MAXIMUM AND ANTIMAXIMUM PRINCIPLES FOR THE p -LAPLACIAN WITH WEIGHTED STEKLOV BOUNDARY CONDITIONS

. We study the maximum and antimaximum principles for the p - Laplacian operator under Steklov boundary conditions with an indeﬁnite weight


Introduction
Let Ω be a bounded domain of R N of class C 2,α for some 0 < α < 1, N ≥ 1.We consider the quasilinear problem −∆ p u + |u| p−2 u = 0 in Ω, |∇u| p−2 ∂u ∂ν = λm(x)|u| p−2 u + h(x) on ∂Ω. (1.1) Here ∆ p u := div(|∇u| p−2 ∇u) is the well known p-Laplacian operator, 1 < p < ∞; m and h are given functions in C r (∂Ω) for some 0 < r < 1.The weight m can change sign, and h ≥ 0, h ≡ 0. We denote by ν = ν(x) the outer normal at x, defined for all x ∈ ∂Ω and by σ the restriction to ∂Ω of the (N − 1)-Hausdorff measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular enough.All the integral along ∂Ω will be understood with respect to the measure σ.
Problems of the form (1.1) appears in several branches of pure and applied mathematics, such as the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary, non-Newtonian fluids, reaction diffusion problems, flow through porous media, nonlinear elasticity, glaciology, etc.
The maximum and antimaximum principles for problem (1.1) with m ≡ 1, have been studied in [3].The authors proved that every solution of (1.1) is positive if λ ∈ (0, λ 1 ) (maximum principle) and that there exists δ = δ(h) > 0 such that, if λ ∈ (λ 1 , λ 1 + δ), then every weak solution is negative (antimaximum principle).Here λ 1 denotes the smallest eigenvalue of the eigenvalue problem associated with (1.1) with m ≡ 1.The authors in [3] also characterize the interval of validity of the uniform antimaximum principle.A uniform antimaximum principle has also been proved in [4,10,11] for the p-Laplacian operator with Neumann boundary conditions.
The norm • 1,p stand here for the natural norm of W 1,p (Ω).We prove in Theorem 4.4 that the real number λ1 (m) provides an interval of validity of the uniform antimaximum principle for (1.1) to the right of λ 1 (m), where λ 1 (m) is the first positive eigenvalue of the eigenvalue problem associated with (1.1).We point out here that λ1 (m) ≤ λ1 (m), where λ1 (m) is the real number found in [3] for the validity of the uniform antimaximum principle in the case m ≡ 1, given by λ1 (m) := inf u p 1,p ; ∂Ω |u| p = 1 and u vanishes in a ball of Ω .
We will also prove that λ1 (m) = λ 1 (m) if 1 < p ≤ N and λ1 (m) > λ 1 (m) if p > N .Furthermore, we prove in Theorem 4.5 that the value λ1 (m) is the greater number σ > λ 1 (m) such that the uniform antimaximun principle holds for any λ ∈ (λ 1 (m), σ).This article is organized as follows.In Section 2, we recall some basic definitions and we review some properties of the principal eigenvalues of the p-Laplacian under Steklov boundary conditions with an indefinite weight.We prove in Section 3 some results concern in maximum principle, existence of solutions and nonexistence of positive solutions for (1.1).We conclude this paper in Section 4 with some results on the antimaximum principle and on the uniformity for this principle.Our mean results of this section are Theorem 4.4 and Theorem 4.5.We finish with some example in dimension 1.
We denote by W The weak convergence will be denoted by and the strong one by →.Here we will denote by p * := N p/(N − p) + the classical critical Sobolev's exponent, and by p * := (N − 1)p/(N − p) + the critical Sobolev's exponent for the trace inclusion.We are interested in the weak solutions of (1.1), i.e., functions u ∈ W 1,p (Ω) such that holds for all v ∈ W 1,p (Ω).
Let us summarize some properties of the principal eigenvalues of the eigenvalue problem associated with problem (1.1), A real number λ is said to be an eigenvalue of (2.1) if and only if there exists u ∈ W 1,p (Ω) \ {0}, called eigenfunction associated with λ, satisfying for all v ∈ W 1,p (Ω).It is proved in [5] (see also [7] and [12] for a more general problem) that (2.1) admits two principal eigenvalues which are characterized by where I : (2.7) See section 5, where we discuss the case N = 1, and let us agree to write λ 2 (m) = +∞ (resp.λ −2 (m) = −∞ if no eigenvalues greater than λ 1 (m) (resp.λ −1 (m) ) exist.Every eigenfunction u associated with a positive (resp.negative) eigenvalue ) for some constant κ 1 > 0 independent of u, see [5].The following result is a simple consequence of the characterizations (2.3) and (2.4).Lemma 2.2.Assume that h ≥ 0, h ≡ 0. If λ ∈ (λ −1 (m), λ 1 (m)) then there exists a constant κ > 0 such that Proof.Assume, by contradiction, that there exists a sequence (u n ) n∈N * ⊂ W 1,p (Ω) with u n 1,p = 1 such that (2.9) Since u n 1,p = 1, then (u n ) n∈N * is bounded in W 1,p (Ω) and there exists a function u such that u n u in W 1,p (Ω) and strongly in L p (∂Ω).Then we obtain and then λI(u) ≥ 1.In particular I(u) = 0.It follows from (2.9) and the weak lower semicontinuity of the norm that Consequently, if I(u) > 0, it follows from (2.3) and (2.10) that which is a contradiction.If I(u) < 0, it follows from (2.4) and (2.10) that which is also a contradiction.

Maximum principle and existence of positive solutions
In this section we prove some results on maximum principle for problem (1.1), existence and uniqueness of solutions, and nonexistence of positive solutions.Remark 3.1.Let u ∈ W 1,p (Ω) be a nonnegative weak solution of (1.1) with h ≥ 0. Using Harnack's inequality (see [13, Theorems 5,6 and 9 pages 264-270]) and Hopf maximum principle (see [15]), its follows that u > 0 a.e. in Ω.
Next, let us recall Picone's identity, see [1].Let v > 0 and u ≥ 0 be two differentiable functions a.e. in Ω and denote (iii) L(u, v) = 0 in Ω if and only if u = kv for some constant k.As a consequence of these identities we have the following result.
for all bounded φ ∈ W 1,p (Ω).Moreover the equality holds if and only if φ is scalar multiplier of u.
Proof.By Picone's identity we obtain and (3.1) holds.Moreover, from assertion (iii) of Picone's identity we have the equality in (3.2) if and only |φ| = cu, for some constant c.In particular φ is of constant sign in Ω.
The following result states the maximum principle for problem (1.1) for the usual range of λ.Theorem 3.3.Assume h ≥ 0, h ≡ 0. Then the maximum principle for (1.1) Proof.Assume by contradiction that u − ≡ 0 and take v = u − as test function in (1.1).We have which implies that λ −1 (m) ≥ λ, and we have a contradiction.Hence, in all cases, we obtain that u ≥ 0 and the conclusion follows from Remark 3.1.
Let us now prove the following uniqueness result.We stress here that the existence result is well known for any h ∈ L q (∂Ω) if q > (p * ) .We give here the proof for the sake of completeness.
Proof.Let us prove that the energy functional K associated with (1.1) is coercive and weakly lower semicontinuous.Since λ ∈ (λ −1 (m), λ 1 (m)), it follows from Lemma 2.2 that where κ 1 = c h ∞ with c > 0 the constant from the embedding of W 1,p (Ω) in L 1 (∂Ω).We conclude that K is coercive.Now assume that u n is a sequence in W 1,p (Ω) such that u n u for some u in W 1,p (Ω).Then, from the compact embedding of W 1,p (Ω) into L q2 (∂Ω), for all q 2 ∈ [1, p * ) we can assume that u n → u in L p (∂Ω) and in L 1 (∂Ω).Then from the lower semicontinuity of the norm we obtain and the result follows.Since K is coercive and weakly lower semicontinuous, then the inf{K(u), u ∈ W 1,p (Ω)} is achieved, providing us with a weak solution of (1.1) (i.e. a critical point of K).
To prove the uniqueness of the solution, assume that u, v ∈ W 1,p (Ω) are two solutions of (1.1) for a fixed λ ∈ (λ −1 (m), λ 1 (m)).From Theorem 3.3 we have that u and v are positive and from Lemma 3.2 that Interchanging u and v we also have By adding (3.5) and (3.6), we have Thus from (3.7) we obtain Then it follows from the Lemma 3.2 that v = cu for some constant c > 0. Since u = v on {x ∈ ∂Ω; h(x) = 0} then c = 1 and we obtain the desired result.
Next we prove that there are no positive solutions when the parameter λ lies outside the interval (λ −1 (m), λ 1 (m)).

Antimaximum principle
In Theorem 4.4 we will prove, for the case p > N , the existence of an interval of uniformity of this principle for problem (1.1).
, then w k ∞,∂Ω = 1 and it follows (using Remark 2.1 with ) that w k lies in C 1,α (Ω).Moreover there exists a constant C > 0 such that w k C 1,α (Ω) ≤ C. Thus, there exists a function w such that, for a subsequence, w n → w in C 1 (Ω).In particular w ≡ 0 since w ∞,∂Ω = 1.Hence, passing to the limit, we obtain that w is an eigenfunction associated with the eigenvalue λ 1 (m) of (2.1).Consequently w > 0 or w < 0 in Ω.If w > 0, then for k large enough we have u k > 0 and this contradicts Theorem 3.5 (1).If w < 0, then for k large enough we have u k < 0 which contradicts the existence of x k .
Notice that, a priori, the value δ of Theorem 4.1 depends of the function h.If this is not so, we say that the antimaximum principle is uniform on (λ 1 (m), λ 1 (m) + δ).
In the following, we study the validity of the uniform antimaximum principle and we will give a variational characterization of the greatest value δ for which the uniform antimaximum principle holds in (λ 1 (m), λ 1 (m) + δ) if p > N .
Following [3,4,10] we introduce the values λ1 (m) and λ−1 (m): λ1 (m) := inf u p 1,p ; where In the following two lemmae we discuss whenever λ 1 (m) is different or equal to λ1 (m).Proof.We distinguish two cases: Case (i) p < N .We define, as in [3] or [4], the sequence of functions y k defined for all x ∈ R N by for some constant C.
Proof.We only give the proofs that concern λ1 (m).
(b) The proof is standard and uses the compact embedding of W 1,p (Ω) in C(Ω) to assure that a weak limit of any minimizing sequence must vanish somewhere on ∂Ω.
(e) Let ϕ 2 be an eigenfunction associated with λ 2 (m).By (2.8) we know that ϕ 2 vanishes somewhere on ∂Ω.Thus ϕ 2 is an admissible function in the definition (4.4) of λ1 (m) and then If λ1 (m) = λ 2 (m) then ϕ 2 would be a minimiser in (4.4) and therefore it must have a constant sign on ∂Ω, according to (c), a contradiction.
With the previous results in hand, we can give an interval where the uniform antimaximum principle holds.EJDE-2020/21 Theorem 4.4.Let p > N and let h ≥ 0, h ≡ 0. If u is a solution of (1.1)In particular I(u − ) > 0. Let us first show that u < 0 on ∂Ω.Indeed, if λ < λ1 (m), we have from (4.10) So u − ∈ A and we conclude that u − does not vanish anywhere on ∂Ω, that is, u < 0 on ∂Ω.If λ = λ1 (m) and we assume by contradiction that u − vanish somewhere on ∂Ω, hence, from the one hand u − is a minimizer for λ1 (m) according to Proposition 4.3(a) and from the other hand, using (4.10) we have We deduce from this relation that u − vanishes on the set of positive measure {x ∈ ∂Ω; h(x) > 0} which is a contradiction with Proposition 4.3(c) (minimizers of λ1 (m) vanish only once).
Next we prove that u < 0 in Ω.Since u < 0 on ∂Ω one has that u + ∈ W 1,p 0 (Ω).Take then v := u + in the weak form of (1.1) to obtain Consequently u + ≡ 0 in Ω and so u ≤ 0 in Ω.Using the well know Harnack's inequality [13,Theorem 5] we deduce that u < 0 in Ω and then u < 0 in Ω.
Finally we prove that the value λ1 (m) (resp.λ−1 (m).) is optimal in the sense that the antimaximum principle holds to the right of λ1 (m) and that the uniform antimaximum principle fails to the right of λ1 (m) + δ for any δ > 0.
Similar results can be stated to the left of λ−1 (m).
Proof.(1) We assume here that p > N as in the case 1 < p ≤ N , λ1 (m) = λ 1 (m) and the result is proved in Theorem 4.1.The proof follows the same pattern of the one in the proof of Theorem 4.1 and we just indicate the changes needed in the contradiction argument.In alternative (a), passing to the limit in (P λ k ,h ) one gets that u is a weak solution of (1.1) for λ = λ1 (m).It follows from Theorem 4.4 that u < 0 in Ω and consequently u k < 0 in Ω for k large enough (since the convergence is in C 1 (Ω)), a contradiction with the existence of x k .In alternative (b), passing to the limit we obtain that w is an eigenfunction associated with λ1 (m).Since w ∞,∂Ω = 1 then w ≡ 0 and therefore λ1 (m) is an eigenvalue of (1.1) and w an eigenfunction associated with λ1 (m), a contradiction with Proposition 4.3 (e).
Hence the only eigenvalues of the eigenvalue problem e − e −1 = β = λ1 (m).Furthermore, if h > 0 is a function defined on the boundary of Ω = (0, 1) by (5.4) Moreover, using the fact that u (t) < 0 for all t ∈ (0, Some properties of Φ p and Λ p can be found in [8,9,14]