The Robin problem for

The Robin problem for

singular p ( x ) -Laplacian equation in a cone

Mikhail Borsuk

Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, 10-957 Olsztyn-Kortowo, Poland

Received 7 August 2018, appeared 1 December 2018 Communicated by Maria Alessandra Ragusa

Abstract. We study the behavior near the boundary angular or conical point of weak solutions to the Robin problem for an elliptic quasi-linear second-order equation with the variable p(x)-Laplacian.

Keywords: p(x)-Laplacian, angular and conical points.

2010 Mathematics Subject Classification: 35J20, 35J25, 35J70.

1 Introduction

The aim of our article is the investigation of the behavior of the weak solutions to the Robin problem for quasi-linear elliptic second-order equations with the variable p(x)-Laplacian in a neighborhood of an angular or a conical boundary point of the bounded cone. Boundary value problems for elliptic second order equations with a non-standard growth in function spaces with variable exponents have been an active investigations in recent years. We refer to [7] for an overview and the recent papers [1,9,10] and reference therein. Differential equa- tions with variable exponents-growth conditions arise from the nonlinear elasticity theory, electrorheological fluids, etc. There are many essential differences between the variable expo- nent problems and the constant exponent problems. In the variable exponent problems, many singular phenomena occurred and many special questions were raised. V. Zhikov [11,12]

has gave examples of the Lavrentiev phenomenon for the variational problems with variable exponent.

Most of the works devoted to the quasi-linear elliptic second-order equations with the variable p(x)-Laplacian refers to the Dirichlet problem in smooth bounded domains (see [7]).

Concerning the Robin problem for such equations we know only a few articles [2,5,6,8], but in these works a domain is smooth and lower order terms depend only on (x,u) and do not depend on |∇u|. Our article [3] is deduced to the Robin problemin a cone for such equations with a singular p(x)-power gradient lower order term. The present article is the continuation of [3]. Here we describe qualitatively the behavior of the weak solution near

BEmail: borsuk@uwm.edu.pl

a conical point, namely we derive the sharp estimate of the type |u(x)| = O(|x|^κ) (cf. §3.1) for the weak solution modulus (for the solution decrease rate) of our problem near a conical boundary point. As well as, we establish the comparison principle for weak solutions.

The Robin boundary conditions appear in the solving Sturm–Liouville problems which are used in many contexts of science and engineering: for example, in electromagnetic problems, in heat transfer problems and for convection-diffusion equations (Fick’s law of diffusion). The Robin problem plays a major role in the study of reflected shocks in transonic flow. Important applications of this problem is the capillary problem.

LetCbe an open cone inRⁿ, n≥2, with the vertex at the origin Oand letBrbe an open ball with radiusr centered atO. We use the following standard notations:

• Sⁿ⁻¹: a unit sphere inRⁿ centered atO;

• (r,ω)_,ω = (ω₁,ω₂, . . . ,ω_n−1): the spherical coordinates ofx ∈_Rⁿ_{with pole} O_: x₁ =rcosω₁,

x₂ =rcosω₂sinω₁, ...

x_n−1 =rcosω_n−1sinω_n−2. . . sinω₁, x_n=rsinω_n−1sinω_n−2. . . sinω₁;

• Ω : a domain on the unit sphere Sⁿ⁻¹ with the smooth boundary ∂Ω obtained by the intersection of the coneCwith the sphereSⁿ⁻¹;

• ∂Ω=_∂C∩Sⁿ⁻¹;

• G₀^d ≡ _C∩B_d ={(r,ω)|0<r <d; ω∈ _Ω};

• Γ^d₀ ≡∂C∩B_d= {(r,ω)|0<r <d; ω∈ ∂Ω};

• Ωd= G₀^d∩ {|x|=d}.

We investigate the behavior in a neighborhood of the origin O of solutions to the Robin problem with the boundary condition on the lateral surface of the cone:

(−4_p(x)u+b(u,∇u) = f(x), x ∈G^d₀⁰,

|∇u|^{p(x)−2 ∂u}

∂−→

n + ^γ

|x|^p(x)−1u|u|^p(x)−2=_{0, x} ∈_Γ^d₀⁰, (RQL) where 0<d₀1 (d₀is fixed) and

4_p(x)u≡div

|∇u|^p(x)−2∇u

. (1.1)

We will work under the following assumptions:

(i) 1< p−≤ p(x)≤ p+ = p(0)<n, ∀x ∈G₀^d0;

(ii) the Lipschitz condition: p(x)∈C^0,1(G₀^d0) =⇒ 0≤ p+−p(x)≤ L|x|, ∀x ∈G^d₀⁰; where Lis the Lipschitz constant for p(x)_.

(iii) |f(_x)| ≤ f0|x|^β(x), f0 ≥ 0, β(_x) > _p^p+−1

+−1+µ(p(_x)−1)λ − p(_x)_; ∀x ∈ G^d₀⁰; γ = const ≥ 1, 0 ≤ µ < 1 andλ is the least positive eigenvalue of problem (NEVP) (see below);

(iv) the functionb(u,ξ)is differentiable with respect to theu,ξvariables inM=_R×_Rⁿand satisfy inMthe following inequalities:

(iv)_a |b(u,ξ)| ≤δ|u|⁻¹|ξ|^p(x)+b₀|u|^p(x)−1, 0≤δ< µ; ifµ>0;

(_iv)_b b(u,ξ)≥ν|u|⁻¹|ξ|^p(x)−b₀|u|^p(x)−1, ν>_{0; if}µ=_0;

(_iv)_c s n

i∑=1

^∂b

(u,ξ)

∂ξ_i

2 ≤b₁|u|⁻¹|ξ|^p(x)−1; ^∂b(_∂u^u,ξ) ≥b₂|u|⁻²|ξ|^p(x); b₀≥0, b₁≥0, b₂≥0;

(iiv) the spherical region Ω⊂Sⁿ⁻¹ is invariant with respect to rotations inSⁿ⁻². We consider the functions class

N^1,p_−1,∞^(x)(G₀^d0) = (

u

u(x)∈ L_∞(G₀^d0)and Z

G₀^d0

D|x|^−p(x)|u|^p(x)+|u|⁻¹|∇u|^p(x)Edx<_∞ )

which was introduced in [4]. It is obvious that N^1,p_−1,∞^(x)(G₀^d0)⊂W^1,p(x)(G₀^d0).

Definition 1.1. The functionuis called a weak bounded solution of problem (RQL) provided that u(x)∈N^1,p_−1,∞^(x)(G₀^d0)and satisfies the integral identity

Q(u,η):≡

Z

G₀^d0

D|∇u|^p(x)−2u_xiη_xi +b(u,∇u)η E

dx+γ Z

Γ₀^d0 r^1−p(x)u|u|^p(x)−2ηdS

−

Z

Ωd0

|∇u|^p(x)−2∂u

∂rηdΩd=

Z

G^d₀⁰ f(x)η(x)dx (II)

for all η(x)∈N^1,p_−1,∞^(x)(G₀^d0).

Remark 1.2. It is easy to verify that the above assumptions(i), (iii), (iv)ensure the existence of integrals over G₀^d andΓ^d₀. Therefore, Definition1.1is correct.

Main result is the following statement.

Theorem 1.3. Let u be a weak bounded solution of problem(RQL), M0 = sup

x∈G^d₀⁰|u(x)|(see [3]) and let λ be the least positive eigenvalue of problem (NEVP) (see Section 2). Suppose that(i)–(iiv) hold. Then there exist de∈ (_0,d₀)and a constant C₀ > ₀depending only on λ,d₀,M₀,p+,p−,L,n, (µ−δ),ν,b₀,f₀and such that

|u(x)| ≤C₀|x|^κ, κ= ^p+−1

p+−1+µλ; ∀x ∈G^d₀^e. (1.2)

2 Nonlinear eigenvalue problem

To prove the main result we shall consider the nonlinear eigenvalue problem for ψ(ω)∈C²(_Ω)∩C¹(_Ω):











−div_ω

(λ²ψ²+|∇_ωψ|²)^(p+−2)/2∇_ωψ

=λ(λ(p+−1) +n−p+) (λ²ψ²+|∇_ωψ|²)^(p+−2)/2ψ, ω∈_Ω, (λ²ψ²+|∇_ωψ|²)^{(p+−2)/2∂ψ}

∂−→ ν +γ

p+−1+µ p+−1

p+−1

·ψ|ψ|^p+−2 =0, ω∈_∂Ω,

(NEVP)

where|∇_ωψ|denotes the projection of the vector∇ψonto the tangent plane to the unit sphere at the pointω:

∇_ωψ= 1

√q₁

∂ψ

∂ω₁, . . . , 1

√qn−1

∂ψ

∂ω_n−1

,

|∇_ωψ|²=

n−1 i

∑

1 q_i

∂ψ

∂ω_i 2

, q₁=1, q_i = (sinω₁· · ·sinω_i−1)², i≥2 and−→

ν denotes the exterior normal to∂Cat points of∂Ω.

If we rename ω = ω₁, ω⁰ = (ω2, . . . ,ω_n−1), then, by assumption (iiv), we can assume that ψ(ω₁,ω⁰) does not depend on ω⁰. Therefore, our problem (NEVP) is equivalent to the following















λ²ψ²+ (p+−1)ψ⁰²

ψ⁰⁰(ω) + (n−2)cotω λ²ψ²+ψ⁰² ψ⁰(ω) +λ(λ(2p+−3) +n−p+)ψ⁰²ψ(ω)

+λ³(λ(p+−1) +n−p+)ψ³(ω) =0, ω ∈ −^ω₂⁰,^ω₂⁰ ,

±(λ²ψ²+ψ⁰²)^(p+−2)/2ψ⁰(ω) +γ _p

+−1+µ p+−1

p+−1

·ψ(ω)|ψ(ω)|^p+−2

ω=±^ω₂⁰=0.

(OEVP)

2.1 Properties of the(OEVP)eigenvalue and corresponding eigenfunction

First of all, we note that any two eigenfunctions are scalar multiples of each other if they solve problem for the sameλ. Therefore, without loss of generality we can assumeψ ^ω₂⁰

=1.

Next, we observe that the following two cases are possible: either ψ(−ω) = −ψ(ω) or ψ(−ω) = ψ(ω). In Section 2 [4], it was shown that we obtain the least positive eigenvalueλ^∗ ifψ(−ω) =ψ(ω); thenψ⁰(−ω) =−ψ⁰(ω) =⇒ ψ⁰(0) =0, ψ(0)6=0, as well as the following inequalities for eigenvalue and for the corresponding eigenfunction:

λ(λ(p+−1) +n−p+)>0; 0< λ^∗ < ^π

ω₀; (2.1)

1≤ψ(ω)≤ψ₀ =const(n,p,λ,ω₀) (2.2) hold.

Now we define the function y(ω) = ^ψ0(ω)

ψ(ω), ψ(0) 6= 0 and let y0 = y ^ω₂⁰

. From(OEVP) we obtain the Cauchy problem











(p+−1)y²+λ²

y⁰(ω) + (p+−1)y⁴+ (n−2)cotω y²+λ² y(ω)

+λ(2λ(p+−1) +n−p+)y²+λ³(λ(p+−1) +n−p+) =0, ω∈ 0,^ω₂⁰ , y(0) =0.

(CP)

and the following equation forλ: y₀

λ²+y²₀

p+−2

2 =−γ

p+−1+µ p+−₁

p+−1

. (λ)

Since

(p+−1)y⁴+λ(2λ(p+−1) +n−p+)y²+λ³(λ(p+−1) +n−p+)

= (p+−1)(y²+λ²)

y²+λ²+ ⁿ−p+

p+−1λ

,

the (CP) equation can be rewritten as follows y⁰(ω)

y²+λ²

=− (n−2)cotω (p+−1)y²+λ²

·y(ω)−(p+−1)(y²+λ²) + (n−p+)λ

(p+−1)y²+λ² , ω ∈_0,^ω0 2

. (2.3) By Lemma 2.2 [4], we have

y(ω)≤0, |y(ω)| ≤z₀ =const(n,λ,ω₀,p+), ∀ω∈ [0,ω₀/2]. (2.4) Proposition 2.1. If assumption(i)is satisfied andγ≥1(see assumption(iii)), then

κ λ

q

λ²+y²₀

p(x)−p(0)

≤1, ∀x∈_Γ^d₀, (2.5)

whereκis defined by(1.2).

Proof. We rewrite (λ) with regard to (1.2):







|y₀|= ^γλ

κ , if p+=2;

q

λ²+y²₀ =_|y^γ

0|

_p ¹

+−2

· ^λ

κ

p+−1

p+−2 , if p+6=2. (2.6) Case p+ =_2.

The inequality (2.5) is true if p(x)≡2. Now, let 1 < p(x)≤ p+ = 2, ∀x ∈ _Γ^d₀. From (λ) we have

|y₀|= γ(1+µ) = ^γλ

κ =⇒ κ λ

q

λ²+y²₀ ≥ κ

λ|y₀|=γ≥1 (p(x)−p+)ln

κ λ

q

λ²+y²₀

≤0 =⇒ (2.5)is true.

Case p+ >2.

From (λ) and (2.6) it follows that

|y₀| ≤ ^λ κ ·γ

1

p+−1 and

q

λ²+y²₀≥ ^λ κ ·γ

1

p+−1 =⇒

(p(x)−p+)ln κ

λ q

λ²+y²₀

≤ ^p(x)−p+

p+−1 lnγ≤0 =⇒ (2.5)is true.

Case p+ <_2.

From (λ) and (2.6) we obtain that

|y₀|=γ λ

κ

p+−1q

λ²+y²₀ 2−p+

≥γ λ

κ p+−1

|y₀|^2−p+ |y₀| ≥ ^λ κ ·γ

p+1−1 =⇒

and q

λ²+y²₀= γ

1 p+−2

λ κ

^p+

−1 p+−2

|y0|^2−1p+ ≥ ^λ κ ·γ

1

p+−1 =⇒

(p(x)−p+)ln κ

λ q

λ²+y²₀

≤ ^p(x)−p+

p+−1 lnγ≤0 =⇒ (2.5)is true.

3 Comparison principle

InG₀^d ⊂Gwe consider the second order quasi-linear degenerate operator Qof the form Q(u,η)≡

Z

G^d₀

DA_i(x,ux)ηx_i +b(x,u,ux)η(x)^Edx+

Z

Γ^d₀

γ(ω)

r^p(x)−1u|u|^p(x)−2η(x)ds

−

Z

Ωd

A_i(x,ux)cos(r,x_i)η(x)_dΩd; γ(ω)≥γ0 >0 (3.1)

foru(x) ∈ N^1,p_−1,∞^(x)(G₀^d) and for all non-negative η(x)belonging to N^1,p_−1,∞^(x)(G₀^d)under the fol- lowing assumptions:

functions A_i(x,ξ),b(x,u,ξ) are Carathéodory, continuously differentiable with respect to the u,ξ variables inM=G×_R×_Rⁿand satisfy inMthe following inequalities:

(i) ^∂A_∂ξ^i(x,ξ)

j ζ_iζ_j ≥κp|ξ|^p(x)−2ζ², ∀ζ ∈_Rⁿ\ {0}; κp >0;

(ii) s

∑n i=1

∂b(x,u,ξ)

∂ξi

2

≤ b₁|u|⁻¹|ξ|^p(x)−1; ^∂b(x,u,ξ_∂u ⁾ ≥b₂|u|⁻²|ξ|^p(x); b₁ ≥0, b₂≥0;

(iii) p(x)≥ p− >1.

Proposition 3.1. Let Q satisfy assumptions (i)–(iii) and functions u,w ∈ N^1,p_−1,∞^(x)(G^d₀) satisfy the inequality

Q(u,η)≤Q(w,η) (3.2)

for all non-negativeη∈N^1,p_−1,∞^(x)(G^d₀).Assume also that the inequality

u(x)≤w(x)onΩd (3.3)

holds. Then u(x)≤ w(x)in G₀^d.

Proof. Let us define z= u−wandu^τ =τu+ (1−τ)w, τ∈[0, 1]. Then we have 0≥Q(u,η)−Q(w,η)

=

Z

G₀^d

* η_xiz_xj

Z ₁

0

∂A_i(x,u^τ_x)

∂u^τ_xj dτ+ηz_xi Z ₁

0

∂b(x,u^τ,u^τ_x)

∂u^τ_xi dτ+ηz Z ₁

0

∂b(x,u^τ,u^τ_x)

∂u^τ dτ +

dx

−

Z

Ωd

Z ₁

0

∂A_i(x,u^τ_x)

∂u^τ_xj dτ

!

cos(r,x_i)·z_xjη(x)dΩd

+

Z

Γ^d₀

γ(ω) r^p(x)−1

Z ₁

0

∂(u^τ|u^τ|^p(x)−2)

∂u^τ dτ

!

z(x)η(x)ds (3.4)

for all non-negativeη∈N^1,p_−1,∞^(x)(G₀^d).

Now, we introduce the sets

(G₀^d)⁺:={x∈ G₀^d | u(x)>w(x)} ⊂G₀^d, (_Γ^d₀)⁺:={x∈ _Γ^d₀ | u(x)>w(x)} ⊂_Γ^d₀

and assume that (G₀^d)⁺ 6= _∅, (_Γ^d₀)⁺ 6= _{∅. Let} k ≥ 1 be any odd number. We choose η=max{(u−w)^k, 0} as a test function in the integral inequality (3.4). We have

Z ₁

0

∂(u^τ|u^τ|^p(x)−2)

∂u^τ dτ= (p(x)−1)

Z ₁

0

|u^τ|^p(x)−2dτ>0.

Then, by assumptions(i)–(iii)andη

Ωd=0, we obtain from (3.4) that Z

(G₀^d)⁺

kκpz^k−1 Z ₁

0

|∇u^τ|^p(x)−2dτ

|∇z|²+b2z^k+1 Z ₁

0

|u^τ|⁻²|∇u^τ|^p(x)dτ

dx

≤ b₁·

Z

(G₀^d)⁺

z^k Z ₁

0

|u^τ|⁻¹|∇u^τ|^p(x)−1dτ

|∇z|dx. (3.5)

By the Cauchy inequality, b₁z^k|∇z||u^τ|⁻¹|∇u^τ|^p(x)−1=

|u^τ|⁻¹z^k+21|∇u^τ|^p(2x)

·

b₁z^k−21|∇z||∇u^τ|^p(2x)−1

≤ ^ε

2|u^τ|⁻²z^k+1|∇u^τ|^p(x)+ ^b

21

2εz^k−1|∇z|²|∇u^τ|^p(x)−2, ∀ε>0.

Hence, takingε=2b₂, we obtain from (3.5) the inequality

kκp− ^b

2 1

4b₂ Z

(G₀^d)⁺

z^k−1|∇z|² Z ₁

0

|∇u^τ|^p(x)−2dτ

dx ≤0. (3.6)

Choosing the odd number k ≥ max 1;_2b^b21

2κp

, in view of z(x) ≡ 0 on ∂(G₀^d)⁺, we get from (3.6) that z(x) ≡ 0 in (G₀^d)⁺. We got a contradiction to our definition of the set (G^d₀)⁺, this completes the proof.

Remark 3.2. For the p(x)-Laplacian assumption(i)is satisfied with

κp=

(1, if p(x)≥2;

p−−1, if 1< p−≤ p(x)<2.

3.1 Barrier function and eigenvalue problem(OEVP)

We shall study thebarrierfunctionw(r,ω)6≡0 as a solution of the auxiliary problem:











−4_p+w=µw⁻¹|∇w|^p+, x∈ G₀^d,

|∇w|^p+−2∂w

∂−→

n + ^γ

|x|^p+−1w|w|^p+−2 =_0, x∈ _Γ^d_0, 0<d≤d₀.

(BFP)

By direct calculations, we derive a solution of this problem in the form w=w(r,ω) =r^κψ^κ/λ(ω), κ= ^p+−1

p+−1+µλ, (BF)

where(λ,ψ(ω))is the solution of the eigenvalue problem (OEVP). For this function we cal- culate with regard toy(ω) = ^ψ0(ω)

ψ(ω):

∂w

∂r =κ^{rκ−1ψκ/λ}(ω); ∂w

∂ω = κ

λr^κψ^κλ−1(ω)ψ⁰(ω);

|∇w|= κ

λr^κ−1ψ^κλ−1(ω) q

λ²ψ²(ω) +ψ⁰²(ω) = κ

λr^κ−1ψ^κλ(ω) q

λ²+y²(ω).

(3.7)

Proposition 3.3. w∈N^1,p_−1,∞^(x)(G₀^d).

Proof. From (BF) and (2.2) it follows thatw∈L_∞(G₀^d). Next, Z

G₀^d

r^−p(x)w^p(x)dx =

Z

G₀^d

r^(κ−1)p(x)ψ^κλp(x)(ω)dx.

Byr≤d1 and assumption(i), we have

r^(κ−1)p(x) ≤r^(κ−1)p−, ifκ ≥1;

r^(κ−1)p(x) ≤r^(κ−1)p+, ifκ ≤1;

ψ^κλp(x)(ω)≤ψ₀^p+ in virtue of (2.2) andκ ≤λ.

Hence it follows that Z

G^d₀

r^−p(x)w^p(x)dx≤ψ₀^p+measΩ·







d^{(κ−1)p−+n}

(κ−1)p−+n, ifκ≥1;

d^{(κ−1)p+ +n}

(κ−1)p++n, ifκ≤1.

(3.8) From (3.7) with regard to (2.4) we obtain that

Z

G^d₀

w⁻¹|∇w|^p(x)dx=

Z

G^d₀

κ λ

p(x)

r^{(p(x)−1)κ−p(x)}ψ^{(p(x)−1)κλ−p(x)}(ω)λ²ψ²(ω) +ψ⁰²(ω)^p(x)/2dx

≤ψ₀^p+−1 Z

G₀^d

r^{(p(x)−1)κ−p(x)} λ²+y²(ω)^p(x)/2dx

≤ψ₀^p+−1 Z

G₀^d

r^{(p(x)−1)κ−p(x)} λ²+z²₀p(x)/2

dx.

Since q

λ²+z²₀=const(n,λ,ω0,p+)andp(x)∈ [p−,p+], we have λ²+y²(ω)^p(x)/2 ≤C₁=const(n,λ,ω₀,p+,p−). From the above inequality we obtain that

Z

G^d₀

w⁻¹|∇w|^p(x)dx≤C₁ψ₀^p+−1 Z

G^d₀

r^{(p(x)−1)κ−p(x)}dx

=C₁ψ₀^p+−1 Z

G^d₀

r^{(κ−1)(p(x)−p+)}·r^{(κ−1)p+−κ}dx. (3.9)

Now, by assumptions(i)–(ii)andr1, we derive

r^{(κ−1)(p(x)−p+)}≤

(r^(1−κ)Lr, ifκ >1, 1, ifκ ≤1.

Using the well known inequality

r^α|lnr| ≤ ¹

αe, ∀α>0, 0<r<1, (3.10) wheree is the Euler number, we establish forκ>1 the inequality

r^(1−κ)Lr ≤e^{L(κ−e 1)}, 0<r<1.

Thus, from (3.9) it follows that Z

G₀^d

w⁻¹|∇w|^p(x)dx≤ C₁ψ₀^p+−1e^{L(κ−e 1)}measΩ· ^dκ

(p+−1)+n−p+

κ(p+−1) +n−p+.

4 The proof of the main Theorem 1.3.

Let A > 1, and let w(r,ω) be the barrier function defined above. By the definition of the operatorQin(I I), we have

Q(Aw,η)≡

Z

G₀^d

D

A^p(x)−1|∇w|^p(x)−2wx_iηx_i +b(Aw,A∇w)η(x)^Edx

+γ Z

Γ^d₀

A^p(x)−1r^1−p(x)w^p(x)−1η(x)dS−

Z

Ωd

A^p(x)−1|∇w|^p(x)−2∂w

∂r η(x)_dΩd

(4.1)

for all d∈(0,d₀)and all non-negativeη∈N^1,p_−1,⁽_∞^x)(G^d₀). Integrating by parts, we obtain that Z

G^d₀

A^p(x)−1|∇w|^p(x)−2w_xiη_xidx

= −

Z

G^d₀

d dx_i

D

A^p(x)−1|∇w|^p(x)−2wx_i

E

η(x)dx+

Z

Γ^d₀

A^p(x)−1|∇w|^p(x)−2dw

dnη(x)dS +

Z

Ωd

A^p(x)−1|∇w|^p(x)−2∂w

∂rη(x)dΩd.

Hence and (4.1), with regard to problem (BFP), it follows that Q(_Aw,η) =_JGd

0 +_JΓd

0, (4.2)

where

J_Gd

0 ≡

Z

G^d₀

*

µA^p(x)−1w⁻¹|∇w|^p(x)−A^p(x)−1|∇w|^p+−2w_xid|∇w|^p(x)−p+ dx_i

− ^∂A

p(x)−1

∂x_i w_xi|∇w|^p(x)−2+b(Aw,A∇w) +

η(x)dx;

J_Γd

0 ≡γ

Z

Γ^d₀

Aw r

p(x)−1* 1−

r|∇w| w

p(x)−p++

η(x)dS.

(4.3)

At first, we assert that J_Γd

0 ≥0. Indeed, by (3.7), r|∇w|

w

Γ^d0

= κ λ

q

λ²+y²₀

and desired inequality follows from Proposition2.1. Thus, from (4.2) it follows that Q(Aw,η)≥ J_Gd

0. (4.4)

Further, we proceed to the estimating of integralJ_Gd

0. SettingW(x) =|∇w|^p(x)−p+, we calculate lnW(x) = (p(x)−p+)ln|∇w|, =⇒ ¹

W(x)·^∂W

∂x_i = ^∂p

∂x_i ln|∇w|+ ^p(x)−p+

|∇w| ·^d|∇w|

dx_i =⇒ d

dx_i

|∇w|^p(x)−p+= |∇w|^p(x)−p+ ∂p

∂x_iln|∇w|+ ^p(x)−p+

|∇w| ·^d|∇w| dx_i

. Similarly,

d dx_i

A^p(x)−1

= A^p(x)−1∂p

∂x_i lnA.

By (4.3), we obtain that J_Gd

0 ≥

Z

G₀^d

(

A^p(x)−1|∇w|^p(x)−2

*

µw⁻¹|∇w|²−(∇p· ∇w)(lnA+ln|∇w|)

− ^p(x)−p+

|∇w| ·wx_id|∇w| dx_i

+

+b(Aw,A∇w) )

η(x)dx.

(4.5)

Passing to polar coordinates, we calculate w_xid|∇w|

dx_i = ^∂w

∂r ·^∂|∇w|

∂r + ¹ r²

∂w

∂ω ·^∂|∇w|

∂ω . Now, by (4.4)–(4.5) with regard to assumption(iv), we obtain that

Q(Aw,η)≥

Z

G^d₀

A^p(x)−1 (

|∇w|^p(x)−2

*

σw⁻¹|∇w|²−(∇p· ∇w)(lnA+ln|∇w|)

− ^p(x)−p+

|∇w| · ∂w

∂r ·^∂|∇w|

∂r + ¹ r²

∂w

∂ω ·^∂|∇w|

∂ω +

−b₀w^p(x)−1 )

η(x)dx,

(4.6)

with

σ= (

µ−δ, ifµ>0;

ν, ifµ=0.

Taking into account (3.7), the Lipschitz condition ofp(x)(ii)and ^ψ_ψ⁰₍⁽_ω^ω₎⁾ =y(ω), we directly calculate:

1)

|(∇p· ∇w)(lnA+ln|∇w|)| ≤ |∇p| · |∇w| ·(lnA+|ln|∇w||)

≤ L|∇w| ·(_lnA+|_ln|∇w||)_; in virtue of (3.7), (2.2), (2.4), we derive:

|ln|∇w|| ≤lnκ λ

+|κ−1| · |lnr|+κ

λψ^κλ−1lnψ+ ¹

2|ln(λ²+y²(ω))|

≤ln λ

κ +lnψ₀+¹

2|ln(λ²+z²₀)|+|κ−1| · |lnr|

=lnC₁(n,p+,λ,ω₀) +|κ−1| · |lnr|; (note that C₁ ≥ 1 : indeed, by virtue of (BF) and (2.2) ^λ

κψ₀ ≥ 1; from (13) [4] and (2.5) it follows that

q

λ²+z²₀ ≥^qλ²+y²₀ ≥ ^λ

κ ≥ 1 =⇒ C₁ = ^λ

κψ₀ q

λ²+z²₀ ≥ 1); using inequality (3.10) with α= ¹₂, we get that

|∇w| · |ln|∇w|| ≤ κ

λr^κ−1ψ^κλ(ω) q

λ²+y²(ω)·(lnC₁+|κ−1| ·lnr)

= κ

λr^κ−2ψ^κλ(ω) q

λ²+y²(ω)· rlnC₁+ (|κ−1|√ r)·√

rlnr

≤ κ

λr^κ−2ψ^κλ(ω) q

λ²+y²(ω)·(rlnC₁+|κ−1|√ r); from which we obtain that

|(∇p· ∇w)(lnA+ln|∇w|)| ≤ Lκ

λr^κ−2ψ^κλ(ω) q

λ²+y²(ω)·(rln(AC₁) +|κ −1|√ r). 2)

∂|∇w|

∂r = κ−1 r |∇w|;

∂|∇w|

∂ω = |∇w| κ

λ + ^y

0(ω) λ²+y²(ω)

·y(ω) =|∇w|

p+−1

p+−1+µ+ ^y

0(ω) λ²+y²(ω)

·y(ω), and by (BF) it follows that

p+−p(x)

|∇w| · ∂w

∂r ·^∂|∇w|

∂r + ¹ r²

∂w

∂ω·^∂|∇w|

∂ω

=κ(p+−p(x))^w r²

κ−1+ ^y

2

λ

p+−1

p+−1+µ+ ^y

0(ω) λ²+y²(ω)

; using (2.3)–(2.4), we establish that

p+−1

p+−1+µ+ ^y

0(ω)

λ²+y²(ω) ≥ −λ (n−p+)

(p+−1)y²+λ² −µ p+−1

p+−1+µ· ^y

2+λ² (p+−1)y²+λ²

≥ −λ (n−p+)

(p+−1)y²+λ² −µ y²+λ² (p+−1)y²+λ²;