Singular periodic solutions for the p -laplacian ina punctured domain

Abstract. In this paper we are interested in studying singular periodic solutions for the p -Laplacian in a punctured domain. We find an interesting phenomenon that there exists a critical exponent p c = N and a singular exponent q s = p -1. Precisely speaking, only if p > p c can singular periodic solutions exist; while if 1 p ≤ p c then all of the solutions have no singularity. By the singular exponent q s = p -1, we mean that in the case when q = q s , completely different from the remaining case q ≠ q s , the problem may or may not have solutions depending on the coefficients of the equation.


Introduction.
Let Ω be a bounded domain of R N containing the origin with smooth boundary. We are concerned with the existence of positive periodic solutions of the following evolutionary p-Laplacian in the punctured domain ∂u ∂t − div |∇u| p−2 ∇u = m(x, t)u q , x ∈ Ω , t ∈ R, with the homogeneous Dirichlet boundary condition and the periodicity condition where Ω = Ω\{0}, and p > 1 and q ≥ 0 are constants. At the singular boundary point x = 0, an additional boundary condition, namely may be required depending on the value of p. Here, m(x, t) and M (t) are appropriately smooth and positive functions which are ω-periodic in time with ω > 0. The problem about singular solutions is an important issue in the study of PDEs, which comes from many practical problems arising in physics, geometry, etc. Interest in the study of the removability of isolated singularities of differential equations has been aroused from various applications in mathematical physics in recent years, see [2,8,9,14,15,17] and the references therein. To the best of our knowledge, it was Serrin [14,15] who first considered the removable singularities of solutions of second-order partial differential equations having the form div A(x, u, ∇u) = B(x, u, ∇u) in a punctured domain D \ {0}, where the domain D is a connected open set in R N . Serrin's model contains the p-Laplacian as a typical form. For the case 1 < p ≤ N , Serrin showed that x = 0 is a removable singular point; while for the case p > N , he pointed out that even though the solution is bounded, x = 0 can also be an unremovable singular point, which is completely different from the case 1 < p ≤ N . In other words, it is natural to consider the problem in a punctured domain in some cases. In [2], Brezis and Véron considered the equation with N > 2, and proved that for q ≥ N N −2 any isolated singularity of (5) is removable, which is not true when 1 < q < N N −2 . Later, Vàzquez and Véron [17] extended the results of [2] by considering two types of typical differential operators, the p-Laplacian and the mean curvature operator, namely −div A(∇u) + g(·, u) = 0, with g satisfying some assumption of power-like growth. Recently, Liskevich and Skrypnik [8,9] established the best possible conditions for isolated singularities to be removable for general quasilinear equations, supposing that the lower-order terms satisfy certain nonlinear Kato-type conditions and extending the results of Brezis and Véron. However, as far as we know, there are no results on periodic solutions of parabolic equations with singularity at the isolated singular point x = 0.
The comparison of time-periodic Dirichlet boundary value problem in punctured domain with corresponding problems in regular domains is meaningful. We list the following related problems, say, problem in regular domain Ω: u(x, t) = 0, x ∈ ∂Ω, t ∈ R, u(x, t) = u(x, t + ω), x ∈ Ω, t ∈ R, problem in punctured domain Ω with boundary condition at {0}: and problem in regular domain Ω ε = Ω\B ε (0): Detailed definitions of solutions for the above problems are listed in Section 2. For the sake of convenience, the sets of positive solutions are denoted by U, U , U , U ε , with positive solutions being denoted by u, u , u , u ε , respectively. Clearly, U ⊂ U , U ⊂ U for any M (t), and u ∈ U for some M (t). The limit as ε → 0 of problems P ε varies with the exponent p. If 1 < p ≤ N , we will show that U = U . Known results show that U is empty for 1 < p < N , q ≥ N p/(N − p) − 1 and uniformly convex Ω, while U ε is nonempty for any ε > 0 and q = p − 1. Thus, generally speaking, the solutions u ε ⊂ U ε do not possess uniform estimates with respect to ε. If p > N and q = p − 1, using the Sobolev-Poincaré-type inequality and the barrier function technique, we will prove that U is nonempty and u ε converges to u for any M (t). Therefore, u ∈ U for some M (t) and u ε converges to u for those M (t).
Meanwhile, there exists infinitely many singular periodic solutions u ∈ U . Our interest here lies in seeking those periodic solutions with singularity at the isolated singular point x = 0. For this purpose, it is worth mentioning that the exponent p has a critical value p c = N in the sense that only if p > N can those solutions exist. In other words, if 1 < p ≤ N , then the problem (P ) is always wellposed with solutions having no singularity, see Theorem 2.5 below. So, throughout the subsequent discussion for singular periodic solutions, we shall always assume that p > N . Another interesting phenomenon is revealed for the other exponent q. Precisely speaking, the exponent q has a singular value q s = p − 1, since in this case, completely different from the remaining case q = q s , the problem may or may not have solutions depending on the coefficients of the equation.
The rest part of this paper is organized as follows. In Section 2, we present some preliminaries and our main results. In Section 3, we prove the existence and nonexistence for the singular exponent q = p − 1. The existence for q = p − 1 is demonstrated in Section 4. In the last section, we show the non-singularity for 1 < p ≤ N .
2. Preliminaries and the main results. We fix some terminologies and notations which will be frequently used in this paper. Let τ ∈ R be fixed and set Assume that p > 1, q ≥ 0, 0 < m ≤ m(x, t) ≤ m < ∞, and 0 < M (t) ∈ L ∞ (τ, τ +ω). Denote by E, E 0 and E 0 M the following reasonable solution spaces, E ={u ∈ L q+1 (Q ω ); u t ∈ L 2 (Q ω ), ∇u ∈ L p (Q ω )}, 376 S. JI, Y. LI, R. HUANG AND J. YIN E 0 ={u ∈ E; u ∂Ω×(τ,τ +ω) = 0}, We shall show that only in the case p > N can we impose the condition (4) for the periodic problem in the punctured domain. We first give the definitions of weak solutions for 1 < p ≤ N and p > N respectively in the following distributional sense where ϕ is a test function which will be specified later in different cases.
Theorem 2.5. For 1 < p ≤ N , there does not exist a singular periodic solution of the problem (P ) in a punctured domain.
Since the periodic problem in regular domains has been intensively studied (we refer the reader to [1,3,4,5,12,13,19,20]), we shall only consider the suppercritical case p > N in the following sections and leave the proof of Theorem 2.5 in the last section.
We will use the upper and lower solution method to show the existence of positive periodic solutions of the problem (1), (2)-(3) with (4) in the case p > N and 0 ≤ q < p − 1. Here, we first give the following definition of upper and lower solutions.
Definition 2.6. For p > N , a function u ∈ E is called a weak ω-periodic upper solution of the problem (1), (2)-(4) provided that for any nonnegative function Replacing "≥" by "≤" in the above inequalities, it follows the definition of a weak lower solution.
It is clear that for p > N , if u ∈ E is a weak upper solution as well as a weak lower solution, then u belongs to E 0 M and u is a weak periodic solution of the problem (1), (2)-(4) as defined in Definition 2.2.
The following lemma will be used later to show the existence of weak solutions.
where u 0 = u. The domain Ω is not regular, since there exists a singular boundary point 0. The existence and uniqueness of solutions for the above problem (8) can be deduced by a domain regularization approach, similar to the proof of Lemma 4.4 below. For the sake of convenience we omit the proof here. Thus, u n is well defined. Then we have The comparison principle shows that u 1 ≥ u 0 . Other order relations can be verified similarly. By the proof of the unique solvability of the problem (8) using the domain regularization approach (see the proof of Lemma 4.4 or the estimates (23) below), we see that , u L ∞ (Qω) and m. 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 , u(x, τ ) = u(x, τ + ω), u ≤ u ≤ u, and for any r ≥ 1, as n → ∞, which implies that u ∈ E 0 M with u ≤ u ≤ u is a bounded weak periodic solution of the problem (1), (2)-(4). By the boundary condition u(0, t) = M (t) ≥ min M (t) > 0, we see that there exists ε > 0 and δ > 0 such that u(x, t) ≥ ε for x ∈ B δ and t ∈ (τ, τ + ω). The strong maximum principle of p-Laplacian on the regular domain (Ω\B δ ) × (τ, τ + ω) implies that u > 0, since u ≡ 0 is not a solution. Thus, u is positive on Q ω .
3. Existence and nonexistence for q = p − 1. In this section, we consider the existence and nonexistence of singular periodic solutions for the singular case q = p − 1. We shall show that the existence of positive periodic solutions depends on the value of m(x, t).
The following Sobolev-Poincaré-type inequality will be used frequently in this paper.
where α = 1 − N p and C 0 is a constant independent of u.
For the evolutionary p-Laplacian, we present the following comparison principle and strong maximum principle.
Proof. We follow the approach of the strong maximum principle for uniformly parabolic operator [11]. Since ∂Ω is smooth, there exists a ball B R (y) ⊂ D with Define the following auxiliary function for for (x, t) ∈ K × (0, ω). Therefor, the conclusion holds for t 0 ∈ (0, ω). Since the auxiliary function v(x, t) is independent of t and u(x, t) is time periodic, it follows the conclusion for t 0 ∈ [0, ω].
Without loss of generality, a positive solution u(x) of −u = mu on (0, R) with u(0) = M and u(R) = 0 is equivalent to a trajectory starting from the positive u -axis and finishing at some point on the line u = M with u > 0 and the time where E ≥ 1 2 mM 2 and mU 2 = 2E. The range of T 1 (E) for E ≥ 1 2 mM 2 is (0, T 1 ( 1 2 mM 2 )], while the range of T 2 (E) for E ≥ 1 2 mM 2 is [T 1 ( 1 2 mM 2 ), 2T 1 ( 1 2 mM 2 )). Thus, the problem (1), (2)-(4) admits a positive periodic solution if and only if That is m < π 2 R 2 . The proof is completed.  Proof. Choose R to be sufficiently large such that Ω ⊂ B R/2 . Let λ 1 be the first eigenvalue of the p-Laplacian with the homogeneous Dirichlet boundary condition on B R , and let ϕ > 0 with ϕ L ∞ (B R ) = 1 be the eigenfunction corresponding to λ 1 . That is ϕ satisfies The solvability of the above eigenvalue problem is well known, see for example [7]. Furthermore, there exists a constant δ > 0 such that ϕ ≥ δ for x ∈ Ω. Then u ≡ 0 and u = κϕ are a pair of weak periodic lower and upper solutions to the problem (1), (2)-(4) provided Lemma 2.7 implies that the problem (1), (2)-(4) admits a periodic solution u ∈ E 0 M with u ≤ u(x, t) ≤ u.
Although the upper and lower solution method may be applicable for small M (t) L ∞ in the case of q > p − 1, we use the fixed point approach for the sake of consistency.
Lemma 4.2 (The Leray-Schauder Fixed Point Theorem). Let E be a Banach space, and T (u, σ) be a mapping from E × [0, 1] to E satisfying: (1) T is a completely continuous mapping; (2) T (u, 0) = 0, for any u ∈ E; (3) There exists a constant C > 0 such that u E ≤ C for all u ∈ E and σ ∈ [0, 1] satisfying u = T (u, σ). Then the mapping T (·, 1) has a fixed point, that is, there exists a u ∈ E such that u = T (u, 1).
Proof. For any x 1 , x 2 ∈ Ω, if the segment x 1 x 2 ⊂ Ω\{0}, then there exists a constant ε > 0 such that x 1 x 2 ⊂ Ω\B ε . Thus, Otherwise, the segment x 1 x 2 lies across 0. For simplicity, we may assume that x 1 , x 2 lie in the x 1 -axis and In the following equations we omit the variable x . Choose δ > 0 sufficiently small, then we have Letting δ → 0, by the continuity u ∈ C(Ω), we see that the conclusion holds.
For p > N , define an operator G by is the weak periodic solution of the following problem First we assert the unique solvability of the above problem (13).
Proof. The uniqueness is trivial, thus we only prove the existence. For 0 < ε < ε 0 with B ε0 ⊂⊂ Ω, consider the following regularized problem Letψ ∈ E 0 be the weak solution of the following p-Laplacian The existence and uniqueness of the above problem (15) is classical. It is clear that 0 <ψ ∈ L ∞ (Q ω ). Take ψ = κψ with κ > 0 sufficiently large so that ψ(x, t) ≥ M (t) for (x, t) ∈ B ε0 × (τ, τ + ω). We see that ψ 0 ≡ 0 and ψ are a pair of bounded lower and upper solutions of the problem (14). By means of the comparison principle, we conclude that problem (14) admits a unique weak solution u ε such that where C is independent of ε and depends on M (t) L ∞ (τ,τ +ω) , v L ∞ (Qω) and m. Now we show the uniform estimate for the C α -norm of u ε . Let ξ(x) ∈ C 1 (Ω) such that ξ(x) = 1 for x ∈ B ε0 , ξ(x) = 0 for x ∈ ∂Ω, 0 ≤ ξ(x) ≤ 1 and |∇ξ| ≤ C.
Define Ω ε = Ω\B ε and Q ε ω = (Ω\B ε ) × (τ, τ + ω). Taking as the test function in problem (14), we have By the periodicity of u ε , the first term in the above inequality satisfies that We also have Combining the last three inequalities, we find that where C is independent of ε. Taking
It is clear that u ≥ 0. Now, we verify the compactness and continuity of the operator G.
. Then the operator G is completely continuous and G(v, 0) = 0 for any v ∈ L ∞ ω ((τ, τ + ω), C α (Ω)). Proof. By the uniform estimates (17), (19) and the continuity (22), we see that for where C is a constant depends on M (t) C 1 ([τ,τ +ω]) , v L ∞ (Qω) and m. The compactness of the operator G follows from the estimate (23) and compact embedding theorems. It is easy to obtain the continuity of G by a similar procedure. The condition G(v, 0) = 0 for any v ∈ L ∞ ω ((τ, τ + ω), C α (Ω)) is easy to verify. Here we recall the following Liouville type results on the exterior domain R N \{0}.
has no solution.
It remains to verify the condition (3) in Lemma 4.2. That is, we reduce the problem of finding nontrivial solutions to the problem of establishing the following a priori estimate.
Take ϕ = w n η 2p (y), where with 0 ≤ η ≤ 1 sufficiently smooth and |∇η| ≤ C R . Then for sufficiently large n, we have ρ q−(p−1) p n |x 0 | > 2R, which yields that B 2R ⊂ Ω n . Thus, we have That is for sufficiently large R > 0. Then there exists a functionŵ ∈ W 1,p loc (R N ) such that, passing to a subsequence if necessary, as n → ∞ m n (y) →m(y), uniformly on B R , ∇w n ∇ŵ in L p (B R ), w n →ŵ in L r (B r ) for any r > 1.
Then we have Taking the ball B R larger and larger, and repeating the argument for the subsequence w n obtained at the previous step, we get a Cantor diagonal subsequence, for simplicity we still denote it by w k , which converges in W 1,p loc (R N ) to a function We can now conclude that (31) is a contradiction. Indeed, thanks to the Liouville type result, Lemma 4.6, we see that (31) has no solution. It remains to prove the case x 0 = 0, in which the rescaling coincides with the singular point 0. Similar to the case x 0 = 0, instead of Eq. (29), we have Ωn |∇w n | p−2 ∇w n · ∇ϕ dy = Ωnm n |w n | q ϕ dy, for any ϕ ∈ C 1 0 (Ω n \{0}). Taking ϕ = (w n − ρ −1 n σ nM )η 2p (y) as the test function, where η(y) is the function defined in the proof of (30) andM is the limit ofM j (s j ), we see that η 2p−1 (w n − ρ −1 n σ nM )|∇w n | p−2 ∇w n · ∇η dy + B 2Rm n |w n | q (w n − ρ −1 n σ nM )η 2p dy |w n | q w n η 2p dy That is Then there exists a functionŵ ∈ W 1,p loc (R N ) such that, passing to a subsequence if necessary, as n → ∞m n (y) →m(y), uniformly on B R , ∇w n ∇ŵ in L p (B R ), w n →ŵ in L r (B r ) for any r > 1.
Then we have    B R |∇ŵ| p−2 ∇ŵ · ∇ϕ dy = B Rm (y)ŵ q ϕ dy, for any ϕ ∈ C 1 0 (B R \{0}), Similarly, taking the ball 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 \{0}) to a function w ∈ W 1,p loc (R N \{0}) satisfying    R N |∇w| p−2 ∇w · ∇ϕ dy = R Nm (y)w q ϕ dy, for any ϕ ∈ C 1 0 (R N \{0}), w L ∞ (R N ) = 1, w > 0, y ∈ R N . (32) Again, we can conclude that (32) is a contradiction by the Liouville-type result Lemma 4.6. The above contradictions imply that u n L ∞ is uniformly bounded. Now we can apply Lemma 4.2 to show the following existence result.
Proof. This result follows from Lemma 4.7, Lemma 4.5, and the Leray-Schauder fixed point theorem Lemma 4.2.
Since u ∈ E 0 and 1 < p ≤ N , we have the following estimate