and the homotopy invariant of Leray–Schauder degree, we have 0 = deg(u σ − G((m − Φ[u σ ])u + σ ) − γ G(θ ), Br , 0) = deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0). The proof is completed.
Summing up the results above, we can see when Φ[u σ ] and m satisfy the conditions (H1) and (H2), there exist constants r, R > 0, such that deg(u σ − G((m − Φ[u σ ])u + σ ), B R , 0) = 1, deg(u σ − G((m − Φ[u σ ])u + σ ), Br , 0) = 0, that is deg(u σ − G((m − Φ[u σ ])u + σ ), Σ , 0) = 1. By the theory of Leray–Schauder degree, we can conclude that the problem (2.2)–(2.4) has a non-trivial periodic solution in Σ . Then by Lemma 2.1, we can see that the problem (2.2)–(2.4) has a nonnegative non-trivial periodic solution in Q T . Finally, we can get our main result Theorem 2.1. The problem (1.1)–(1.3) has a nonnegative non-trivial periodic solution u. Proof. Using Lemma 2.2 and the proof of Lemma 2.3, we know that all the solutions u σ ∈ Σ of the problem (2.2)–(2.4) satisfy r < u σ L ∞ (Q T ) < R, where r and R are all constants which are independent of σ . In the proof of Lemma 2.2, we have ∇u σ L ∞ (Q T ) ≤ C, where C is independent of σ . Combining this with the conditions (2.3) and (2.4), it follows that 1, p
u σ ∈ L p (0, T ; W0 (Ω )) ∩ C T (Q T ).
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σ In order to get the uniform estimate of ∂u ∂t , we multiply (u σ )t by Eq. (2.2), and integrate it over Q T . By the periodicity of u σ , the condition (H1) and Cauchy’s inequality, (u σ )2t dtdx QT = (m − Φ[u σ ])u σ (u σ )t dtdx − A(∇u σ )∇u σ ∇(u σ )t dtdx
QT
QT
holds. Noticing the periodicity of u σ , we have A(∇u σ )∇u σ ∇(u σ )t dtdx QT
T
= 0
∂ ∂t s
Ω
B(∇u)dx dt = 0,
where B(s) = 0 A(ξ )ξ dξ . So (u σ )2t dtdx QT = (m − Φ[u σ ])u σ (u σ )t dtdx − A(∇u σ )∇u σ ∇(u σ )t dtdx QT QT = (m − Φ[u σ ])u σ (u σ )t dtdx QT 1 1 ≤ ((m − Φ[u σ ])u σ )2 dtdx + (u σ )2t dtdx. 2 2 QT QT By the boundedness of Φ[ω] and u σ , we have (u σ )2t dtdx ≤ C, QT
with constant C independent of σ . So we can get the uniform estimate of ∂u σ ≤ C. ∂t 2 L (Q T )
∂u σ ∂t
,
By the above estimates of u σ , and noticing that the constant C is independent of σ , we can conclude there exist a subsequence {u σi } in Σ and u ∈ Σ , which satisfy weak in L p (Q T ), |∇u σi | p−2 ∇u σi |∇u| p−2 ∇u, ∂u σi ∂u , weak in L 2 (Q T ), ∂t ∂t in C(Q T ). u σi → u, It is easy to see that u is the nonnegative non-trivial periodic solution of the problem (1.1)–(1.3). The proof is completed. Acknowledgments This work was partially supported by the National Science Foundation of China (Grant Nos 10401006 and 10426018), partially supported by a Specific Foundation for Ph.D. Specialities of
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the Educational Department of China and partially supported by the 985 project. The authors would like to thank the referees for their valuable suggestions for the revision of the manuscript. References [1] W. Allegretto, P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal term, IMA J. Math. Control Inform. 16 (1) (1999) 43–58. [2] Y. Wang, J. Yin, Z. Wu, Periodic solutions of evolution p-Laplacian equations with nonlinear sources, J. Math. Anal. Appl. 219 (1) (1998) 76–96. [3] Z.Q. Wu, Q.N. Zhao, J.X. Yin, H.L. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001. [4] E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [5] Y. Wang, Z. Wu, J. Yin, Periodic solutions of evolution p-Laplacian equations with weakly nonlinear sources, Int. J. Math. Game Theory Algebra 10 (1) (2000) 67–77. [6] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [7] A. Calsina, C. Perello, Equations for biological evolution, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 939–958. [8] A. Calsina, C. Perello, J. Saldana, Non-local reaction–diffusion equations modelling predator–prey coevolution, Publ. Mat. 32 (1994) 315–325. [9] J. Crema, J.L. Boldrini, More on forced periodic solutions of quasi-parabolic equations, Cadernos de Matem´atica 01 (2000) 71–88. [10] J.L. Boldrini, J. Crema, On forced periodic solutions of superlinear quasi-parabolic problems, Electron. J. Differential Equations 1998 (14) (1998) 1–18. [11] J. Crema, J.L. Boldrini, Periodic solutions of quasilinear equations with discontinuous perturbations, Cadernos de Matem´atica 01 (2000) 53–69. [12] J. H´uska, Periodic solutions in superlinear parabolic problems, Acta Math. Univ. Comenianae LXXI (1) (2002) 19–26.