Local boundedness for $p$-Laplacian with degenerate coefficients

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leq\lambda$ and its inverse $\lambda^{-1}$ are allowed to be unbounded. Assuming certain integrability conditions on $\lambda$ and $\lambda^{-1}$ depending on $p$ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $p>1$.


Introduction
In this note, we study local boundedness of weak (sub)solutions of non-uniformly elliptic quasi-linear equations of the form where Ω ⊂ R d with d ≥ 2 and a : Ω × R d → R d is a Caratheodory function.The main example that we have in mind are p-Laplace type operators with variable coefficients, that is, there exist p > 1 and A : Ω → R d×d such that a(x, ξ) = A(x)|ξ| p−2 ξ for all x ∈ Ω and ξ ∈ R d .In order to measure the ellipticity of a, we introduce for fixed p > 1 and suppose that λ and µ are nonnegative.In the uniformly elliptic setting, that is that there exists 0 < m ≤ M < ∞ such that m ≤ λ ≤ µ ≤ M in Ω, solution to (1) are locally bounded, Hölder continuous and even satisfy Harnack inequality, see e.g.classical results of Ladyzhenskaya & Ural'tseva, Serrin and Trudinger [28,35].
In this contribution, we are interested in a nonuniformly elliptic setting and assume that λ −1 ∈ L t (Ω) and µ ∈ L s (Ω) for some integrability exponents s and t.In [5], we studied this in the case of linear nonuniformly elliptic equations, that is a(x, ξ) = A(x)ξ corresponding to the case p = 2, and showed local boundedness and Harnack inequality for weak solutions of (1) provided it holds 1  s + 1 t < 2 d−1 .The results of [5] improved classical findings of Trudinger [36,37] (see also [33]) from the 1970s and are optimal in view of counterexamples constructed by Franchi et.al. in [24].In this manuscript we extend these results to the more general situation of quasilinear elliptic equation with p-growth as described above.More precisely, we show Theorem 1.Let d ≥ 2, p > 1, and let Ω ⊂ R d .Moreover, let s ∈ [1, ∞] and t ∈ (1/(p − 1), ∞] satisfy Let a : Ω × R d → R d be a Caratheodory function with a(•, 0) ≡ 0 such that λ and µ defined in (2) satisfy µ ∈ L s (Ω) and 1 λ ∈ L t (Ω).Then any weak subsolution of (1) is locally bounded from above in Ω.
Remark 1.Note that Theorem 1, restricted to the case p = 2 recovers the local boundedness part of [5,Theorem 1.1].
They proved local boundedness under the relation 1 pt + 1 qs + 1 p − 1 q < 1 d (see also [9] for related results).Considering the specific case f (x, ξ) = λ(x)|ξ| p , the result of [12] implies local boundedness of solutions to ∇ • (λ(x)|∇u| p−2 ∇u) = 0 provided λ −1 ∈ L t (Ω) and λ ∈ L s (Ω) with 1  s + 1 t < p d , which is more restrictive compared to assumption (3) in Theorem 1.It would be interesting to investigate if the methods of the present paper can be combined with the ones of [12] to obtain local boundedness for minimizer of functionals satisfying (4) assuming 1  pt Note that in the specific case s = t = ∞, this follows from [26].
The proof of Theorem 1 is presented in Section 2 and follows a variation of the well-known Moseriteration method.The main new ingredient compared to earlier works [36,12] lies in an optimized choice of certain cut-off functions -an idea that we first used in [5] for linear nonuniformly elliptic equations (see also [1,8,38] for recent applications to linear parabolic equations).As mentioned above, an example constructed in [24] shows that condition (3) is optimal for the conclusion of Theorem 1 in the case p = 2.In the second main result of this paper, we show -building on the construction of [24] -that condition (3) is optimal for the conclusion of Theorem 1 for all p ∈ (1, ∞).More precisely, we have Then there exists λ : B(0, 1) → (0, ∞) satisfying λ ∈ L s (B 1 ) and λ −1 ∈ L t (B 1 ) and an unbounded weak subsolution of satisfying the strict inequalities 1 s + 1 t > p d−1 and t t+1 p < d − 1.In particular, we see that condition (3) is sharp on the scale of Lebesgue-integrability for the conclusion of Theorem 1.We note that in the particularly interesting case p = 2 and d = 3 the construction in Theorem 2 fails in the critical case 1  s + 1 t = p d−1 , see [1] for counterexamples to local boundedness for related problems in d = 3.Let us now briefly discuss a similar but different instance of non-uniform ellipticity which is one of the many areas within the Calculus of Variations, where G. Mingione made significant contributions.Consider variational integrals (6) ˆΩ F (x, ∇u) dx, where the integrand F satisfies (p, q) growth conditions of the form which where first systematically studied by Marcellini in [29,30]; see also the recent reviews [31,32].The focal point in the regularity theory for those functionals is to obtain Lipschitz-bounds on the minimizer.Indeed, once boundedness of |∇u| is proven the unbalanced growth in (7) becomes irrelevant and there is a huge literature dedicated to Lipschitz estimates under various assumptions on F , see e.g. the interior estimates [4,6,7,19] in the autonomous case, [2,11,13,14,15,16,18,25] in the non-autonomous case, [10,17] for Lipschitz-bounds at the boundary, and also examples where the regularity of minimizer fail [3,21,30,20,23].Finally, we explain a link between functionals with (p, q)-growth and (linear) equations with unbounded coefficients.Consider the autonomous case that F (x, ξ) = F (ξ) and let u ∈ W 1,p (Ω) be a local minimizer of (6).Linearizing the corresponding Euler-Largrange equation yield (formally) Assuming (p, q)-growth with p = 2 of the form , which is the currently best known general bound ensuring Lipschitz-continuity of local minimizer of (6) -this reasoning was made rigorous in [6] for p ≥ 2 (see also [7] for the case p ∈ (1, ∞)).

Local boundedness, proof of Theorem 1
Before we prove Theorem 1, we introduce the notion of solution that we consider here.We call u a weak solution (subsolution, supersolution) of (1) in Ω if and only if u ∈ H 1,p (Ω, a) and Moreover, we call u a local weak solution of (1) in Ω if and only if u is a weak solution of (1) in Ω ′ for every bounded open set Ω ′ ⋐ Ω.Throughout the paper, we call a solution (subsolution, supersolution) of (1) in Ω a-harmonic (a-subharmonic, a-superharmonic) in Ω.
The above definitions generalize the concepts of weak solutions and the spaces H 1 (Ω, a) and H 1 0 (Ω, a) discussed by Trudinger [36,37] in the linear case, that is a(x, ξ) = A(x)ξ.We stress that the condition λ −1 ∈ L 1 p−1 (Ω) and Hölder inequality imply and thus, we have that W 1,1 (Ω) ⊂ H 1,p (Ω, a), where we use that by the same computation as above it holds u u H 1,p (Ω,a) and that by definition we have λ ≤ µ.From this, we also deduce that the elements of H 1,p (Ω, a) are strongly differentiable in the sense of [22].In particular this implies that there holds a chain rule in the following sense Remark 3. Let g : R → R be uniformly Lipschitz-continuous with g(0) = 0 and consider the composition ), and it holds ∇F = g ′ (u)∇u a.e.(see e.g.[37,Lemma 1.3]).In particular, if u satisfies u ∈ H 1,p (Ω, a) (or ∈ H 1,p (Ω, a)) then also the truncations Now we come to the local boundedness from above for weak subsolutions of (1).
Then, there exists c = c(d, p, s, t) ∈ [1, ∞) such that for any weak subsolution u of (1) and for any ball .
In the two-dimensional case, we have the following Then, there exists c = c(d, p) ∈ [1, ∞) such that for any weak subsolution u of (1) and for any ball Before we proof Theorem 3 and Proposition 1, we show that they imply the claim of Theorem 1 Proof of Theorem 1.In view of Theorem 3 and Proposition 1 it remains to show that for any weak subsolution u of (1) and for any ball < ∞.This is a consequence of Hölder inequality and the concept of weak subsolution, see Definition 1. Indeed, we have where the right-hand side is finite since u ∈ H 1,p (Ω, a) (note that λ ≤ µ by definition).
For the proof of Theorem 3, we need a final bit of preparation, namely the following optimization lemma . Lemma 1 generalizes [5, Lemma 2.1] from p = 2 to p > 1 and we provide a proof in the appendix.
Proof of Theorem 3. By standard scaling and translation arguments it suffices to suppose that B 1 ⋐ Ω and u is locally bounded in B 1  2 .Hence, we suppose from now on that B 1 ⋐ Ω.In Steps 1-4 below, we consider the case s > 1.We first derive a suitable Caccioppoli-type inequality for powers of u + (Step 1) and perform a Moser-type iteration (Steps 2-4).In Step 5, we consider the case 1 which includes the case s = 1.
Step 1. Caccioppoli inequality.Assuming B ⊂ Ω, for any cut-off function η ∈ C 1 0 (B), η ≥ 0 and any β ≥ 1, there holds For β ≥ 1, we use the weak formulation (8) with 1 Rigorously, we are a priori not allowed to test with u β .Instead, for N ≥ 1 one should modify u β by replacing u β with affine αN α−1 u − (α − 1)N β in the set u ≥ N , obtain the conclusion by testing the weak formulation with this modified function, and subsequently sends N → ∞ -for details, see [5,Page 460].
We have ´(a(x, ∇u) − a(x, ∇u + )) • ∇(η p u + ) = 0, so that we were able to replace u with u + inside a(x, •).Applying Leibniz rule we get from the previous display (10) where to simplify the notation for the rest of this proof we write u instead of u + .Using definition of for any ξ ∈ R d (in fact we use (2) for ξ = 0 and for ξ = 0 the inequality follow from the assumption a(x, 0) = 0), we can bound the r.h.s. in the last math display from above by , where in the second step we applied Hölder inequality with exponents p and p p−1 , respectively.Observe that the last term on the r.h.s.appears on the l.h.s. in (10), so that after absorbing it we get from ( 10) , which after taking the p-th power turns into Step 2. Improvement of integrability.We claim that there exists c = c(d, p, s) ∈ [1, ∞) such that for 1 2 ≤ ρ < σ ≤ 1 and α ≥ 1 it holds (11) ∇(u α ) Let η ∈ C 1 0 (B σ ), η ≥ 0, with η = 1 in B ρ .First, we rewrite the Caccioppoli inequality (9) from Step 1 as inequality for u 1+ β−1 p : Calling v := u 1+ β−1 p , we can estimate the r.h.s. with the help of Lemma 1, yielding Using Hölder inequality with exponents ( t+1 t , t + 1) and the fact that η = 1 in B ρ , we see that Using that 1 2 ≤ ρ ≤ σ ≤ 1, combination of two previous relations yields which after taking p-root turns into with α := 1 + β−1 p .
Step 4. Iteration.We iterate the outcome of Step 3.For ᾱ ≥ 1 and n ∈ N let Step 4 with α := α n has the form Using that L p approximates L ∞ as p → ∞, we see that which for ᾱ = 1 yields the desired claim where we use χ = 1 + δ and s * ≤ tp t+1 .
In particular, we see that ∇φ = 0 a.e. in B r0 , hence φ ≡ 0 and thus 2 Using that pt t+1 > d − 1, which follows from 1 + 1 t < p d−1 , we have by Sobolev embedding that sup Sr 0 u + ≤ c u + W 1, pt t+1 (Sr 0 ) for some c = c(d, p, t) > 0 which by the above choice of r 0 completes the claim.
Proof of Proposition 1.This follows exactly as in Step 5 of the proof of Theorem 3 using that for d = 2 it holds sup Sr 0 u + ≤ c u + W 1,1 (Sr 0 ) .We close this section by deriving from Theoem 3 in the case s > 1 an L ∞ − L γ estimate .
Hence, there exists where the last integral is finite since Next, we show ´B1 λ θ |∇v| p < ∞.For this we compute the gradient of v: Moreover, we compute and for later usage .
In view of ( 24), we need to show that for all γ ∈ i∈N,i≥j For γ ∈ i∈N {4 −i }, we directly observe that Moreover, the definition of η i via (20) ensures that (33) holds as an equality for all γ ∈ i∈N,i≥j { 1 2 4 −i } which finishes the argument.