In [3]–[14] and their references, sharp necessary conditions for existence of global solutions to different classes of second-order elliptic equations with gradient terms were obtained

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

BLOW-UP OF SOLUTIONS TO SYSTEMS OF NONLINEAR INEQUALITIES WITH SINGULARITIES ON UNBOUNDED SETS

EVGENY GALAKHOV, OLGA SALIEVA, LIUDMILA UVAROVA

Abstract. We establish conditions for the blow-up of solutions to several systems of nonlinear differential inequalities, with singularities on unbounded sets.

1. Introduction

In recent years, many mathematicians study global solvability of nonlinear par- tial differential equations and inequalities with singular coefficients. Here nonlinear terms can depend both on the values of the unknown function and on its derivatives.

This problem is not only interesting in its own right but also has important mathe- matical and physical applications. Thus, Liouville type theorems on nonexistence of positive solutions to nonlinear equations in the whole space or in the half-space can be used to obtain a priori estimates of solutions to respective problems in bounded domains [1, 2]. On the other hand, the class under consideration includes, in par- ticular, Hamilton–Jacobi and Korteweg–de Vries equations that play an important role in contemporary mathematical physics [11, 12].

In [3]–[14] and their references, sharp necessary conditions for existence of global solutions to different classes of second-order elliptic equations with gradient terms were obtained. The proofs are based either on ODE techniques for radially sym- metric solutions or on the nonlinear capacity method, which was suggested in [10]

and extensively developed in [9].

In this article we obtain sufficient conditions for nonexistence of solutions for sev- eral classes of systems of inequalities that have singular coefficients on unbounded sets, such as straight lines and planes, as well as smooth curves and surfaces in R^N. Here we obtain nonexistence results in natural functional classes, that is, as- suming only minimal local integrability properties, unlike our previous paper [7]

where lower bounds for local integrals were required. Some further related results for scalar inequalities were included into [8].

The main results of the paper are formulated in section 2. In section 3, a nonex- istence theorem is proven for a system of higher-order elliptic inequalities with a nonlinearity dependent only on the values of the unknown functionu, and in sec- tion 4, a similar result is obtained for a system of inequalities with a nonlinearity

2000Mathematics Subject Classification. 35J47, 35J48, 35J60.

Key words and phrases. System of nonlinear differential inequalities; blow-up; solvability.

c

2014 Texas State University - San Marcos.

Submitted September 14, 2014. Published October 14, 2014.

1

containing |Du|. In sections 5 and 6, the respective statements are extended to systems of second-order elliptic inequalities with a nonlinear principal part.

2. Main results

We assume that the setS satisfies a geometrical condition which is based on an idea from our paper [7], and modified according to our problem setting and func- tional classes under consideration. To formulate it, we need some extra notation.

Letε >0 andx∈R^N. Denoteρ(x) = dist(x, S), Bε(0) ={x∈R^N : |x| ≤ε}, and

Sε={x∈R^N : ρ(x)< ε}.

ForR >0, introduce the set

S^R=SR\S1/R∩BR(0).

Now we can formulate our assumption onS.

(H1) Suppose that there exists a family of functionsξR∈C₀^2k(R^N\S; [0,1]) such that

ξR(x) =

(0 (x∈S_1/(2R)∪(R^N\S_2R)),

1 (x∈SR\S_1/R) (2.1)

and there exists a constantc >0 such that

|D^αξ_R(x)| ≤cρ^−|α| (x∈R^N). (2.2) Example 2.1. We can consider as the set S a hyperplane S = Πn = {x = (x1, . . . , xn)∈R^N:xn = 0}. In that case we can choose ξR(x) = ˜ξR(xn), where

ξ˜_R(x_n) =

(0 (|xn| ≤ _2R¹ or|xn| ≥2R), 1 (_R¹ ≤ |xn| ≤R)

See Figure 1.

ξ˜R

6

· · @

·@· · ·

XXXX XXX

· · -

−2R −R −_R¹−_2R¹ 0 _2R¹ _R¹ R 2R x_n

·1

Figure 1. The function ˜ξ_R(x_n)

Further we assume that the set S satisfies assumption (H1). We formulate our first result for a system of nonlinear elliptic inequalities

−∆pu≥a(x)v^q1 (x∈R^N \S),

−∆qv≥b(x)u^p1 (x∈R^N\S), u(x), v(x)≥0 (x∈R^N \S),

(2.3)

where p, q, p1, q1 >1, p−1 < p1, q−1 < q1, a, b ∈ C(R^N \S) are nonnegative functions such thata(x)≥a0ρ^−α|x|^β,b(x)≥b0ρ^−γ|x|^δ forx∈R^N \S,a0, b0>0, α, β, γ, δ∈R,ρ(x) = dist(x, S). Introduce the quantities

σ1=(|α| −β)(q−1) + (|γ| −δ−q)q1)(p−1)−pp1q1

p1q1−(p−1)(q−1) , σ2= (|γ| −δ)(p−1) + (|α| −β−p)p1)(q−1)−qp1q1

p₁q₁−(p−1)(q−1) .

(2.4)

Theorem 2.2. Let N + min{σ1, σ2} ≤ 0. Then system (2.3) has no nontrivial (nonzero) solution.

Example 2.3. For a system of inequalities (2.3) withp=q= 2:

−∆u≥a(x)v^q1 (x∈R^N \S),

−∆v≥b(x)u^p1 (x∈R^N\S), u(x), v(x)≥0 (x∈R^N \S),

(2.5)

the quantities defined in (2.4) take the form

σ₁=(α−β+ (γ−δ−2)q₁)−2p₁q₁ p1q1−1 , σ2= (γ−δ+ (α−β−2)p1)−2p1q1

p1q1−1 .

(2.6)

Further we consider the system

−∆_pu≥a(x)|Dv|^q1 (x∈R^N \S),

−∆qv≥b(x)|Du|^p1 (x∈R^N \S). (2.7) For this system one has the following result.

Theorem 2.4. Let

max{(p−1)(α(q−1) +q1(β+ 1)),(q−1)(β(p−1) +p1(α+ 1))}

≥N(p₁q₁−(p−1)(q−1))−p₁q₁.

Then system (2.7)has no nontrivial (non-constant) solution.

We also consider the systems of higher-order differential inequalities (−∆)^ku≥a(x)|Dv|^q (x∈R^N \S),

(−∆)^lv≥b(x)|Du|^p (x∈R^N \S) (2.8) and

(−∆)^ku≥a(x)v^q (x∈R^N\S), (−∆)^lv≥b(x)u^p (x∈R^N\S),

u≥0, v≥0 (x∈R^N),

(2.9) wherek, l∈N,p, q >1.

For system (2.8), there holds the following theorem.

Theorem 2.5. Let maxn

|α|+β+ (|γ|+δ+ 2l−1 + (2k−1)p)q,

|γ|+δ+ (|α|+β+ 2k−1 + (2l−1)q)po

≥N(pq−1).

Then system (2.8)has no nontrivial (non-constant) solution.

For system (2.9), one has Theorem 2.6. Let

maxn

|α|+β+ (|γ|+δ+ 2l+ 2kp)q,|γ|+δ+ (|α|+β+ 2k+ 2lq)po

≥N(pq−1).

Then system (2.9)has no nontrivial (nonzero) solution.

3. Proof of Theorem 2.2

Suppose that there exists (u, v) – a nontrivial solution of system (2.3). Let ϕR∈C₀^∞(R^N;R+) be a family of test functions to be specified below.

Multiplying the first inequality (2.3) by u^λ_εϕR and the second one by v^λ_εϕR, whereuε=u+ε,vε=v+ε,ε >0 and max{1−p,1−q}< λ <0, we obtain

Z

a(x)v^q1u^λ_εϕ_Rdx≤cλ Z

|Du|^pu^λ−1_ε ϕ_Rdx+ Z

|Du|^p−1|Dϕ_R|u^λ_εdx, (3.1) Z

b(x)u^p1v_ε^λϕRdx≤cλ Z

|Dv|^qv^λ−1_ε ϕRdx + Z

|Dv|^q−1|DϕR|v_ε^λdx. (3.2) Application of Young’s inequality to the first terms on the right-hand sides of the obtained relations results in

Z

a(x)v^q1u^λ_εϕ_Rdx+c|λ|

2 Z

|Du|^pu^λ−1_ε ϕ_Rdx≤c_λ

Z |DϕR|^p

ϕ^p−1_R u^λ+p−1_ε dx, (3.3) Z

b(x)u^p1v^λ_εϕRdx+c|λ|

2 Z

|Dv|^qv^λ−1_ε ϕRdx≤dλ

Z |DϕR|^q

ϕ^q−1_R v_ε^λ+q−1dx, (3.4) where the constants cλ and dλ depend only on p, q, and λ. Further, multiplying each differential inequality (2.3) byϕR and integrating by parts, we arrive at

Z

a(x)v^q1ϕ_Rdx≤Z

|Du|^pu^λ−1_ε ϕ_Rdx^p−1_p Z |Dϕ_R|^p

ϕ^p−1_R u^{(1−λ)(p−1)}_ε dx^1/p , (3.5) Z

b(x)u^p1ϕRdx≤Z

|Dv|^qv^λ−1_ε ϕRdx^q−1_q Z |DϕR|^q

ϕ^q−1_R v^{(1−λ)(q−1)}_ε dx1/q

. (3.6) Combining (3.3)–(3.6) and takingε→0, we obtain a priori estimates

Z

a(x)v^q1ϕ_Rdx≤D_λZ |DϕR|^p

ϕ^p−1_R u^λ+p−1dx^p−1_p Z |DϕR|^p

ϕ^p−1_R u^{(1−λ)(p−1)}dx1/p

, (3.7) Z

b(x)u^p1ϕRdx≤Eλ

Z |DϕR|^q

ϕ^q−1_R v^λ+q−1dx^q−1_q Z |DϕR|^q

ϕ^q−1_R v^{(1−λ)(q−1)}dx1/q

, (3.8) whereD_λ, E_λ>0 depend only onp, q, andλ.

Applying the H¨older inequality with exponentrto the first integral on the right- hand side of (3.7), we obtain

Z |DϕR|^p

ϕ^p−1_R u^λ+p−1dx^p−1_p

≤Z

b(x)u^(λ+p−1)rϕRdx^p−1_pr Z b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^p−1_pr0

,

(3.9)

where ¹_r+_r¹0 = 1.

Choosing the exponentrso that (λ+p−1)r=p₁, from (3.7) and (3.9) we have Z

a(x)v^q1ϕRdx≤Dλ

Z

b(x)u^p1ϕRdx^p−1_pr Z b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^p−1_pr0

×Z |DϕR|^p

ϕ^p−1_R u^{(1−λ)(p−1)}dx1/p

.

(3.10)

Applying the H¨older inequality with exponent y > 1 to the last integral on the right-hand side of (3.10), we obtain

Z |Dϕ_R|^p

ϕ^p−1_R u^{(1−λ)(p−1)}dx≤Z

b(x)u(1−λ)(p−1)yϕ_Rdx^1/y

×Z b^−y

0

y(x)|DϕR|^py0

ϕ^py_R⁰⁻¹ dx1/y⁰

,

(3.11)

where ¹_y +_y¹0 = 1.

Choosingyin (3.11) so that (1−λ)(p−1)y=p1and taking into account (3.10), we reach the estimate

Z

a(x)v^q1ϕ_Rdx≤D_λZ

b(x)u^p1ϕ_Rdx^p−1_pr Z b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^p−1_pr0

×Z

b(x)u^p1ϕ_Rdx_py¹ Z b^−y

0

y (x)|DϕR|^py0 ϕ^py_R⁰⁻¹ dx_py¹0

; i.e.,

Z

a(x)v^q1ϕRdx

≤Dλ

Z

b(x)u^p1ϕRdx^p−1_pr +_py¹ Z b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^p−1_pr0

×Z b^−y

0

y(x)|Dϕ_R|^py0 ϕ^py_R⁰⁻¹ dx_py¹0

,

(3.12)

where the exponentsrandy are chosen so that 1

y + 1

y⁰ = 1, (1−λ)(p−1)y=p1, 1

r+ 1

r⁰ = 1, (λ+p−1)r=p1.

(3.13)

Note that such choice of r and y is possible due to our hypotheses on p and p₁ provided thatλ <0 is small enough in absolute value. Similarly, choosingsandz

such that

1 z + 1

z⁰ = 1, (1−λ)(q−1)z=q1, 1

s+ 1

s⁰ = 1, (λ+q−1)s=q1,

(3.14) and estimating the right-hand side of (3.8) by the H¨older inequality, we obtain

Z

b(x)u^p1ϕRdx

≤Eλ

Z

a(x)v^q1ϕRdx^q−1_qs +_qz¹Z a^−s

0

s(x)|DϕR|^qs0

ϕ^qs_R⁰⁻¹ dx^q−1_qs0

×Z a^−z

0

z(x)|Dϕ_R|^qz0 ϕ^qz_R⁰⁻¹ dx_qz¹0

.

(3.15)

Combining (3.12) and (3.15), we finally arrive at Z

a(x)v^q1ϕRdx1−mn

≤DλE_λ^NZ a^−s

0

s(x)|DϕR|^qs0

ϕ^qs_R⁰⁻¹ dx^n(q−1)_qs0 Z a^−z

0

z(x)|DϕR|^qz0 ϕ^qz_R⁰⁻¹ dx_qzⁿ0

×Z b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^p−1_pr0Z b^−y

0

y(x)|DϕR|^py0 ϕ^py_R⁰⁻¹ dx_py¹0

(3.16)

and Z

b(x)u^p1ϕRdx1−mn

≤E_λD_λ^mZ b^−r

0

r(x)|DϕR|^pr0

ϕ^pr_R⁰⁻¹ dx^m(p−1)_pr0 Z b^−y

0

y(x)|DϕR|^py0 ϕ^py_R⁰⁻¹ dx_py^m0

×Z a^−s

0

s(x)|Dϕ_R|^qs0

ϕ^qs_R⁰⁻¹ dx^q−1_qs0 Z a^−z

0

z(x)|Dϕ_R|^qz0 ϕ^qz_R⁰⁻¹ dx_qz¹0

,

(3.17)

n:= p−1 pr + 1

py, m:= q−1 qs + 1

qz. (3.18)

Simple calculations taking into account (3.13) and (3.14) give explicit values ofm andn, namely,

m=q−1

q₁ , n=p−1

p₁ . (3.19)

Our assumptions imply that the exponent on the left-hand side of (3.16), (3.17) is such that

1−mn=p₁q₁−(p−1)(q−1) p1q1

>0.

Thus from (3.17) and our assumptions onaandbwe have Z

(S^R\S^1/R)∩BR(0)

a(x)v^q1dx≤CR^N+σ1, Z

(S^R\S^1/R)∩BR(0)

b(x)u^p1dx≤CR^N+σ2.

(3.20)

Taking R → ∞ in (3.20), under condition N + min{σ1, σ2} ≤ 0 we come to a contradiction, which completes the proof of Theorem 2.2.

4. Proof of Theorem 2.4

Multiplying inequalities (2.7) by the test functionϕ_R∈C₀¹(R^N; [0,1]) and inte- grating by parts, we obtain

Z

R^N

a(x)|Dv|^q1ϕ_R(x)dx≤ Z

R^N

(|Du|^p−2Du, Dϕ_R)dx, Z

R^N

b(x)|Du|^p1ϕR(x)dx≤ Z

R^N

(|Dv|^q−2Dv, DϕR)dx, which because of relations

(|Du|^p−2Du, DϕR)≤ |Du|^p−1|DϕR|, (|Dv|^q−2Dv, DϕR)≤ |Dv|^q−1|DϕR| and the H¨older inequality, results in

Z

R^N

a(x)|Dv|^q1ϕ_R(x)dx

≤Z

R^N

b(x)|Du|^p1ϕ_R(x)dx^p−1_p

1

×Z

R^N

b⁻

p−1

p1−p+1(x)|DϕR|^p1−p+1p1 ϕ¹⁻

p1 p1−p+1

R (x)dx^p1−p+1_p1 ,

(4.1)

Z

R^N

b(x)|Du|^p1ϕR(x)dx

≤Z

R^N

a(x)|Dv|^q1ϕR(x)dx^q−1_q1

×Z

R^N

a^{−q1q−1−q+1}(x)|DϕR|^q1−q+1q1 ϕ¹⁻

q1 q1−q+1

R (x)dx

q1−q+1 q1

.

(4.2)

Substituting (4.1) into (4.2) and vice versa, we obtain Z

R^N

a(x)|Dv|^q1ϕR(x)dx1−^{(p−1)(q−1)}_p

1q1

≤Z

R^N

a^{−q1q−1−q+1}(x)|DϕR|^q1−q+1q1 ϕ¹⁻

q1 q1−q+1

R (x)dx^(p−1)(q_p^1−q+1)

1q1

×Z

R^N

b⁻

p−1

p1−p+1(x)|DϕR|^p1−p+1p1 ϕ¹⁻

p1 p1−p+1

R (x)dx^p1

−p+1 p1

,

Z

R^N

b(x)|Du|^p1ϕR(x)dx1−^{(p−1)(q−1)}_p

1q1

≤Z

R^N

a⁻

q−1

q1−q+1(x)|DϕR|^q1−q+1q1 ϕ¹⁻

q1 q1−q+1

R (x)dx^q1

−q+1 q1

×Z

R^N

b⁻

p−1

p1−p+1(x)|DϕR|^p1−p+1p1 ϕ¹⁻

p1 p1−p+1

R (x)dx

(q−1)(p1−p+1) p1q1

;

i. e., Z

R^N

a(x)|Dv|^q1ϕR(x)dx

≤Z

R^N

a⁻

q−1

q1−q+1(x)|DϕR|^q1−q+1q1 ϕ¹⁻

q1 q1−q+1

R (x)dx_p^{(p−1)(q1−q+1)}

1q1−(p−1)(q−1)

×Z

R^N

b^{−p1p−1−p+1}(x)|Dϕ_R|^p1−p+1p1 ϕ¹⁻

p1 p1−p+1

R (x)dx_p ^{q1 (p1−p+1)}

1q1−(p−1)(q−1)

,

(4.3)

Z

R^N

b(x)|Du|^p1ϕR(x)dx

≤Z

R^N

a^{−q1q−1−q+1}(x)|DϕR|^q1−q+1q1 ϕ¹⁻

q1 q1−q+1

R (x)dx

p1 (q1−q+1) p1q1−(p−1)(q−1)

×Z

R^N

b⁻

p−1

p1−p+1(x)|DϕR|^p1−p+1p1 ϕ¹⁻

p1 p1−p+1

R (x)dx

(q−1)(p1−p+1) p1q1−(p−1)(q−1)

.

(4.4)

Choosing test functionϕR∈C₀¹(R^N; [0,1]) so that ϕR(x) =

(1 (|x| ≤R), 0 (|x| ≥2R), and

|DϕR(x)| ≤cR⁻¹ (x∈R^N), (4.5) we obtain

Z

B_R(0)

a(x)|Dv|^q1dx≤cR^N−

(p−1)((|α|+β)(q−1)+q1 (|γ|+δ+1))+p1q1 p1q1−(p−1)(q−1) , Z

BR(0)

b(x)|Du|^p1dx≤cR^N−

(q−1)((|γ|+δ)(p−1)+p1 (|α|+β+1))+p1q1 p1q1−(p−1)(q−1) . TakingR→ ∞, we complete the proof similarly to Theorem 2.2.

5. Proof of Theorem 2.5

Multiplying inequalities (2.8) by the test function ϕ_R ∈ C₀^2k−1(R^N; [0,1]) and integrating by parts, we obtain

Z

R^N

a(x)|Dv|^qϕR(x)dx≤ Z

R^N

(Du, D((−∆)^k−1ϕR))dx, Z

R^N

b(x)|Du|^pϕR(x)dx≤ Z

R^N

(Dv, D((−∆)^l−1ϕR))dx, which by relations

(Du, D((−∆)^k−1ϕR))≤ |Du| · |D((−∆)^k−1ϕR)|, (Dv, D((−∆)^l−1ϕ_R))≤ |Dv| · |D((−∆)^l−1ϕ_R)|,

and the H¨older inequality, results in Z

R^N

a(x)|Dv|^qϕR(x)dx

≤Z

R^N

b(x)|Du|^pϕR(x)dx^1/p

×Z

R^N

b^−p−11 (x)|D((−∆)^k−1ϕ_R)|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_p ,

(5.1)

Z

R^N

b(x)|Du|^pϕ_R(x)dx

≤Z

R^N

a(x)|Dv|^qϕ_R(x)dx1/q

×Z

R^N

a^−q−11 (x)|D((−∆)^l−1ϕ_R)|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_q .

(5.2)

Substituting (5.1) into (5.2) and vice versa, we obtain Z

R^N

a(x)|Dv|^qϕR(x)dx1−_pq¹

≤Z

R^N

a^−q−11 (x)|D((−∆)^l−1ϕ_R)|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_pq

×Z

R^N

b^−p−11 (x)|D((−∆)^k−1ϕ_R)|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_p ,

Z

R^N

b(x)|Du|^pϕ_R(x)dx1−_pq¹

≤Z

R^N

a^−q−11 (x)|D((−∆)^l−1ϕR)|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_q

×Z

R^N

b^−p−11 (x)|D((−∆)^k−1ϕR)|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_pq

; i.e.,

Z

R^N

a(x)|Dv|^qϕ_R(x)dx

≤Z

R^N

a^−q−11 (x)|D((−∆)^l−1ϕR)|^q−1q ϕ⁻

1 q−1

R (x)dx_pq−1^q−1

×Z

R^N

b^−p−11 (x)|D((−∆)^k−1ϕR)|^p−1p ϕ⁻

1 p−1

R (x)dx^(p−1)q_pq−1 ,

(5.3)

Z

R^N

b(x)|Du|^pϕR(x)dx

≤Z

R^N

a^−q−11 (x)|D((−∆)^l−1ϕR)|^q−1q ϕ⁻

1 q−1

R (x)dx^p(q−1)_pq−1

×Z

R^N

b^−p−11 (x)|D((−∆)^k−1ϕR)|^p−1p ϕ⁻

1 p−1

R (x)dx_pq−1^p−1 .

(5.4)

Denote K = max{k, l}. Choosing the test function ϕR ∈C₀^2K−1(R^N; [0,1]) so that

ϕR(x) =

(1 (|x| ≤R), 0 (|x| ≥2R), and

|D^µϕR(x)| ≤cR⁻¹ (x∈R^N; 0≤ |µ| ≤2K−1), (5.5) we have

Z

B_R(0)

a(x)|Dv|^qdx≤cR^N−|α|+β+(|γ|+δ+2l−1+(2k−1)p)q

pq−1 ,

Z

BR(0)

b(x)|Du|^pdx≤cR^N−|γ|+δ+(|α|+β+2k−1+(2l−1)q)p

pq−1 .

TakingR→ ∞, we complete the proof similarly to Theorems 2.2 and 2.4.

6. Proof of Theorem 2.6

Multiplying inequalities (2.9) by the test functionϕR ∈C₀^2K(R^N; [0,1]), where K= max{k, l}, and integrating by parts, we obtain

Z

R^N

a(x)v^qϕR(x)dx≤ Z

R^N

u·(−∆)^kϕRdx, Z

R^N

b(x)u^pϕR(x)dx≤ Z

R^N

v·(−∆)^lϕRdx.

Taking into account that

u·(−∆)^kϕ_R≤u· |(−∆)^kϕ_R)|, v·(−∆)^lϕR≤v· |(−∆)^lϕR)|

and using the H¨older inequality, we arrive at Z

R^N

a(x)v^qϕR(x)dx≤Z

R^N

b(x)u^pϕR(x)dx^1/p

×Z

R^N

b^−p−11 (x)|(−∆)^kϕR|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_p ,

(6.1)

Z

R^N

b(x)u^pϕR(x)dx≤Z

R^N

a(x)v^qϕR(x)dx^1/q

×Z

R^N

a^−q−11 (x)|(−∆)^lϕR|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_q .

(6.2)

Substituting (6.1) into (6.2) and vice versa, we obtain Z

R^N

a(x)v^qϕ_R(x)dx1−_pq¹

≤Z

R^N

a^−q−11 (x)|(−∆)^lϕ_R|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_pq

×Z

R^N

b^−p−11 (x)|(−∆)^kϕ_R|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_p , Z

R^N

b(x)u^pϕR(x)dx1−_pq¹

≤Z

R^N

a^−q−11 (x)|(−∆)^lϕR|^q−1q ϕ⁻

1 q−1

R (x)dx^q−1_q

×Z

R^N

b^−p−11 (x)|(−∆)^kϕR|^p−1p ϕ⁻

1 p−1

R (x)dx^p−1_pq

;

i.e.,

Z

R^N

a(x)v^qϕR(x)dx

≤Z

R^N

a^−q−11 (x)|(−∆)^lϕR|^q−1q ϕ⁻

1 q−1

R (x)dx_pq−1^q−1

×Z

R^N

b^−p−11 (x)|(−∆)^kϕR|^p−1p ϕ⁻

1 p−1

R (x)dx^(p−1)q_pq−1 ,

(6.3)

Z

R^N

b(x)u^pϕR(x)dx

≤Z

R^N

a^−q−11 (x)|(−∆)^lϕR|^q−1q ϕ⁻

1 q−1

R (x)dx^p(q−1)_pq−1

×Z

R^N

b^−p−11 (x)|(−∆)^kϕR|^p−1p ϕ⁻

1 p−1

R (x)dxpq−1^p−1

.

(6.4)

Choosing the test functionϕR∈C₀^2K(R^N; [0,1]) so that ϕR(x) =

(1 (|x| ≤R), 0 (|x| ≥2R) and estimate (5.5) holds, we obtain

Z

BR(0)

a(x)|Dv|^qdx≤cR^N−|α|+β+(|γ|+δ+2l+2kp)q

pq−1 ,

Z

B_R(0)

b(x)|Du|^pdx≤cR^N−|γ|+δ+(|α|+β+2k+2lq)p

pq−1 .

TakingR→ ∞, we complete the proof similarly to Theorems 2.2, 2.4 and 2.5.

Acknowledgements. This research was supported by grants of Russian Founda- tion of Basic Research 13-01-12460-ofi-m and No. 14-01-00736 and by a President grant for government support of the leading scientific schools of the Russian Fed- eration No. 4479.2014.1.

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Evgeny Galakhov

Peoples’ Friendship University of Russia, Miklukho-Maklaya str. 6, Moscow, 117198, Russia

E-mail address:[email protected]

Olga Salieva

Moscow State Technological University “Stankin”, Vadkovsky lane 3a, Moscow, 125994, Russia

E-mail address:[email protected]

Liudmila Uvarova

Moscow State Technological University “Stankin”, Vadkovsky lane 3a, Moscow, 125994, Russia

E-mail address:[email protected]