POSITIVE PERIODIC SOLUTIONS OF THE WEIGHTED p -LAPLACIAN WITH NONLINEAR SOURCES

. In this paper we study the existence of time periodic solutions for the evolutionary weighted p -Laplacian with a nonlinear periodic source in a bounded domain containing the origin. We show that there is a critical exponent q c = q c ( α,β ) = ( N + β ) p N + α − p − 1 and a singular exponent q s = p − 1: there exists a positive periodic solution when 0 < q < q c and q (cid:54) = q s ; while there is no positive periodic solution when q ≥ q c . The case when q = q s is completely diﬀerent from the remaining case q (cid:54) = q s , the problem may or may not have solutions depending on the coeﬃcients of the equation.


(Communicated by Hirokazu Ninomiya)
Abstract. In this paper we study the existence of time periodic solutions for the evolutionary weighted p-Laplacian with a nonlinear periodic source in a bounded domain containing the origin. We show that there is a critical exponent qc = qc(α, β) = (N +β)p N +α−p − 1 and a singular exponent qs = p − 1: there exists a positive periodic solution when 0 < q < qc and q = qs; while there is no positive periodic solution when q ≥ qc. The case when q = qs is completely different from the remaining case q = qs, the problem may or may not have solutions depending on the coefficients of the equation.
1. Introduction. We are concerned with the existence of positive periodic solutions of the following weighted p-Laplacian with a nonlinear periodic source ∂u ∂t = div |x| α |∇u| p−2 ∇u + m(x, t)|x| β u q , (x, t) ∈ Ω × R, subject to the homogeneous boundary condition and the periodic condition u(x, t) = u(x, t + ω), where Ω ⊂ R N is a bounded domain with smooth boundary and 0 ∈ Ω, p > 1, α ≥ 0, β ≥ 0, q ≥ 0 are constants, m(x, t) ∈ C 1 (Ω × R) is a positive function which is ω-periodic in time with ω > 0.
Periodic parabolic problems of non-weighted case have been extensively studied since the last century. Among the earliest works of this aspect, Seidman [22] (1975) studied the related special case of (1), namely ∂u ∂t = div |∇u| p−2 ∇u + m(x, t), (x, t) ∈ Ω × R.
He established the existence of nontrivial periodic solutions for p ≥ 2 and m(x, t) ≡ 0. It is worth mentioning the work of Beltramo and Hess [3] (1984), who considered the linear case ∂u ∂t = ∆u + m(x, t)u, (x, t) ∈ Ω × R.
It was shown that only for some special m(x, t) can the equation have nontrivial periodic solutions. The nonlinear case was verified by Esteban [6] (1986) (see also [7] (1988)). Her results imply the existence of positive periodic solutions for q > 1 with N ≤ 2 or 1 < q < N N −2 with N > 2, and the nonexistence of positive periodic solutions for q ≥ N +2 N −2 with N > 2 and star-shaped domains. The gap N N −2 ≤ q < N +2 N −2 with N > 2 was filled by Quittner [20], in which he proved the existence with some restrictions on the structure of m(x, t). For the evolution p-Laplacian, Wang et al. [25] established the existence of positive periodic solutions for the case p − 1 < q < p − 1 + p N and p ≥ 2. A rather complete characterization, in terms of the parameters p > 1 and q > 0, of whether or not the positive periodic solutions exist was given by Yin et al. [27].
A special case of the steady state of equation (1) was studied by Song et al. [24] recently, where m ≡ 1 and Ω is the unit N -ball without the origin. They considered the unbounded, nonnegative, singular solutions and gave a classification of the behavior of these solutions at the isolated singularity, the origin.
To the best of our knowledge, the existence and nonexistence results of positive periodic solutions for the weighted case are not fully established, especially in the case of weighted flux. The main interest in the past decades lies in the periodic parabolic eigenvalue problem with weights in the source, where L is a linear uniformly elliptic operator, m(x, t) is a given weight function, see [5,9,10,13] and the original work of Beltramo and Hess [2,3]. We are quite interested in the case of weighted flux, in which (1) may have degeneracy or singularity due to the weight |x| α and the p-Laplacian when p = 2. We shall give a complete characterization of whether or not the positive periodic solutions exist for the parameters p > 1 and q > 0. Indeed, there is a singular exponent q s = p − 1 and a critical exponent q c = q c (α, β) = (N +β)p N +α−p − 1, such that under certain conditions: (i) the problem (1)-(3) admits positive periodic solutions for 0 < q < q c with q = q s ; (ii) the problem (1)  This paper is organized as follows. In Section 2, we discuss the sublinear case 0 ≤ q < p − 1 and establish the existence result. In Section 3, we consider the superlinear case q > p − 1, in which we will show the existence for p − 1 < q < q c and nonexistence for q ≥ q c . The last section is devoted to the singular case q = p−1.
2. The sublinear case. In this section, we shall show the existence of positive periodic solutions in the case 0 ≤ q < p − 1. Let τ ∈ R be fixed and set The weighted space L r (Ω; |x| α ) for r ≥ 1 and α ≥ 0 is defined as the set of all functions u = u(x) that are measurable on Ω and satisfy Ω u r |x| α dx < +∞, with the norm Other weighted spaces such as L r (Q ω ; |x| α ) and W 1,p (Ω; |x| α ) are defined similarly.
Denote by E and E 0 the reasonable solution spaces, namely u(x, t) = 0 for x ∈ ∂Ω in the sense of trace}. Because of the singularities of p-Laplacian and the weight |x| α , the problem (1)-(3) might not have classical solutions in general; hence we consider the following weak solutions.
Definition 2.1. A function u ∈ E is called a weak ω-periodic upper solution of the problem (1)-(3) provided that for any nonnegative function ϕ ∈ C 1 (Q ω ) with ϕ(x, t) = 0 for x ∈ ∂Ω, there holds Replacing "≥" by "≤" in the above inequalities, it follows the definition of a weak lower solution. Furthermore, if u is a weak upper solution as well as a weak lower solution, then we call it a weak solution of the problem (1)-(3).
We use the method of upper and lower solutions to show the existence of weak solutions. First we present the following comparison principle.
Proof. Define a function sequence {u n } ∞ n=0 by the following iteration scheme

2415
where u 0 = u. The existence and uniqueness of solutions for the above problem is classical, so u n is well defined. Then we have In fact, by the iteration scheme and the definition of upper and lower solutions, we have x ∈ Ω.
The comparison principle Lemma 2.2 shows that u 1 ≥ u 0 . Other order relations can be verified similarly. By the monotonicity of u n with respect to n, there exists a function u such that u n (x, t) tends to u a.e. in Q ω , and u(x, τ ) = u(x, τ + ω), u ≤ u ≤ u. Moreover, multiplying the first equation of (4) by u n , and integrating over Q ω , we have Furthermore, multiplying the first equation of (4) by ∂u n /∂t yields Qω ∂u n ∂t By the periodicity, we obtain for some constant C independent of n. It follows that for any r ≥ 1, as n → ∞, which implies that u ∈ E 0 with u ≤ u ≤ u is a bounded weak periodic solution of the problem (1)-(3). We need the following weighted Sobolev inequality, see for example [4,21]. Throughout this paper, we define Then for any ϕ ∈ W 1,p 0 (Ω; |x| α ), the following inequality holds where p < r < p * and C 0 = C 0 (N, r, p, α, β, Ω) is independent of ϕ.
In what follows, we shall show that all periodic solutions of the problem (1)-(3) are uniformly bounded in the case 0 ≤ q < p − 1. Proof. We shall give the proof in four steps.
Step 1. Let u ∈ E 0 be a solution of the problem (1)-(3). Multiplying (1) by u r with r > 0 being arbitrary, then integrating the resulting equation over Ω yields Here we have assumed the integrability of u in L q+r (Ω; |x| β ); otherwise we can pass through a regularization approach. In fact, we can multiply (1) by u r M instead of u r , and derive uniform estimates Here and in what follows we continue to use the letter C as a generic positive constant, whose particular value depends on the context and may change within the proof. Noticing that q+r p+r−1 < 1, we have Applying Lemma 2.4 again, we obtain that

It follows
since q+r p+r−1 < 1. By (6) and the arbitrariness of r > 0, there exists a t 0 ∈ (τ, τ + ω) such that Combining the last inequality with (7) and using the periodicity of u(x, t), we obtain that We further have Step 2. Multiplying (1) by (u − k) r + χ [t1,t2] (t), with k > 0 being arbitrary and χ [t1,t2] (t) the characteristic function of the interval [t 1 , t 2 ], then integrating over Ω × (t 1 , t 2 ) yields Assume that the absolutely continuous function I k (t) attains its maximum at ρ ∈ [τ, τ + ω]. Since I k (τ ) = I k (τ + ω), we can take ρ > τ . Set t 1 = ρ − ε, t 2 = ρ, with ε > 0 small enough such that t 1 > τ . Therefore, we have I k (ρ) ≥ I k (ρ − ε), which implies Letting ε → 0 + and recalling (8), we have Applying Lemma 2.4 and Hölder's inequality to the above inequality yields Step 3. If m(x, t) is independent of t, then the periodic solution of the problem (1)-(3) must be a steady state. In fact, multiplying (1) by u t and integrating over which, together with the periodicity of u(x, t), implies that Qω (9) can be interpreted as In addition, we note that Combining the last inequality with (10) yields [26] and (11), we conclude that Step 4. If m(x, t) depends on t, we have assumed that β = 0. Then (9) and Hölder's inequality yield provided that r ≥ r + 1. We also have Combining the above two inequalities yields Take r as defined in (12) for β = 0, then r > r + 1 if we choose r appropriately large since α < p, Step 3, we conclude that µ C = 0 and This completes the proof. Next we consider the eigenvalue problem which is closely related to the problem (1)-(3). We note that the eigenvalue problem (13) might not admit its principal eigenvalue nor the principal eigenfunction. In fact, the weighted embedding inequality is not valid for any C 0 independent of ϕ if α > β + p.
Lemma 2.6. Assume that the hypotheses of Lemma 2.4 hold. Then the eigenvalue problem (13) admits its first eigenvalue λ 1 and the first eigenfunction ϕ 1 (x) that satisfy Proof. By the weighted Sobolev inequality Lemma 2.4, we see that (14) holds for Similar to the well known results of non-weighted p-Laplacian eigenvalue problems, see for example [15], using the same variational approach, we can verify that the infimum can be taken as a minimum. It follows that µ is actually the first eigenvalue λ 1 of the problem (13) and there exists an eigenfunction 0 < ϕ 1 ∈ W 1,p 0 (Ω; |x| α ). Since the eigenvalue problem (13) is homogeneous with respect to ϕ, for any given r > 0, we can take ϕ 1 such that Replacing (8) by the above inequality in the proof of Theorem 2.5, then repeating Step 2 and Step 3 therein, we obtain ϕ 1 ∈ L ∞ (Ω). Proof. Choose R to be appropriately large such that Ω ⊂ B R/2 . Let λ 1 , λ 1 be the first eigenvalues of the weighted p-Laplacian with homogeneous Dirichlet boundary conditions on Ω and B R respectively, and let ϕ, ϕ with ϕ L ∞ (Ω) = ϕ L ∞ (B R ) = 1 be the eigenfunctions corresponding to λ 1 and λ 1 respectively. Precisely speaking, ϕ and ϕ satisfy Lemma 2.6 provides the existence of the first eigenvalues and the first eigenfunctions. Furthermore, there exists a constant δ > 0 such that ϕ ≥ δ for x ∈ Ω. Then Φ = εϕ with ε = (m/λ 1 ) 1/(p−1−q) is a periodic lower solution of the problem (1)-(3), and Ψ = κ ϕ with κ = (m/ λ 1 ) 1/(p−1−q) /δ is a periodic upper solution of the same problem. The monotonic dependence of the first eigenvalue with respect to the domain shows that 3. The superlinear case. In this section, we prove the existence and nonexistence results of positive periodic solutions. We begin with some Liouville type results, which will be used to show the existence of periodic solutions of the problem (1)-(3).
First, we show the asymptotic behavior of solutions of the following inequality which is a simple modification of Lemma 2.3 in [23].
Then v is a fundamental solution of −div(|x| α |∇u| p−2 ∇u) = 0, so by applying the comparison lemma (a similar version as Lemma 2.2) in the domain |x| > 2R we get (16).
Next, we prove the following nonexistence result of the weighted differential inequality −div(|x| α |∇u| p−2 ∇u) ≥ m(x)|x| β u q , the proof is similar to the proof of non-weighted cases in [16,27].
Rather than the weighted differential inequality in Lemma 3.2, we shall show that the nonexistence result can be extended to a larger range of exponents. For the Liouville type results of non-weighted differential equations or inequalities associated with the p-Laplacian, we refer the readers to [18,23] and references therein. We only prove the one dimensional case and the radially symmetric case below.
Proof. According to Lemma 3.2, we only need to consider the case p < N + α, α < β + p, and p − 1 < (N +β)(p−1) N +α−p < q < (N +β)p N +α−p − 1. Suppose that u > 0 is a weak solution for N = 1 or a radially symmetric solution for N ≥ 2 of (21). For the case when N = 1, we set r = x; while for the radially symmetric case, we may assume that u(x) = u(r) with r = |x|. Then we have The condition (N +β)(p−1) , we obtain that u is bounded in 0 < r < 1, and w is bounded in a neighborhood of s = +∞. Thus v > 0 for t ∈ R and lim Define y(t) = |v + bv| p−2 (v + bv). It follows that the pair (v, y) solves the ordinary differential system v = |y| We may assume that µ = 1, since a simple rescaling of u = κu in (21) can make change of the coefficient m 0 = κ −(q−(p−1)) m 0 . We claim that the singular dynamical system (22) admits no solution (v, y) such that v(t) > 0 for all t ∈ R and lim t→+∞ v(t) = 0. We only need to consider the half plane {(v, y); v ≥ 0}. Denote the vector field of the system (22) by (Φ 1 , Φ 2 ) and define two curves Bendixson's negative criterion shows that there exists no periodic orbit since div(Φ 1 , There are two equilibrium points: The linearization matrix at P 1 is .
We can verify that traceA = A standard analysis of the system (22) shows that v < 0 and y < 0 in the fourth quadrant. Thus, the trajectory (v(t), y(t)) denoted by Γ that satisfies v(t) > 0 for all t ∈ R and lim t→+∞ v(t) = 0 can not run through the y-axis or leave from the sink P 1 . Then the Poincaré-Bendixson theorem implies that Γ must enter the equilibrium point P 0 from the first quadrant and can only behave like one of the following cases: (i) lim can be excluded by an argument similar to the proof of Lemma 2.2 in [12]. In the case (ii), we see that Γ does not intersect Γ 1 ; otherwise, suppose the last point that Γ intersects Γ 1 before entering (0, 0) is (v(t 0 ), y(t 0 )), then v turns to positive for t < t 0 and v(t) ≤ v(t 0 ), which is a contradiction. Therefore, Γ is contained in the region G = {(v, y); v > 0, y > 0, |y| (2−p)/(p−1) y < bv}.
Similar to the proof of Lemma 2.5 in [12], we complete this proof by constructing a curve denoted by Γ 0 that does not intersect Γ, and then Γ is contained in a bounded domain, which contradicts to the case (ii). Define Γ 0 = {(v, y); y = k(v a − c), v > 0, y > 0}, where p − 1 < a < q and k, c are positive constants. The condition a > p − 1 ensures that Γ 0 intersects Γ 1 , and then Γ 0 divides G into a bounded part and an unbounded part. The normal vector of curve Γ 0 is (kav a−1 , −1). We only need to show that (kav a−1 , Since a < q, this condition is valid for appropriate constants k and c. Lemma 3.3 is sharp in the following sense: there exist bounded positive radially symmetric solutions if q = (N +β)p N +α−p −1 and α < β+p for p < N +α. A straightforward calculation shows that (21) has solutions of the form with N +α−p p−1 C p−1 = m0 (n+β)κ and κ > 0.
The Liouville type results for the non-weighted case are needed in the following proofs. Here we recall the following Lemma.

Lemma 3.4 ([23]
). Assume that q > p − 1 for p ≥ N , or p − 1 < q < N p N −p − 1 for p < N , and m(x) is an appropriately smooth function with 0 < m 0 ≤ m(x) ≤ M 0 . Then the problem   has no solution.
To prove the existence of periodic solutions, we reduce the problem of finding nontrivial solutions to the problem of establishing a priori estimates by the following lemma, which can be found in [1].
Here we would like to give an intuitive explanation of the reason why k is introduced in G. We are concerned with the operator G(0, ·) that admits an isolated zero fixed point and we try to prove the existence of another non-zero fixed point. The idea is to connect those two points together in the (k, u) space by introducing a parameter k. Now the zero fixed point of G(0, ·) is no longer isolated in the (k, u) space and the continuous component of fixed points (k, u) must intersect with the {0} × E with some positive fixed point provided that this component is a priori bounded in the (k, u) space.
Define an operator It is clear that u ≥ 0. First, we verify the compactness and continuity of the operator G.
Using (28) and a similar argument as in establishing (27), we further obtain that The compactness of the operator G follows from (27), (29) and (30). It is easy to obtain the continuity of G by a similar procedure. This completes the proof. Next, we check the conditions of G in Lemma 3.5. Proof. Let u = G(0, 0). Multiplying (24) with v = 0 and k = 0 by u, integrating over Q ω yields Qω |∇u| p |x| α dxdt = 0.
By the weighted Sobolev inequality Lemma 2.4, we see that Qω u p |x| β dxdt ≤ C Qω |∇u| p |x| α dxdt = 0, since α < β + p, which implies that u = 0 a.e. in Q ω . Next, we shall show that there exists an R > 0 such that if u = G(0, u) and sup t u(·, t) L q (Ω;|x| β ) < R, then u ≡ 0. Taking k = 0 and v = u in (24), then multiplying it by u q−q and integrating over Ω yield By virtue of Lemma 2.4, we get which means that u = 0 a.e. in Q ω . It remains to show that there exists an R < R 0 such that deg(id − G(0, ·), B(0, R), 0) = 1.
Consider the following problem where σ ∈ [0, 1]. Construct a homotopic mapping , by T (σ, v) = u. Similar to Lemma 3.6, we see that T is completely continuous. Assume that sup where R ≤ R 0 is to be determined. Multiplying the first equation of (31) by u r , integrating over Ω, we get (25) with k = 0 and m(x, t) replaced by σm(x, t), that is Similar to the proof of (26), we have where C is independent of σ, v, u and R. Furthermore, we can take r in the above inequality large as r, such that − p−1 . By the integral mean value theorem, we see that there exists a t σ ∈ [τ, τ + ω) such that Then the periodicity of u and a similar argument as Theorem 2.5 (Step 1) lead to For appropriately large r, Hölder's inequality gives T (1, ·), T (0, ·), B(0, R), 0) = 1.
This completes the proof. Now we can apply Lemma 3.5 to show the following existence result.
then the problem (1)-(3) admits at least one positive periodic solution. Proof. According to Lemma 3.5, we only need to verify the boundedness of the set L , which is shown by a contradiction argument. Suppose L is not bounded, then there exist two sequences, k n and u n , such that u n = G(k n , u n ) and which implies that The rest of the proof is rather complicated and we present an outline as follows: (i) show that k n / u n L ∞ (Qω) → 0 and u n L ∞ (Qω) → +∞ as k → ∞; (ii) find the accumulation point such that x n → x 0 and t n → t 0 as k → ∞; (iii) if x 0 = 0, rescale the time and spatial variables together with the solutions similar to the non-weighted case; (iv) if x 0 = 0, the rescaling coincides with the weights; (v) in both cases (iii) and (iv), derive energy estimates to provide locally uniform convergence to a Liouville type problem; (vi) finish the contradiction argument by utilizing the Liouville type results in both non-weighted case and weighted case.
Suppose to the contrary, there exists a constant C > 0 and a subsequence denoted by the same symbol such that u n L ∞ (Qω) /k n ≤ C. Note that if k n is bounded, then (33) shows that u n L ∞ (Qω) → ∞, which means (34). Thus, without loss of generality, we may assume that 0 < k n → +∞. Making change of variable v n = u n k n , we have For any ϕ ∈ C 1 ω (Q ω ) with ϕ| ∂Ω = 0, we have n Qω m(x, t)(v n + 1) q ϕ|x| β dxdt.
Then there exists a function w n ∈ W 1,p (Ω n ; |x(y)| α ) with w n L ∞ = 1 such that as j → ∞ (passing to a subsequence if necessary) ∇w nj ∇w n in L p (Ω n ; |x(y)| α ), w nj → w n in L r (Ω n ) for any r > 1, and m nj (y, s j ) → m n (y) locally uniformly, m n (y) ∈ C γ (Ω n ) for some 0 < γ < 1.
In fact, we note that t) is ω-periodic with respect to time. Further, according to the uniform continuous of m(x, t) on Ω × [τ, τ + ω], we find locally uniformly on Ω n as j → ∞, provided that r j → r 0 (passing to a subsequence). Since {r j } ⊂ [τ, τ + ω], we know that the above convergence of time sequence is valid. Then we obtain that Ωn |∇w n | p−2 ∇w n · ∇ϕ|x(y)| α dy = Ωn m n (w n + k n ρ −1 n ) q ϕ|x(y)| β dy.
Taking balls B R larger and larger, and repeating the argument for the subsequence w n obtained in the previous step, we get a Cantor diagonal subsequence, still denoted by w k for convenience, which converges in W 1,p loc (R N ) to a function We can now conclude that (38) is a contradiction. Indeed, thanks to the nonweighted Liouville type result Lemma 3.4, for p − 1 < q < p * (α, β) ≤ p * (0, 0), we see that (38) has no solution.
Similar to the case x 0 = 0, we repeat the argument therein by replacing x(y) = y for y ∈ Ω n and some other modifications. In fact, instead of (36), we have sup s Ωn Then (37) can be replaced by Then there exists a functionŵ ∈ W 1,p loc (R N ; |y| α ) such that, passing to a subsequence if necessary, as n → ∞ m n (y) → m(y), |x(y)| → |x 0 | uniformly on B R , ∇w n ∇ŵ in L p (B R ; |y| α ), w n →ŵ in L r (B r ) for any r > 1.
Then we have    B R |∇ŵ| p−2 ∇ŵ · ∇ϕ|y| α dy = B R m(y)ŵ q ϕ|y| β dy, for any ϕ ∈ C 1 0 (B R ), Similarly, taking balls B R larger and larger, and repeating the above argument, we get a Cantor diagonal subsequence, still denoted by w k , which converges in W 1,p loc (R N ; |y| α ) to a function w ∈ W 1,p loc (R N ; |y| α ), such that (39) Again, we can now conclude that (39) is a contradiction. Indeed, for the case q > p − 1 with p ≥ N + α and the case p − 1 < p ≤ (N +β)(p−1) N +α−p with 1 < p < N + α, the weighted Liouville type result Lemma 3.2 implies that (39) has no solution. Next, for the case (N +β)(p−1) N +α−p < q < (N +β)p N +α−p − 1 with 1 < p < N + α and N = 1, the same contradiction follows from Lemma 3.3. Last, for the case (N +β)(p−1) N +α−p < q < (N +β)p N +α−p − 1 with 1 < p < N + α and N ≥ 2, Ω = B R0 (0), we may restrain ourselves in searching for radially symmetric solutions. That is, the operator G is replaced by as G(k, v) = u, where u is the solution of the problem (24) andL q (B R0 (0); |x| β ) is the radially symmetric subset of L q (B R0 (0); |x| β ). A simple phase plane analysis similar to the proof of Lemma 3.3 shows that the maximum of u n L ∞ (Qω) = u n (x n , t n ) must be attained at x n = 0. Thus the only accumulation point of {x n } is x 0 = 0 and the functions obtained by the change of variable are also radially symmetric. Therefore, the weighted Liouville type result Lemma 3.2 in radially symmetric case implies that (39) has no solution. The above contradictions imply that k n + u n L ∞ is uniformly bounded. This completes the proof. The results of Theorem 3.8 are sharp in the sense of the following nonexistence results.
Proof. If m is independent of t, then the periodic solution of the problem (1)-(3) must be a steady state, as proved in the Step 3 of Theorem 2.5. Therefore, there is no positive solution if Ω is star-shaped.
If m < λ 1 , there exists λ with m < λ < λ 1 and a domain Ω with Ω ⊂⊂ Ω such that λ is the first eigenvalue of the weighted p-Laplacian eigenvalue problem (13) on Ω, and correspondingly, ψ is the first eigenfunction with ψ L ∞ = 1. Further, there exists a constant δ > 0 such that ψ ≥ δ for x ∈ Ω. A simple calculation shows that K ψ is an upper solution of the problem (42) for appropriately large K > 0. Since the source m(x, t)|x| β u q−1 is locally Lipschitz continuous for u ∈ (0, +∞) and Kδ ≤ K ψ ≤ K for x ∈ Ω, the comparison principle Lemma 2.2 holds. Then we have u ≤ K ψ. Let w be the solution of the following problem x ∈ ∂Ω, t > 0, u(x, 0) = K ψ(x), x ∈ Ω. (43) We conclude that w(x, t) is decreasing in t and u(x, t) ≤ w(x, t). Thus, there exists a function w(x) such that w(x) = lim t→∞ w(x, t). It follows that w(x) is a steady state of the first equation of (43) with homogeneous Dirichlet boundary condition. Clearly, we have w(x) = 0 since m(x, t) < λ 1 , which means that u(x, t) tends to 0 uniformly as t → ∞. Next, we consider the case m > λ 1 . TakeΩ ⊂⊂ Ω such that the first eigenvaluê λ of the weighted p-Laplacian eigenvalue problem (13) onΩ satisfying λ 1 <λ < m. Letφ be the first eigenfunction and φ be its zero-extension to Ω.
Denote byû the solution of the above problem. Clearly, the solution u of problem (42) is an upper solution of problem (44) onΩ. We note that there exists a function δ(T ) > 0 such that u(x, t) ≥ δ(T ) for x ∈Ω and t ∈ [0, T ]. By the comparison principle, we have u ≥û. Similar to the non-weighted result of [14], we see thatû blows up in finite time if p > 2 since m >λ, which means that u blows up in finite time. When 1 < p < 2, let u = g(t)φ, where g(t) satisfies that      g (t) = (m −λ)g p−1 (t), t > 0, g(t) > 0, t > 0, g(0) = 0.
Since u(x, t) ≥ κϕ(x) and m(x, t) >λ > λ 1 , we see that u(x, t) and κϕ(x) are upper and lower solutions of the periodic problem (45). By Lemma 2.3, the problem (45) admits a bounded positive periodic solution, which is a steady state sinceλ is a constant. It follows that the eigenvalue problem (13) admits a bounded positive eigenfunction and a eigenvalueλ > λ 1 , which is a contradiction.