Monotonicity and symmetry of positive solution for 1-Laplace equation

: In this paper we deal with a Dirichlet problem for an elliptic equation involving the 1-Laplace operator. Under suitable assumptions on the nonlinearity we show that there exists a symmetric, monotonic and positive solution via the moving plane method. We shall show a priori estimates for some positive solutions


Introduction and main results
We are interested in the symmetry and monotonicity of solutions to the problem where , Ω is a smooth bounded domain in R N , N ≥ 2, and strictly convex.The purpose of the paper is to investigate a priori estimates and symmetric properties of the solutions when the domain is assumed to have symmetric properties and f is supposed to satisfy the following conditions (H 1 ), (H 2 ) and (H 4 ).We also assume that f satisfies the following conditions (H 3 ) and (H 5 ) to use mountain pass lemma to get a nontrivial solution.(H 1 ): f : [0, +∞) is a locally Lipschitz continuous function and f (s) ≥ 0 for ∀ s ∈ [0, +∞).(H 2 ): f (s) ≤ C 1 (1 + s 1 * −1 ), for ∀ s ∈ [0, +∞), with 1 * = N N−1 and a constant C 1 > 0. (H 3 ): There exists θ > 1, and k 0 > 0 such that 0 < θF(s) ≤ s f (s), s ≥ k 0 .
(H 4 ): There exists a constant C 2 > 0 such that lim inf where F(s) = s 0 f (t)dt.(H 5 ): There exists a constant α ∈ (0, 1  N−1 ) such that We point out that the similar p-Laplace problems (p > 1) have many applications and have been studied for a long time, more precisely, Dirichlet problems for the p-Laplace operator, in Ω, u = 0, on ∂Ω. (1.2) In the case p = 2, the problem (1.2) −∆ p u = f (u) has been widely studied.Gidas and Spruck [27] prove a priori bounds for nonlinearities f for N ≥ 3 behave as a subcritical power at infinity, introducing the blow up method together with Liouville type theorems for solutions in R N .Figneiredo, Lions and Nussbaum [19] consider the existence and a priori estimates of positive solutions of the problem (1.2) when f satisfies the superlinear grow at infinity.They prove a priori bound for positive solutions of the problem (1.2) under the hypothesis lim s→∞ f (s) = 0, together with the monotonic results by Gidas, Ni and Nirenberg [28] obtained by the Alexandrov-Serrin moving plane method [37].The moving plane method has been improved and simplified by Beresticky and Nirenberg [7] with the aid of the maximum principle in small domain.With the help of the blow up procedure, Azizieh and Clément [5] prove a priori estimates for the problem (1.2) in the case of Ω being a strictly convex domain and f satisfying some suitable assumption.Damascelli and Pacella [14,15] apply the moving plane method to prove some monotonic and symmetric results for the p-Laplace equation in the singular case 1 < p < 2, also see [6,13].The results are later extended to the case p > 2 in the papers [12,17,18].Damascelli and Pardo [16] used the technique introduced in [19] that allowed to give the a priori estimates for solutions in case 1 < p < N, case p = N, and case p > N. Esposito, Montoro and Sciunzi [24] study symmetric and monotonic properties of singular positive solutions to the problem (1.2) via moving plane method under suitable assumptions on f .However, all the above mentioned papers can not deal with the case p = 1.In this paper, we can extend the case p > 1 to the case p = 1.
Obviously, the problem of ∆ 1 is different from ∆ p (p > 1).When p = 1, it is necessary to replace W 1,1 by BV, the space of functions of bounded variation.A function u ∈ L 1 (Ω) is called a function of bounded variation, whose partial derivatives in the sense of distribution are Radon measures.We point out that the space W 1,p (Ω) is reflexive, however, the space BV(Ω) is not reflexive, so that we can not follow the arguments on ∆ p .The 1-Laplace operator ∆ 1 introduces some extra difficulties and special features.The first difficulty occurs by defining the quotient Du |Du| , Du being just a Radon measure.To deal with the 1-Laplacian operator, we need the theory of pairing of L ∞ divergence measure vector fields (see the pioneering works [3,4,8]).
Demengel [21] is concerned with existence of solution in BV(Ω) to the problem [22] is devoted to the elliptic equations with 1-Laplacian operator and introduces the concept of locally almost 1-harmonic functions in Ω.The comparison principle, the first eigenvalue and related eigenfunctions for the 1-Laplacian operator are established in [22].Kawohl and Schuricht [30] consider a number of problems that are associated with the 1-Laplace operator ∆ 1 , the formal limit of the p-Laplace operator as p → 1, by investigating the underlying variational problem.Since the corresponding solution typically belongs to BV and not to W 1,1 , they have to study the minimizers of the functionals containing the total variation.In particular, they look for constrained minimizers subject to a prescribed L 1 norm which can be considered as an eigenvalue problem for the 1-Laplace operator.Degiovanni and Magrone [20] are concerned with the problem (1.3) with 3) admits a nontrivial solution by the non-standard linking methods.Salas and Segura de León [35] study the problem (1.3) with f (x, u) satisfying subcritical growth; i.e., | f (x, u)| ≤ C(1 + |u| q ) with 0 < q < 1 * − 1.They prove that for the problem (1.3) there exists at least two nontrivial solutions, one nonnegative and one nonpositive, by using known existence results for the p-Laplacian (p > 1) and considering the limit as p → 1 + .De Cicco, Giachetti, Oliva and Petitta [9] study the existence and regularity of special distributional nonnegative solutions to the boundary value singular problem (1.3) with f (x, u) = h(u)g(x).They show existence of nonnegative solutions to (1.3) with u max{1,γ} ∈ BV(Ω).These solutions are obtained as a limit as p → 1 + of nonnegative solutions of the p-Laplacian problems −∆ p u p = h(u p )g with u p = 0 on ∂Ω.We also refer to [33][34][35][36]38] for the a priori estimates and gradient estimates of solutions.In this paper we can study the monotonicity and symmetry of positive solution to the 1-Laplace problem and show the a priori estimates for the solution.
By the theory of pairing of L ∞ divergence measure vector fields, we introduce the following definition of solutions to the problem (1.1).Definition 1.1.We say that u ∈ BV loc (Ω), u > 0, is a solution to problem (1.1) if there exists a vector field z ∈ DM ∞ (Ω) with z L ∞ ≤ 1 such that where γ is the unit exterior normal on ∂Ω, and the spaces BV loc (Ω) and DM ∞ (Ω) are given in Section 2.
To state more precisely some known result about the monotonicity and symmetry of solutions of the problem (1.1), we need some notations.Let ν be a direction in R N .For a real number µ we define If µ > a(ν) then Ω ν µ is nonempty, thus we set Following [6] and [12][13][14][15][16][17][18], we observe that µ − a(ν) small then (Ω ν µ ) is contained in Ω and will remain in it, at least until one of the following occurs: (A) (Ω ν µ ) becomes internally tangent to ∂Ω. (B) T ν µ is orthogonal to ∂Ω.Let Π 1 (ν) be the set of those µ > a(ν) such that for each η < µ none of the conditions (A) and (B) holds and define Since Ω is supposed to be smooth, note that neither Π 1 (ν) nor Π 2 (ν) are empty and Π 1 (ν) ⊂ Π 2 (ν), so that µ 1 (ν) ≤ µ 2 (ν).We deal with solutions to the problem (1.1) in the sense of Definition 1.1.Our main result is stated as follows.Theorem 1.2.Let Ω be a smooth bounded domain in R N , N ≥ 2, which is strictly convex.Assume the nonlinearity f satisfies the conditions (H 1 ) − (H 5 ).Then there exists a nontrivial positive solution u to the problem (1.1) in the sense Definition 1.1, bounded in L ∞ (Ω) (i.e., u ∈ L ∞ (Ω)), and for any direction ν and for µ in the interval (a(ν), µ 1 (ν)], where a(ν) and µ 1 (ν) are given by (1.7) and (1.8) respectively.If f is locally Lipschitz continuous in the closed interval [0, +∞), the condition (1.10) holds for any µ in the interval (a(ν), µ 2 (ν)].Corollary 1.3.Let the smooth bounded domain Ω ⊂ R N , N ≥ 2, be strictly convex with respect to a direction ν and symmetric with respect to the hyperplane Assume that the nonlinearity f satisfies the conditions (H 1 ) − (H 5 ), which is locally Lipschitz continuous in the closed interval [0, +∞) and strictly positive in (0, +∞).Then there exists a nontrivial positive solution u to the problem (1.1) in the sense Definition 1.1, bounded in L ∞ (Ω), almost everywhere symmetric, i.e., u(x) = u(x ν 0 ) and nondecreasing in the ν-direction a.e. in Ω ν 0 .Remark 1.4.Since the moving plane procedure can be performed in the same way but in the opposite direction, then it is obvious that Corollary 1.3 is obtained by Theorem 1.2 (see Corollary 2.4 of [16]).

Preliminaries on BV space
Throughout this paper, Ω denotes an bounded subset of R N with Lipschitz boundary.The symbol |Ω| stands for its N dimensional Lebesgue measure and H N−1 (E) for the N − 1 dimensional Hausdorff measure of a set E ⊂ R N .An outward normal with vector γ = γ(x) is defined for H N−1 a.e.x ∈ ∂Ω.We will denote by W 1,p 0 (Ω) the usual Sobolev space, of measureable functions having weak gradient in L p (Ω; R N ) and zero trace on ∂Ω.If 1 < p < N, denote by p * = N p N−p its critical Sobolev exponent.BV(Ω) will denote the space of functions of bounded variation where Dv : Ω → R N is the distributional gradient of u.It is endowed with the norm by where BV(Ω) is a Banach space which is non-reflexive and non-separable.The notion of a trace on the boundary can be extended to functions v ∈ BV(Ω) and this fact allows us to write v| ∂Ω .Moreover, the trace defines a linear bounded operator i : BV(Ω) → L 1 (∂Ω) which is onto.By the trace, we have an equivalent norm on BV(Ω) where H N−1 denotes the N − 1 dimensional Hausdorff measure.We will often use this norm in what follows.In addition, the following continuous embeddings hold which are compact for 1 ≤ m < N N−1 (see for instance [25,41]).We denote by M(Ω) the space of Radon measures with finite total variation over Ω, by , Ω ⊂⊂ Ω}.The theory of L ∞ divergence measure vector fields is due to Anzellotti [4] and Chen and Frid [8].We define the following distribution (z, Dv) for ∀ ϕ ∈ C 1 c (Ω).In Anzellotti's theory we need some compatibility conditions, such as divz ∈ L 1 (Ω) and v ∈ BV(Ω) ∩ L ∞ (Ω) or divz a Radon measure with finite total variation and v Then the distribution (z, Dv) defined in (2.1) previously satisfies for all open set U ⊂⊂ Ω and all ϕ ∈ C 1 c (U). Lemma 2.2 ( [34,35]).The distribution (z, Dv) is a Radon measure.It and its total variation |(z, Dv)| are absolutely continuous with respect to the measure |Dv| and holds for all Borel sets B and for all open sets U such that B ⊂ U ⊂ Ω. Lemma 2.3 ( [10,11,34]).Let z ∈ DM ∞ loc (Ω) and let v ∈ BV(Ω) ∩ L ∞ (Ω).Then zv ∈ DM ∞ loc (Ω).Moreover, the following formula holds in the sense of measures It follows from Anzellotti's theory that every z ∈ DM ∞ (Ω) has a weak trace on ∂Ω of the normal component of z which is denoted by [z, γ] with γ the unit exterior normal on ∂Ω, which satisfies for all z ∈ DM ∞ (Ω) and v ∈ BV(Ω) ∩ L ∞ (Ω).Lemma 2.4 (Green formula [10,11,34]).Let z ∈ DM ∞ loc (Ω), = divz and v ∈ BV(Ω) and assume v ∈ L 1 (Ω, µ).Then vz ∈ DM ∞ (Ω) and the following holds

Weak solution to p-Laplacian problem
Let p 0 := min{θ, N N−1 }, with θ > 1 given by (H 3 ).For each 1 < p < p 0 , let us consider the following problem where Ω is a bounded smooth domain in R N , N ≥ 2, 1 < p < p 0 and f : [0, +∞) → R satisfies the conditions (H 1 ) − (H 5 ).We need the following propositions and a priori estimates of p-Laplace equation to prove Theorem 1.2.Definition 3.1.We say u p ∈ W 1,p 0 (Ω), u p ≥ 0, is a weak solution to the problem (3.1) in the sense that for ∀ ϕ ∈ W 1,p 0 (Ω).If u p ∈ W 1,p (Ω) is a weak solution of the problem (3.1) with f satisfying the critical growth, then u p ∈ C 1,α (Ω) with α ∈ (0, 1) (see [23,31,40]), so that we suppose from the beginning a C 1 regularity for the solution.Next, we recall some results on the monotonicity and estimates of solutions for the p-Laplace equation.One can refer to [1,16,19,29,32] for the proof of the following Proposition 3.2-3.7.
a continuous function which is locally Lipschitz continuous in (0, ∞) and strictly positive in (0, ∞) if p > 2. Let w ∈ C 1 (Ω) be a weak solution of (3.1).Then for any direction ν and for µ in the interval (a(ν), µ 1 (ν)], we have If f is locally Lipschitz continuous in the closed interval [0, +∞), then (3.3) holds for any µ in the interval (a(ν), µ 2 (ν)], where a(ν), µ 1 (ν) and µ 2 (ν) are given by (1.7), (1.8) and (1.9).Proposition 3.3 ( [16,19]).Let Ω be a strictly convex bounded smooth domain, and define Then the following result holds for a weak solution w ∈ C 1 (Ω) of the problem (3.1) with f satisfying the condition (H 1 ) I x is a part of a cone K x with vertex in x, where all the K x are congruent to a fixed cone K, and if where θ is given by (H 3 ).Then, λ 1 is the first eigenvalue of the operator −∆ p 0 (λ 1 ≤ λ for any eigenvalue λ), it is simple, i.e., there is only an eigenfunction up to multiplication by a constant, and it is isolated.Moreover a first eigenfunction does not change sign in Ω and by the strong maximum principle it is in fact either strictly positive or strictly negative in Ω.So we can select a unique eigenfunction φ 1 such that Ω φ p 0 1 dx = 1, and φ 1 > 0 in Ω.
The following extension of the Picone's identity for the p-Laplacian has been proved in [1].Proposition 3.5 (Picone's identity [1]).Let v 1 , v 2 ≥ 0 be differentiable functions in an open set Ω, with v 2 > 0 and p > 1. Set As a consequence we have The following extension of the Pohozaev's identity for the p-Laplacian has been given by [29].Proposition 3.6 (Pohozaev's identity for p-Laplace [29]).Let w ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), p > 1, be a weak solution of the problem where where γ is the unit exterior normal on ∂Ω.
We need also local W 1,∞ (Ω) result at the boundary.This result follows from the global estimates by Lieberman [31] extending the local interior estimates by Dibenedetto [23].Proposition 3.7 ( [16]).Let Ω be a smooth bounded domain in R N , N ≥ 2, and w ∈ C 1 (Ω) be a solution of the problem Then there exists a constant C > 0 only depending on M and δ such that Next, we will give the estimate of the solution for the problem (3.1).Theorem 3.8.If u p is a weak solution to the problem (3.1) and f satisfies the conditions (H 2 ) − (H 4 ), then u p satisfies where the constant C > 0 is not dependent on p. Proof.By 1 < p < p 0 , Proposition 3.4, Proposition 3.5 with v 2 = u p , v 1 = φ 1 and Young's inequality, we have By the condition (H 3 ), there exists a constant C 3 > 0 such that that is where k 1 = max{k 0 , 1} and k 0 is given by (H 3 ).Indeed, from (H 3 ), it holds Setting k 1 = max{k 0 , 1} and integrating the above inequality (3.7) with respect to t on the interval [k 1 , s], one has That is Considering (3.9) and s f (s) ≥ θF(s), for s ≥ k 1 , one gets the inequality (3.6).Now, taking into account (3.5), (3.6) with s = u p and Young's inequality, we get where λ 1 +|Ω| is given by (3.5) and φ p 1 dx ≤ 0 is given by the condition (H 1 ) ( f (s) ≥ 0, for all s ≥ 0) respectively, and the last inequality is given by Proposition 3.4 with Ω φ p 0 1 dx = 1 and k 1 = max{k 0 , 1} ≥ 1.By Proposition 3.3 and (3.10), for any x ∈ Ω \ Ω δ , we have that σ( inf where the constant C 5 may be depend on C 4 , σ, θ, p 0 and φ 1 by (3.11), but are independent of p. Estimate (3.11) gives the uniform L ∞ bounds near the boundary: ∃ δ > 0 and C 5 > 0 such that for ∀ u p ∈ W 1,p 0 (Ω) satisfying the problem (3.1).On the other hand, from the condition (H 2 ) and (3.12), we have ) is given by the condition (H 2 ) and Sobolev embedding.By Proposition 3.7, (3.12) and (3.13), we get where the constant C 7 > 0 is only depending on C 5 , C 6 and δ.By Proposition 3.6 (Pohozaev's identity) and (H 4 ), there exists a large enough constant as s ≥ k 2 , so that by the condition (H 2 ) and taking s = u p in (3.15) From the definitions of C 5 , C 6 , C 7 and C 8 , i.e., (3.11)-(3.14)and (3.16), we obtain that the constant C is not dependent on p.The proof of Theorem 3.8 is completed.
The following existence result holds.Theorem 3.9.Let f satisfy the conditions (H 1 ), (H 2 ), (H 3 ) and (H 5 ).Then there exists a nontrivial positive solution u p to the problem (3.1).Proof.By the conditions (H 1 ), (H 2 ), (H 3 ) and (H 5 ), it is well known that there exists a nontrivial solution u p ≥ 0 to the problem (3.1).The positive solution u p is obtained using the mountain pass lemma by Ambrosetti and Rabinowitz [2] for the following truncated functional J + p : W 1,p 0 (Ω) → R given by where We claim that J + p satisfies the structure of mountain pass lemma and the (P − S ) condition.Indeed, by the condition (H 5 ), 0 is a local minimum of J + p .From the condition (H 3 ), there exist two constants C, C > 0, such that for all s ∈ [0, +∞) with θ > 1.This implies that for ∀ w ∈ W 1,p 0 (Ω).We can choose a w 0 ∈ W 1,p 0 (Ω) and w 0 W 1,p 0 = 1 such that as t → +∞, with 1 < p < p 0 := min{θ, N N−1 }.Whence there exists a large number t 0 > 0 such that We set e := t 0 w 0 ∈ W 1,p 0 (Ω).Since (H 2 ) and the embedding Considering (3.21) and (H 3 ), J + p satisfies the (P − S ) condition.

The proof of Theorem 1.2
In this section we prove our main results concerning the case p = 1, namely Theorem 1.2.Under the same assumption of Theorem 1.2, we divide the proof into few steps.
Step 1. Existence of a solution u and a field z.
Step 4. The monotonicity of solution u.
Step 1. Existence of a solution u for the problem (1.1) and existence of a field z ∈ DM ∞ (Ω) satisfying (1.4) and z L ∞ ≤ 1. Proof of Step 1: From Theorem 3.8, we obtain that u p is bounded in u p → u strongly in L m (Ω), (4.1) as p → 1 + .Next, we will show that there exists a vector field z satisfying (1.4).Recalling Theorem 3.8, we obtain that {u p } is bounded in W 1,p 0 (Ω) ⊂ BV(Ω).So that for 1 ≤ r < p = p p−1 , we have and thus where the constant C 8 is given by (3.16).This implies that |∇u p | p−2 ∇u p is bounded in L r (Ω; R N ) with respect to p. Then there exists z r ∈ L r (Ω; R N ) such that as p → 1 + .A standard diagonal argument shows that there exists a unique vector field z which is defined on Ω independently of r, such that as p → 1 + .By applying the semicontinuity of the L r norm the previous inequality (4.4) implies Using ϕ ∈ C 1 c (Ω) with ϕ ≥ 0 as a test function in (3.1), we have Taking p → 1 + in the left hand side of (4.7) and by (4.6), we get lim for ∀ ϕ ∈ C 1 c (Ω).On the other hand, thanks to (4.2) and f (s) a locally Lipschitz continuous function, we have f (u p (x)) → f (u(x)), a.e.x ∈ Ω.
Before proving (z, Du) = |Du|, we need the following lemma for which one can refer to [9].Lemma 4.1 ( [9]).Under the same assumptions of Theorem 1.2, the following identity holds for By Young's inequality and Fatou's Lemma, we estimate the first integral term in (4.12) On the other hand, by (4.6) we have lim and the Dominated Convergence Theorem, we obtain the right hand side of (4.12) is as follows From (4.12)-(4.15),we have By (4.16) and Lemma 4.1, we also have Therefore, by (2.1), we get The arbitrariness of ϕ implies that |Du| ≤ (z, Du) as measures in Ω.On the other hand, since z L ∞ ≤ 1, and as measures in Ω, we have |Du| = (z, Du).
Step Since u p ∈ W 1,p 0 (Ω) is bounded, by the fact that u p = 0 on ∂Ω and Young's inequality, we get We use the lower semicontinuity (4.18) to pass to the limit as p → 1 + and obtain where the last equality is given by the Dominated Convergence Theorem and Step 4. The monotonicity of the solution u of problem (1.1).
Step 5.The boundedness of the solution u, i.e., u ∈ L ∞ (Ω).Before proving u ∈ L ∞ (Ω), we need to prove the following lemma.Lemma 4.2.For every ε > 0 there exists k 3 > 0 which does not depend on p, such that for every k ≥ k 3 and ∀ p ∈ (1, p 0 ), with Proof of Lemma 4.2: Using Sobolev embedding W 1,p 0 (Ω) ⊂ BV(Ω) → L N N−1 (Ω), Theorem 3.8 and Holder's inequality, we obtain that where S 1 is given by the best Sobolev constant see [26,39]  From (4.28), for ∀ ε > 0, ∃ k 5 > 0 large enough (not depend on p) and δ > 0 small enough such that as k ≥ k 5 , we have |A k | < δ and From (4.26) and (4.29), we obtain for all k ≥ k 3 := max{k 4 , k 5 }.The proof of Lemma 4.2 is completed.Proof of Step 5: Next, we would like to use Stampacchia truncation [38] to prove the boundedness of the positive solution u.For every k > 0, we define the auxiliary function By Lemma 4.2 and taking ε = 1 (2C 1 S 1 ) N , there exists k 3 > 0 which does not depend on p, such that for all k ≥ k 3 and p ∈ (1, p 0 ).Consequently, from (4.32) and (4.33) we obtain Since u p (x) → u(x) a.e.x ∈ Ω and Fatou's Lemma, we can pass to the limit on p → 1 + in (4.34), to conclude that Step 6. u is nontrivial.Proof of Step 6: For ∀ v ∈ BV(Ω), we define the functional J + : BV(Ω) → R as where F + (s) = s 0 f + (t)dt and f + is given by (3.18).We will say that v where γ is the unit exterior normal on ∂Ω.The critical points of J + coincide with solutions to the problem (1.1) in the sense Definition 1.1, for which one can refer to [9] or [35].We shall show that 0 is a local minimum of J + .Indeed, by the condition (H 5 ), there exists small enough δ > 0 such that for ∀ |s| ∈ (0, δ) and for some constant C 9 > 0 with α ∈ (0, 1 N−1 ).Moreover, by the definition of F + (s), we have where the last inequality is given by the embedding BV(Ω) → L 1+α (Ω), α ∈ (0, 1 N−1 ).Choosing a positive constant ρ < min{δ, ( 1 2C 10 ) 1 α }, we obtain for ∀ v ∈ BV(Ω) and v BV ≤ ρ.This implies that 0 is a local minimum of J + .Now, we introduce the auxiliary functional where J + p is given by (3.17 for all w ∈ W 1,p 0 (Ω) with p < p 0 < θ.Recalling the structure of mountain pass lemma in Theorem 3.9, we can deduce that there exists e = t 0 w 0 ∈ W 1,p 0 (Ω) ⊂ BV(Ω) and e BV > ρ such that J(e) < 0 by (3.20).
Obviously, the critical points of I p are identical with the critical points of J + p .Then u p given by Theorem 3.9 is a critical point of J + p , and also a critical point of I p , which implies that the critical point u p satisfies I p (u p ) = inf )
, and |A k | stands for its N dimensional Lebesgue measure.Inequality (4.25) implies that lim k→∞ |A k | = 0.It holds that for ∀ ε > 0, there exists a large number k 4 > 0 such that .30)Then, choosing G k (u p ) as a test function in (3.1), we get Ω |∇G k (u p )| p dx = Ω f (u p )G k (u p )dx.