Introduction and main results This article concerns quasilinear elliptic equations of the form div g(x,|∇u|) ∇u |∇u

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

HARNACK’S INEQUALITY FOR QUASILINEAR ELLIPTIC EQUATIONS WITH GENERALIZED ORLICZ GROWTH

MARIA A. SHAN, IGOR I. SKRYPNIK, MYKHAILO V. VOITOVYCH

Abstract. We prove Harnack’s inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth con- ditions. Our approach covers new cases of variable exponent and (p, q) growth conditions.

1. Introduction and main results This article concerns quasilinear elliptic equations of the form

div

g(x,|∇u|) ∇u

|∇u|

= 0, x∈Ω, (1.1)

where Ω is a bounded domain inRⁿ,n≥2.

Throughout this article we assume that the function g(x,v) : Ω×R+ → R+, R+= [0,+∞) satisfies the following assumptions:

(A1) g(·,v)∈L¹(Ω) for all v∈R+,g(x,·) is continuous and non-decreasing for almost allx∈Ω, lim_v→+0g(x,v) = 0 and lim_v→+∞g(x,v) = +∞;

(A2) there exist c1>0,q >1 andb0≥0 such that g(x,w)

g(x,v) ≤c1

w v

q−1

(1.2) for allx∈Ω and for all w≥v> b₀;

(A3) there existsp >1 such that g(x,w)

g(x,v) ≥w v

p−1

(1.3) for allx∈Ω and for all w≥v>0;

(A4) for anyK >0 and for any ball B8r(x0)⊂Ω there existsc2(K)>0 such that

g(x₁,v/r)≤c₂(K)e^λ(r)g(x₂,v/r)

for allx₁, x₂∈B_r(x₀) and for all r≤v≤K. Hereλ(r) : (0, r_∗)→R+ is a continuous, non-increasing function, satisfying the conditions described below.

2010Mathematics Subject Classification. 35B65, 35D30, 35J60, 49N60.

Key words and phrases. Elliptic equation; non-standard growth; non-log condition;

bounded solution; Harnack inequality.

c

2021 Texas State University.

Submitted September 7, 2020. Published April 7, 2021.

1

The following functions defined on Ω×R+ satisfy assumptions (A1)–(A4):

g(x,v) = v^p(x)−1+ v^q(x)−1, g(x,v) = v^p(x)−1 1 + ln(1 + v)

, g(x,v) = v^p−1+a(x)v^q−1, a(x)≥0, g(x,v) = v^p−1 1 +b(x) ln(1 + v)

, b(x)≥0,

where the exponents p, q, p(·), q(·), and the coefficients a(·) and b(·) satisfy the following conditions:

(i) 1< p < p(x)≤q(x)< q <+∞for allx∈Ω;

(ii)

|p(x)−p(y)|+|q(x)−q(y)| ≤ λ(|x−y|)

ln|x−y|

, x, y∈Ω, x6=y, (1.4) the functionλ(r)/|lnr|is non-decreasing on (0, r_∗), lim_r→0λ(r)/|lnr|= 0;

(iii)

|a(x)−a(y)| ≤A|x−y|^αe^λ(|x−y|), x, y ∈Ω, x6=y, (1.5) A >0, 0< q−p≤α≤1, the functionr^αe^λ(r)is non-decreasing on (0, r_∗), lim_r→0r^αe^λ(r)= 0;

(iv)

|b(x)−b(y)| ≤ B e^λ(|x−y|)

ln|x−y|

, x, y∈Ω, x6=y, B >0, (1.6) the functione^λ(r)/|lnr|is non-decreasing on (0, r_∗), lim_r→0e^λ(r)/|lnr|= 0.

The study of regularity of minima of functionals with non-standard growth of (p, q)-type was initiated by Zhikov [35, 36, 37, 38, 40], Marcellini [22, 23] and Lieberman [21]. In the last thirty years, the qualitative theory of second order equations with so-called “log-condition” (i.e. if 0 ≤ λ(r) ≤ L < +∞) has been actively developed; see, for instance, [1, 2, 4, 5, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 25, 33]. These classes of equations have numerous applications in physics and have been attracted attention for several decades; see [7, 28, 34] and references therein.

The case when conditions (1.4), (1.5), (1.6) hold differs substantially from the log-case. To the best of our knowledge there are only a few results in this direction.

Zhikov [39] obtained a generalization of the logarithmic condition which guarantees the denseness of smooth functions in a Sobolev spaceW^1,p(x)(Ω). Particularly, this result holds if 1< p≤p(x) and

|p(x)−p(y)| ≤L|ln|ln|x−y| | |

|ln|x−y| | , x, y∈Ω, x6=y, L < p/n.

In the case when the variable exponentp(x) satisfies the condition

|p(x)−p(x₀)| ≤Lln ln ln|x−x₀|⁻¹ ln|x−x0|⁻¹ , 0< L < p/(n+ 1), x, x0∈Ω, |x−x0|<1/27,

(1.7) Alkhutov and Krasheninnikova [3] proved the continuity of solutions to thep(x)- Laplace equation at the pointx₀, and Surnachev [31] established the Harnack in- equality for solutions. The continuity of solutions to thep(x)-Laplace equation up

to the boundary were proved by Alkhutov and Surnachev [6] under the additional condition

Z

0

exp

−Cexp βλ(r)dr

r = +∞, (1.8)

where C and β are some positive constants, depending only upon the data. We note that the functionλ(r) =Lln ln lnr⁻¹,r∈(0, e^−e),Lβ <1, satisfies condition (1.8).

In [30], we attempted to systematize and unify the approach to establish the local regularity of bounded solutions of elliptic and parabolic equations with non- standard growth. For this, we have introduced elliptic and parabolic B1 classes, which generalize the well-known B_p classes (p >1) of De Giorgi, Ladyzhenskaya, Ural’tseva [20] and cover their other numerous and scattered analogues (see ref- erences in [30]). It was proved in [30] that functions from the B_1,g,λ(Ω) class are continuous if conditions (A2), (A4) and (1.8) are fulfilled. In addition, if con- dition (A3) is fulfilled, then the solutions of (1.1) belong to the B_1,g,λ(Ω) class.

At the same time, we do not use the specific properties of the generalized Orlicz and Sobolev-Orlicz spaces, as was done, for example, in the papers of Harjulehto, H¨ast¨o et al [16, 17, 18, 19]. Although it should be noted that in the case when 0 ≤ λ(r) < L < +∞, the assumptions (A2)–(A4) are almost equivalent to the conditions (aDec)^∞_q , (A1-n), (aInc)p from their papers.

Returning to our paper [30], we note that there are no Harnack-type theorems in it. Although, such type results were obtained in [1, 2, 4, 5, 27] in the log- case and in [31] under condition (1.7). Therefore, it is natural to conjecture that the Harnack inequality holds for bounded solutions of (1.1) under the conditions (A1)–(A4). In this paper, we give a positive answer to this hypothesis. This also encompasses the classic results of Moser [26], Serrin [29], Trudinger [32] and Di Benedetto & Trudinger [15] for bounded solutions in the standard growth case, and of course, we use some of the ideas of Moser and Trudinger in our proofs.

Before formulating the main results, let us remind the reader the definition of a weak solution to (1.1). Moreover, throughout the article, we use the well-known notation for sets inRⁿ, spaces of functions and their elements, etc. (see, e.g. [20]).

In particular, we will use the notation –R

Ef dx=|E|⁻¹R

Ef dxfor any measurable set E ⊂ Rⁿ with |E| 6= 0 and f ∈ L¹(E), where |E| denotes the n-dimensional Lebesgue measure ofE. We set

G(x,v) =g(x,v)v forx∈Ω,v≥0 (1.9) and writeu∈W^1,G(Ω) if u∈W^1,1(Ω) andR

ΩG(x,|∇u|)dx <+∞;u∈W_loc^1,G(Ω) if u ∈ W^1,G(E) for any open set E compactly embedding in Ω. We denote by W₀^1,G(Ω) the set of all functionsu∈W^1,G(Ω) which have a compact support in Ω.

Definition 1.1. We say that a function u: Ω → R is a bounded weak solution (subsolution, supersolution) to (1.1) if u ∈ W_loc^1,G(Ω)∩L^∞(Ω) and the integral equality (inequality)

Z

Ω

g(x,|∇u|) ∇u

|∇u|∇ϕ dx= (≤,≥) 0 (1.10) holds for anyϕ∈W₀^1,G(Ω) (for subsolutions and supersolutions, we requireϕ≥0).

We refer to the parametersM = ess sup_Ω|u|,n,p,q,c₁,c₂(M) as our structural data, and we writeγ if it can be quantitatively determined a priori only in terms

of the above quantities. The generic constantγ may vary from line to line. The main result of this paper reads as follows.

Theorem 1.2(weak Harnack inequality). Fix a pointx0∈Ωand consider the ball B8ρ(x0)⊂Ω. Let ube a non-negative bounded weak supersolution to (1.1) under conditions(A1)–(A4). Then for any0< s < n/(n−1) it holds:

– Z

B_5ρ/4(x₀)

g^s

x0,u+ 2(1 +b0)ρ ρ

dx^1/s

≤Λ(γ,3n, ρ)g

x₀,m(ρ) + 2(1 +b₀)ρ ρ

,

(1.11)

where m(ρ) = ess inf_Bρ(x0)u andΛ(c, β, ρ) = exp cexp βλ(ρ)

for any c, β ∈R andρ∈(0, r_∗).

Corollary 1.3. Let u be a non-negative bounded weak solution to (1.1) under conditions (A1)–(A4), and let ρ0 be a sufficiently small positive number such that B8ρ₀(x0)⊂Ω. There exist positive numbers c,β depending only on the data such that ifΛ(c, β, r)≤ ³₂Λ(c, β,2r)for all0< r≤ρ/2< ρ0/2, and additionally

Z

0

Λ(−c, β, r)dr

r = +∞ and lim

r→0rΛ(c, β, r) = 0,

then the solutionuis continuous atx0. Particularly, the functionλ(r) =Lln ln lnr⁻¹, r∈(0, e^−e), satisfies the above conditions if0< L <1/β.

Theorem 1.4 (Moser-type sup-estimate of solutions). Fix a point x0 ∈ Ω and consider the ballB8ρ(x0)⊂Ω. Let conditions(A1)–(A4)be fulfilled, and let ube a non-negative bounded weak solution to (1.1),M(ρ) = ess sup_Bρ(x0)u. Then

g

x0,M(ρ) + 2(1 +b₀)ρ ρ

≤γe^2nλ(ρ)– Z

B_5ρ/4(x₀)

g

x0,u+ 2(1 +b₀)ρ ρ

dx. (1.12) From Theorems 1.2 and 1.4 we arrive at the following theorem.

Theorem 1.5(Harnack inequality). Let all the assumptions of Theorems 1.2, 1.4 be fulfilled. Then there exist positive constantsC,c,β depending only on the data, such that

ess sup_B

ρ(x₀)u≤CΛ(c, β, ρ)

ess inf_Bρ(x0)u+ (1 +b₀)ρ

, (1.13) whereΛ(c, β, ρ)was defined in Theorem 1.2.

The rest of this article contains the proofs of Theorems 1.2 and 1.4.

2. Proof of Theorem 1.2 (weak Harnack inequality)

For proving Theorem 1.2, we need some inequalities and several lemmas. First, we note simple analogues of Young’s inequality:

g(x, a)b≤εg(x, a)a+g(x, b/ε)b ifε, a, b >0, x∈Ω; (2.1) g(x, a)b≤1

εg(x, a)a+ε^p−1g(x, b)b ifε∈(0,1], a, b >0, x∈Ω. (2.2) In fact, ifb≤εa, theng(x, a)b≤εg(x, a)a, and ifb > εa, then since the function v→g(x,v) is increasing we have thatg(x, a)b≤g(x, b/ε)b, which proves inequality (2.1). Using assumption (A3) by similar arguments we arrive at inequality (2.2).

Next, we set

G(x,w) = Z w

0

g(x,v)dv forx∈Ω, w>0. (2.3) Then the following inequalities hold:

G(x,w)≥γ G(x,w) for allx∈Ω, w≥2(1 +b₀), (2.4) G(x,w)≥pG(x,w) for allx∈Ω, w>0. (2.5) Indeed, ifx∈Ω and w≥2(1 +b0) then by (1.2), (1.9), and (2.3), we have

G(x,w) = Z w

0

g(x,v)dv≥ Z w

b₀

g(x,v)dv≥ g(x,w) c1w^q−1

Z w

b₀

v^q−1dv≥ 1−2^−q

c1q G(x,w), which implies (2.4). Now, let x∈Ω and w >0 be arbitrary, then by (1.3), (1.9) and (2.3) we obtain

G(x,w) = Z w

0

g(x,v)dv≤ g(x,w) w^p−1

Z w

0

v^p−1dv =1

pg(x,w)w = 1

pG(x,w), which yields (2.5).

The rest of the lemmas in this section are successive stages in the proof of Theorem 1.2. The proof follows Trudinger’s strategy [32], which we adapted to equation (1.1) under conditions (A1)–(A4).

Lemma 2.1. Let all the assumptions of Theorem 1.2 be fulfilled. Then there exists positive constantγ depending only on the known data such that

exp – Z

B2ρ(x0)

ln u+ 2(1 +b0)ρ dx

≤Λ(γ,3n, ρ)

m(ρ) + 2(1 +b0)ρ

. (2.6) Proof. We fix σ ∈ (0,1), for any ρ ≤ r < r(1 +σ) ≤ 2ρ, we take a function ζ∈C₀^∞(Br(1+σ)(x0)), 0≤ζ≤1,ζ= 1 inBr(x0) and|∇ζ| ≤(σr)⁻¹. Let

w= lnκ

u, u=u+ 2(1 +b₀)ρ, (2.7)

where the constantκis defined by the condition (w)x₀,2ρ= –R

B_2ρ(x₀)w dx= 0, i.e.

κ= exp – Z

B_2ρ(x₀)

lnu dx

. (2.8)

We test (1.10) byϕ= ^u_G(x^(w−k)+

0,u/ρ)ζ^q, (w−k)₊= max{0, w−k},k >0. Since we are dealing with bounded and non-negative solutions (supersolutions), then this and all other test functions used in the paper belong toW₀^1,G(Ω). This is a consequence of conditions (A1) and (A2) and the result of Marcus and Mizel [24, Theorem 2].

So, we have Z

A_k,r(1+σ)

G(x,|∇u|) G(x0, u/ρ)ζ^qdx +

Z

A_k,r(1+σ)

G(x,|∇u|) G(x0, u/ρ)

G(x0, u/ρ)

G(x0, u/ρ) −1 (w−k)₊ζ^qdx

≤ γ σ

Z

A_k,r(1+σ)

g(x,|∇u|) G(x₀, u/ρ)

u

ρ(w−k)+ζ^q−1dx,

whereAk,r=Br(x0)∩ {w > k}. By (2.5), the value in curly brackets is estimated from below as follows:

G(x₀, u/ρ)

G(x0, u/ρ) −1≥p−1, (2.9)

and therefore Z

A_k,r(1+σ)

G(x,|∇u|)

G(x₀, u/ρ)ζ^qdx+ (p−1) Z

A_k,r(1+σ)

G(x,|∇u|)

G(x₀, u/ρ)(w−k)+ζ^qdx

≤γ Z

A_k,r(1+σ)

g(x,|∇u|) G(x0, u/ρ)

u

σρζ(w−k)+ζ^qdx.

(2.10)

We use inequality (2.1) with a=|∇u|, b = _σρζ^u and sufficiently small ε >0, and then (2.4) with w =u/ρ, to estimate from above the right-hand side of (2.10):

γ Z

A_k,r(1+σ)

g(x,|∇u|) G(x₀, u/ρ)

u

σρζ (w−k)+ζ^qdx

≤ p−1 2

Z

A_k,r(1+σ)

G(x,|∇u|)

G(x0, u/ρ)(w−k)+ζ^qdx +γ

σ Z

A_k,r(1+σ)

g x,_σρζ^γu

g(x₀, u/ρ)(w−k)+ζ^q−1dx.

Combining this inequality and (2.10), we obtain Z

A_k,r(1+σ)

G(x,|∇u|)

G(x₀, u/ρ)ζ^qdx≤ γ σ

Z

A_k,r(1+σ)

g x,_σρζ^{γ u}

g(x₀, u/ρ)(w−k)+ζ^q−1dx. (2.11) Since _σρζ^γu ≥ ^u_ρ ≥ b0 and |x−x0| < r(1 +σ) ≤2ρfor x ∈ A_k,r(1+σ), then using conditions (A2) and (A4), we get that for allx∈A_k,r(1+σ), it holds

g x, γ u

σρζ

≤γ(σζ)^1−qg(x, u/ρ)≤γ(σζ)^1−qe^λ(ρ)g(x0, u/ρ). So, from (2.11) we obtain

Z

A_k,r(1+σ)

G(x,|∇u|)

G(x0, u/ρ)ζ^qdx≤γ σ^−qe^λ(ρ) Z

A_k,r(1+σ)

(w−k)+dx. (2.12) To estimate the term on the left-hand side of (2.12), we use (2.1) with ε= 1, a=u/ρ, b=|∇u|, assumption (A4), the definitions of the functionsG, G, w(see equalities (1.9), (2.3) and (2.7), respectively) and (2.5):

Z

A_k,r(1+σ)

|∇w|ζ^qdx= Z

A_k,r(1+σ)

|∇u|

u

g(x, u/ρ) g(x, u/ρ)ζ^qdx

≤1

ρ|Ak,r(1+σ)|+1 ρ

Z

A_k,r(1+σ)

G(x,|∇u|) G(x, u/ρ)ζ^qdx

≤1

ρ|A_k,r(1+σ)|+γe^λ(ρ) ρ

Z

A_k,r(1+σ)

G(x,|∇u|) G(x0, u/ρ)ζ^qdx.

(2.13)

Collecting (2.12) and (2.13), we obtain Z

A_k,r(1+σ)

|∇w|ζ^qdx≤ γ e^2λ(ρ) σ^qρ

|A_k,r(1+σ)|+ Z

A_k,r(1+σ)

(w−k)+dx .

From this, using Sobolev’s embedding theorem and standard iteration arguments (see, for instance [20, Section 2, Theorem 5.3]), and choosingkfrom the condition

k≥γ e^2nλ(ρ) – Z

B_2ρ(x₀)

|w|ⁿ⁻¹ⁿ dxⁿ⁻¹_n + 1, we obtain that

ess sup_Bρ(x0)w≤γ e^2nλ(ρ) – Z

B_2ρ(x₀)

|w|ⁿ⁻¹ⁿ dxⁿ⁻¹_n

+ 1. (2.14)

To estimate the right-hand side of (2.14) we use the Poincar´e inequality by our choice ofκ(see (2.8)) we have

– Z

B_2ρ(x₀)

|w|ⁿ⁻¹ⁿ dxⁿ⁻¹_n

= – Z

B_2ρ(x₀)

|w−(w)_x0,2ρ|ⁿ⁻¹ⁿ dxⁿ⁻¹_n

≤γ ρ¹⁻ⁿ Z

B_2ρ(x₀)

|∇w|dx.

(2.15)

Next, similarly to (2.13), we have Z

B_2ρ(x₀)

|∇w|dx≤ Z

B_4ρ(x₀)

|∇w|ζ^qdx

≤γρⁿ⁻¹+γe^λ(ρ) ρ

Z

B_4ρ(x₀)

G(x,|∇u|) G(x0, u/ρ)ζ^qdx,

(2.16)

here we haveζ ∈C₀^∞(B4ρ(x0)), 0≤ζ ≤1, ζ= 1 inB2ρ(x0), and|∇ζ| ≤2/ρ. In addition, testing (1.10) byϕ= _G(x^uζq

0,u/ρ), similarly to (2.12), we obtain Z

B_4ρ(x₀)

G(x,|∇u|)

G(x0, u/ρ)ζ^qdx≤γρⁿe^λ(ρ). (2.17) Now, collecting (2.14)–(2.17) and taking into account (2.7) and (2.8), we arrive at the required inequality (2.6). The proof is complete.

Lemma 2.2. Under the assumptions of Theorem 1.2 there exists δ0 =δ0(ρ) >0 depending only on the data andρ, such that

– Z

B_3ρ/2(x0)

u+ 2(1 +b0)ρδ0

dx^1/δ0

≤Λ(γ,2n, ρ) exp – Z

B_2ρ(x₀)

ln u+ 2(1 +b0)ρ dx

.

(2.18)

Proof. Let us fix σ ∈ (0,1) and for any 3ρ/2 ≤ r < r(1 +σ) ≤2ρ consider the functionζ ∈C₀^∞ B_r(1+σ)(x0)

, 0≤ζ ≤1, ζ = 1 in Br(x0), |∇ζ| ≤ (σr)⁻¹. We define

v= lnu+ 2(1 +b0)ρ

κ = lnu

κ, vµ= max{v, µ}, µ >0.

Testing (1.10) byϕ= ^v

s−1 µ uζ^l

G(x0,u/ρ),s≥1,l≥q, and using (2.9), we have (p−1)

Z

B_r(1+σ)(x0)

G(x,|∇u|)

G(x0, u/ρ)v_µ^s−1ζ^ldx

≤(s−1) Z

B_r(1+σ)(x0)∩{v>µ}

G(x,|∇u|)

G(x₀, u/ρ)v^s−2_µ ζ^ldx

+γ l Z

B_r(1+σ)(x0)

g(x,|∇u|) G(x₀, u/ρ)

u

σρζ v_µ^s−1ζ^ldx.

Choosing µ from the condition _µ^s = ^p−1₂ and using inequalities (2.1), (2.4) and conditions (A2) and (A4) similarly to the derivation of (2.12), from the previous we obtain

Z

B_r(1+σ)(x₀)

G(x,|∇u|)

G(x0, u/ρ)v_µ^s−1ζ^ldx≤ γ l^γe^λ(ρ) σ^q

Z

B_r(1+σ)(x₀)

v_µ^s−1ζ^l−qdx. (2.19) Estimating the term on the left-hand side of (2.19), similarly to (2.13), we obtain

Z

B_r(1+σ)(x₀)

|∇vµ|v^s−1_µ ζ^ldx≤ Z

B_r(1+σ)(x₀)

|∇u|

u v_µ^s−1ζ^ldx

≤ γ l^γ σ^q

e^2λ(ρ) ρ

Z

B_r(1+σ)(x₀)

v_µ^s−1ζ^l−qdx

≤ γ l^γ σ^q

e^2λ(ρ) ρ

Z

B_r(1+σ)(x₀)

v_µ^sζ^l−qdx.

Using Sobolev’s embedding theorem, from this we have –

Z

Br(x0)

v

sn n−1

µ dx≤γs e^2λ(ρ) σ^q –

Z

B_r(1+σ)(x0)

v^s_µdx_n−1ⁿ

. (2.20)

Forj= 0,1,2, . . ., we define the sequences rj= ρ

2(3 + 2^−j), Bj=Br_j(x0), s_j= n

n−1 ^j+1

, µ_j = 2s_j

p−1, y_j = – Z

B_j

v_µ^sj

jdx.

Then inequality (2.20) can be rewritten in the form y_j+1^1/sj+1 ≤ γ2^jqsje^2λ(ρ)1/sj

y_j^1/sj, j= 0,1,2, . . . , (2.21) and by Sobolev’s inequality and (2.17), we have

y0≤γexp 2nλ(ρ) n−1

. (2.22)

From this by iteration, forj= 0,1,2, . . ., we have y_j+1^1/sj+1≤γ

Pj i=0 1

si2^q

Pj i=1 i

si

n n−1

^Pj_i=0ⁱ⁺¹_si exp

2λ(ρ)

j

X

i=0

1 s_i

y

n−1 n

0

≤γe^2nλ(ρ).

(2.23)

Letm ∈Nbe arbitrary, then there exists j ≥1 such that s_j−1< m≤sj. Using H¨older’s inequality, from (2.23) we obtain

– Z

B_3ρ/2(x₀)

v^m₊ m!dx≤–

Z

B_3ρ/2(x₀)

v_µ^m

j

m! dx≤ γ y_j^m/sj

m! ≤γ^m+1

m! e^2nmλ(ρ)≤γ^m+1e^2nmλ(ρ). Choosingδ0=δ0(ρ) from the condition

δ₀= 1

2γe^−2nλ(ρ), (2.24)

from the previous we have – Z

B_3ρ/2(x0)

(δ0v+)^m

m! dx≤γ2^−m, which implies

– Z

B_3ρ/2(x0)

e^δ0vdx≤– Z

B_3ρ/2(x0)

e^δ0v+dx≤

∞

X

m=0

– Z

B_3ρ/2(x0)

(δ0v+)^m

m! dx≤2γ.

From this, sincee^δ0v = (u/κ)^δ0 we have –

Z

B_3ρ/2(x₀)

u^δ0dx^1/δ0

≤(2γ)^1/δ0κ≤Λ(γ,2n, ρ)κ,

that together with (2.8) yields the desired inequality (2.18). This completes the

proof.

The next lemma is a simple consequence of Lemmas 2.1 and 2.2.

Lemma 2.3. Let all the assumptions of Lemma 2.2 be fulfilled, and set

δ1=δ0/(q−1), (2.25)

whereδ0 is defined by (2.24). Then

– Z

B_3ρ/2(x0)

g^δ1

x0,u+ 2(1 +b0)ρ ρ

dx1/δ1

≤Λ(γ,3n, ρ)g

x0,m(ρ) + 2(1 +b0)ρ ρ

.

(2.26)

Proof. By condition (A2) we have –

Z

B_3ρ/2(x₀)

g^δ1 x₀,^u+2(1+b_ρ ^0)ρ g^δ1 x0,^m(ρ)+2(1+b_ρ ^0)ρdx

≤1 +c^δ₁¹– Z

B_3ρ/2(x0)∩{u>m(ρ)}

u+ 2(1 +b0)ρ m(ρ) + 2(1 +b₀)ρ

δ0

dx.

By Lemmas 2.1 and 2.2 the second term on the right-hand side of this inequality is estimated from above as follows:

– Z

B_3ρ/2(x0)

u+ 2(1 +b0)ρ m(ρ) + 2(1 +b0)ρ

^δ0

dx≤Λ(γ,3n, ρ),

which completes the proof.

To complete the proof of Theorem 1.2 we need the following lemma.

Lemma 2.4 (Inverse H¨older inequality). Let the assumptions of Theorem 1.2 be fulfilled, then for allδ1≤s < n/(n−1) we have

– Z

B_5ρ/4(x0)

g^s

x0,u+ 2(1 +b0)ρ ρ

dx1/s

≤Λ(γ,2n+ 1, ρ) – Z

B_3ρ/2(x0)

g^δ1

x0,u+ 2(1 +b0)ρ ρ

dx1/δ1

.

(2.27)

Proof. We setψ(x,w) = ^G(x,w)_w forx∈Ω, w>0, and note that by (2.4) and (2.5), we have

g(x,w)≤γψ(x,w) for allx∈Ω, w≥2(1 +b0), (2.28) ψ(x,w)≤ 1

pg(x,w) for allx∈Ω, w>0, (2.29) which gives

ψ_w⁰ (x,w)≤γψ(x,w)

w for allx∈Ω, w≥2(1 +b0), (2.30) ψ⁰_w(x,w) = g(x,w)−ψ(x,w)

w ≥(p−1)ψ(x,w)

w for allx∈Ω, w>0. (2.31) We need a Cacciopoli-type inequality for negative powers of ψ(x0, u/ρ). To establish it, we fixσ∈(0,1) andr >0 such that 5ρ/4≤r < r(1 +σ)≤3ρ/2, and take a functionζ∈C₀^∞ B_r(1+σ)(x0)

, 0≤ζ≤1,ζ= 1 in Br(x0),|∇ζ| ≤(σr)⁻¹. Testing (1.10) by ϕ = ψ^−τ(x0, u/ρ)ζ^θ, 0 < τ < 1, θ ≥ q, and using (2.31), we obtain

(p−1)τ Z

B_r(1+σ)(x0)

ψ^−τ(x0, u/ρ)G(x,|∇u|) u ζ^θdx

≤γ θ σρ

Z

B_r(1+σ)(x0)

ψ^−τ(x0, u/ρ)g(x,|∇u|)ζ^θ−1dx,

which by (2.1), (A2), (A4) and (2.28) implies Z

B_r(1+σ)(x₀)

ψ^−τ(x₀, u/ρ)G(x,|∇u|) u ζ^θdx

≤ γ θ^q (στ)^q

e^λ(ρ) ρ

Z

B_r(1+σ)(x₀)

ψ^1−τ(x₀, u/ρ)ζ^θ−qdx.

(2.32)

Based on inequality (2.32), we organize Moser-type iterations for the function ψ(x₀, u/ρ). To do this, we fix 0< t < n/(n−1) andl ≥nq/(n−1), then by the Sobolev inequality and by (2.30) and (2.29), we obtain

Z

B_r(1+σ)(x₀)

ψ^t(x₀, u/ρ)ζ^ldxⁿ⁻¹_n

≤γ Z

B_r(1+σ)(x0)

∇

ψ^t(n−1)n (x₀, u/ρ)ζ^l(n−1)n dx

≤γt Z

B_r(1+σ)(x0)

ψ^{t(n−1)n −1}(x0, u/ρ)g(x0, u/ρ)

u |∇u|ζ^l(n−1)n dx +γ l

σρ Z

B_r(1+σ)(x0)

ψ^t(n−1)n (x0, u/ρ)ζ^{l(n−1)n −1}dx.

(2.33)

Using (2.1), (A4), (2.28) and (2.32) withτ = 1−t(n−1)/n andθ =l(n−1)/n, we estimate the first term on the right-hand side of (2.33) as follows:

Z

B_r(1+σ)(x₀)

ψ^{t(n−1)n −1}(x₀, u/ρ)g(x₀, u/ρ)

u |∇u|ζ^l(n−1)n dx

≤γe^λ(ρ) Z

B_r(1+σ)(x₀)

ψ^{t(n−1)n −1}(x₀, u/ρ)g(x, u/ρ)

u |∇u|ζ^l(n−1)n dx

≤γe^λ(ρ) Z

B_r(1+σ)(x0)

ψ^{t(n−1)n −1}(x0, u/ρ)G(x,|∇u|)

u ζ^l(n−1)n dx +γe^λ(ρ)

ρ Z

B_r(1+σ)(x0)

ψ^{t(n−1)n −1}(x0, u/ρ)g(x, u/ρ)ζ^l(n−1)n dx

≤γ l^q σ^q

1−t(n−1) n

−qe^2λ(ρ) ρ

Z

B_r(1+σ)(x₀)

ψ^t(n−1)n (x0, u/ρ)ζ^{l(n−1)n −q}dx.

(2.34)

Combining (2.33), (2.34), we arrive at

– Z

Br(x0)

ψ^t(x0, u/ρ)dxⁿ⁻¹_n

≤ γ l^q σ^q

1−t(n−1) n

^−q e^2λ(ρ)–

Z

B_r(1+σ)(x₀)

ψ^t(n−1)n (x₀, u/ρ)dx,

(2.35)

for 0< t <_n−1ⁿ andl≥_n−1^nq .

Now, letδ1≤s < n/(n−1), and letj be a non-negative integer such that s n−1

n ^j+1

≤δ₁≤s n−1 n

^j

. (2.36)

Setting in (2.35) l =nq, r =ri = ^ρ₄(6−2⁻ⁱ),r(1 +σ) =ri+1, Bi =Br_i(x0) and t=t_i=s ⁿ⁻¹_n ⁱ

fori= 0,1, . . . , j+ 1, we have

– Z

B_i

ψ^ti(x0, u/ρ)dx^1/ti

≤ γ2^iq

1−n−1 n s−q

e^2λ(ρ)1/ti+1 – Z

Bi+1

ψ^ti+1(x0, u/ρ)dx1/ti+1

. Iterating this relation and using H¨older’s inequality, we obtain

– Z

B_5ρ/4(x₀)

ψ^s(x0, u/ρ)dx^1/s

= – Z

B₀

ψ^t0(x₀, u/ρ)dx^1/t0

≤

j

Y

i=0

h

γ2^iqe^2λ(ρ)

1−n−1

n s−qi1/ti+1 – Z

B_j+1

ψ^tj+1(x₀, u/ρ)dx1/tj+1

≤2^qPji=0i/ti+1h

γe^2λ(ρ)

1−n−1

n s−qi^Pj_i=01/ti+1 γ–

Z

B_3ρ/2(x0)

ψ^δ1(x0, u/ρ)dx1/δ1

,

and by (2.36), (2.25) and (2.24) we have

j

X

i=0

1 t_i+1 ≤ 1

δ₁ n n−1

∞

X

i=0

n−1 n

ⁱ

= n²

δ₁(n−1),

j

X

i=0

i t_i+1 ≤j

j

X

i=0

1

t_i+1 ≤ γ(λ(ρ) + 1) δ₁ .

From this, and recalling the definition ofδ1 (see again (2.25) and (2.24)), we arrive at the required inequality (2.27). This completes the proof.

Combining Lemmas 2.3 and 2.4, we obtain that

– Z

B_5ρ/4(x₀)

g^s

x0,u+ 2(1 +b0)ρ ρ

dx^1/s

≤Λ(γ,3n, ρ)g

x0,m(ρ) + 2(1 +b0)ρ ρ

, which proves Theorem 1.2.

3. Proof of Theorem 1.4 (sup-estimate of solutions)

Let us fixσ, σ₁ ∈(0,1), ρ≤r < r(1 +σσ₁) < r(1 +σ) ≤5ρ/4, and consider a function ζ ∈ C₀^∞ B_r(1+σσ1)(x₀)

such that 0 ≤ ζ ≤ 1, ζ = 1 in B_r(x₀) and

|∇ζ| ≤(σσ1r)⁻¹. Testing (1.10) byϕ=uG^s−1(x0, u/ρ)ζ^l, s≥1,l≥max{q, s/2}, and using (2.5), we have

s Z

B_r(1+σσ

1 )(x₀)

G(x,|∇u|)G^s−1(x0, u/ρ)ζ^ldx

≤l Z

B_{r(1+σσ1 )}(x0)

g(x,|∇u|) u

σσ1ρζG^s−1(x0, u/ρ)ζ^ldx.

(3.1)

Using (2.1) with ε= _2l^s, a=|∇u|, b = _σσ^u

1ρζ, we estimate the right-hand side of (3.1) from above as follows:

l Z

B_{r(1+σσ1 )}(x₀)

g(x,|∇u|) u

σσ1ρζ G^s−1(x₀, u/ρ)ζ^ldx

≤ s 2 Z

B_{r(1+σσ1 )}(x₀)

G(x,|∇u|)G^s−1(x₀, u/ρ)ζ^ldx +l

Z

B_r(1+σσ

1 )(x₀)

g x, u

εσσ1ρζ u

σσ1ρζG^s−1(x0, u/ρ)ζ^ldx,

(3.2)

moreover, since _εσσ^u

1ρζ ≥ ^u_ρ ≥2(1 +b₀), conditions (A2), (A4), inequality (2.4) and ε=s/(2l) give the estimate

l Z

B_{r(1+σσ1 )}(x0)

g x, u

εσσ₁ρζ u

σσ₁ρζ G^s−1(x0, u/ρ)ζ^ldx

≤ c₁l ε^q−1

1 (σσ1)^q

Z

B_{r(1+σσ1 )}(x₀)

g x,u

ρ u

ρG^s−1(x₀, u/ρ)ζ^l−qdx

≤ γ l^qe^λ(ρ) (σσ1)^q

Z

B_r(1+σσ

1 )(x₀)

G^s(x0, u/ρ)ζ^l−qdx.

(3.3)

Combining (3.1), (3.2), (3.3), we obtain s

Z

B_r(1+σσ

1 )(x₀)

G(x,|∇u|)G^s−1(x0, u/ρ)ζ^ldx

≤ γl^qe^λ(ρ) (σσ1)^q

Z

B_{r(1+σσ1 )}(x0)

G^s(x0, u/ρ)ζ^l−qdx.

In turn, using this inequality, as well as (A4), (2.1) and (2.4), we deduce that Z

B_r(1+σσ

1 )(x₀)

∇

G^s(x0, u/ρ)ζ^l dx

≤ s ρ

Z

B_r(1+σσ

1 )(x₀)

G^s−1(x0, u/ρ)g(x0, u/ρ)|∇u|ζ^ldx

+ l

σσ₁ρ Z

B_{r(1+σσ1 )}(x0)

G^s(x0, u/ρ)ζ^l−1dx

≤γse^λ(ρ) ρ

Z

B_{r(1+σσ1 )}(x₀)

G(x,|∇u|)G^s−1(x₀, u/ρ)ζ^ldx + γs l

σσ1ρ Z

B_r(1+σσ

1 )(x₀)

G^s(x0, u/ρ)ζ^l−1dx

≤ γs l^q (σσ1)^q

e^2λ(ρ) ρ

Z

B_{r(1+σσ1 )}(x0)

G^s(x0, u/ρ)ζ^l−qdx.

Combining this and Sobolev’s inequality, we arrive to Z

B_r(x₀)

G^n−1sn (x0, u/ρ)dxⁿ⁻¹_n

≤ Z

B_r(1+σσ

1 )(x₀)

∇

G^s(x0, u/ρ)ζ^l dx

≤ γs l^q (σσ₁)^q

e^2λ(ρ) ρ

Z

B_{r(1+σσ1 )}(x0)

G^s(x0, u/ρ)dx.

(3.4)

Now, fori,j= 0,1,2, . . ., we define the sequences r_i,j= ρ

4(5−2⁻ⁱ) +ρ

82^−i−j, s_j = n n−1

^j

, l_j=q n n−1

^j . Let ζi,j ∈C₀^∞ Br_i,j(x0)

, 0≤ζi,j ≤1, ζi,j = 1 in Br_i,j+1(x0), |∇ζi,j| ≤ ^2i+j+4_ρ . Fori,j= 0,1,2, . . ., we also setri=r_i,∞,Mi= ess sup_B

ri(x0)uand yi,j=

– Z

B_ri,j(x0)

G^sj(x0, u/ρ)dx^1/sj

. From (3.4) we obtain

yi,j+1≤

γ2^(i+j)qe^2λ(ρ)1/sj

yi,j, i, j= 0,1,2, . . . . (3.5) We iterate inequality (3.5) with respect to j and use the fact that ri+1 = ri,0 to obtain

G

x0,M_i+ 2(1 +b₀)ρ ρ

≤γ2^iγe^2nλ(ρ)– Z

B_ri+1(x₀)

G(x0, u/ρ)dx

≤γ2^iγe^2nλ(ρ)Mi+1+ 2(1 +b0)ρ

ρ –

Z

B_ri+1(x₀)

g(x0, u/ρ)dx.

This inequality, (2.2), and (2.4) imply that, for any ε ∈(0,1) and i = 0,1,2, . . ., the following inequalities hold:

g

x0,Mi+ 2(1 +b0)ρ ρ

≤ε^p−1g

x₀,M_i+1+ 2(1 +b₀)ρ ρ

+1 εg

x0,Mi+ 2(1 +b0)ρ ρ

Mi+ 2(1 +b0)ρ Mi+1+ 2(1 +b0)ρ

≤ε^p−1g

x0,Mi+1+ 2(1 +b0)ρ ρ

+γ2^iγ

ε e^2nλ(ρ)– Z

B_5ρ/4(x0)

g(x0, u/ρ)dx.

Iterating the resulting inequality, for eachi≥1 we have g

x₀,M(ρ) + 2(1 +b₀)ρ ρ

=g

x₀,M₀+ 2(1 +b₀)ρ ρ

≤ε^i(p−1)g

x0,Mi+ 2(1 +b0)ρ ρ

+γε⁻¹e^2nλ(ρ)

i−1

X

j=0

(ε^p−12^γ)^j– Z

B_5ρ/4(x0)

g(x0, u/ρ)dx.

Finally, choosing ε from the condition ε^p−12^γ = 1/2 and passing i to infinity, we arrive at

g

x0,M(ρ) + 2(1 +b₀)ρ ρ

≤γe^2nλ(ρ)– Z

B_5ρ/4(x₀)

g

x0,u+ 2(1 +b₀)ρ ρ

dx.

This completes the proof of Theorem 1.4.

Acknowledgments. This work was supported by Ministry of Education and Sci- ence of Ukraine (project numbers 0121U109525 and 0119U100421), and by the Volkswagen Foundation project “From Modeling and Analysis to Approximation”.

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Maria A. Shan

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Gen. Batiouk str. 19, 84116 Sloviansk, Ukraine.

Vasyl’ Stus Donetsk National University, 600-richcha str. 21, 21021 Vinnytsia, Ukraine Email address:shan [email protected]

Igor I. Skrypnik

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Gen. Batiouk str. 19, 84116 Sloviansk, Ukraine.

Vasyl’ Stus Donetsk National University, 600-richcha str. 21, 21021 Vinnytsia, Ukraine Email address:[email protected]

Mykhailo V. Voitovych

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Gen. Batiouk str. 19, 84116 Sloviansk, Ukraine

Email address:[email protected]