Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 216, pp. 1–12.
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 partial 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 mathematical 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 particular, 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 symmetric 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 several 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
RN . Here we obtain nonexistence results in natural functional classes, that is, assuming 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 nonexistence theorem is proven for a system of higher-order elliptic inequalities with a
nonlinearity dependent only on the values of the unknown function u, and in section 4, a similar result is obtained for a system of inequalities with a nonlinearity
2000 Mathematics 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
2
E. GALAKHOV, O. SALIEVA, L. UVAROVA
EJDE-2014/216
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 set S satisfies a geometrical condition which is based on an
idea from our paper [7], and modified according to our problem setting and functional classes under consideration. To formulate it, we need some extra notation.
Let ε > 0 and x ∈ RN . Denote ρ(x) = dist(x, S),
Bε (0) = {x ∈ RN : |x| ≤ ε},
and
Sε = {x ∈ RN : ρ(x) < ε}.
For R > 0, introduce the set
S R = SR \ S1/R ∩ BR (0).
Now we can formulate our assumption on S.
(H1) Suppose that there exists a family of functions ξR ∈ C02k (RN \ S; [0, 1]) such
that
(
0 (x ∈ S1/(2R) ∪ (RN \ S2R )),
ξR (x) =
(2.1)
1 (x ∈ SR \ S1/R )
and there exists a constant c > 0 such that
|Dα ξR (x)| ≤ cρ−|α|
(x ∈ RN ).
(2.2)
Example 2.1. We can consider as the set S a hyperplane S = Πn = {x =
(x1 , . . . , xn ) ∈ RN : xn = 0}. In that case we can choose ξR (x) = ξ˜R (xn ), where
(
1
0 (|xn | ≤ 2R
or |xn | ≥ 2R),
˜
ξR (xn ) =
1
1 ( R ≤ |xn | ≤ R)
See Figure 1.
·
−2R
·
−R
ξ˜
6R
·1
XXX
@
· @·
· ·
1
1 1
− R1 − 2R
0 2R
R
·
R
XXX
X·
2R
xn
Figure 1. The function ξ˜R (xn )
Further we assume that the set S satisfies assumption (H1). We formulate our
first result for a system of nonlinear elliptic inequalities
−∆p u ≥ a(x)v q1
(x ∈ RN \ S),
−∆q v ≥ b(x)up1
(x ∈ RN \ S),
u(x), v(x) ≥ 0
N
(x ∈ R \ S),
(2.3)
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BLOW-UP OF SOLUTIONS
3
where p, q, p1 , q1 > 1, p − 1 < p1 , q − 1 < q1 , a, b ∈ C(RN \ S) are nonnegative
functions such that a(x) ≥ a0 ρ−α |x|β , b(x) ≥ b0 ρ−γ |x|δ for x ∈ RN \ S, a0 , b0 > 0,
α, β, γ, δ ∈ R, ρ(x) = dist(x, S). Introduce the quantities
(|α| − β)(q − 1) + (|γ| − δ − q)q1 )(p − 1) − pp1 q1
,
p1 q1 − (p − 1)(q − 1)
(|γ| − δ)(p − 1) + (|α| − β − p)p1 )(q − 1) − qp1 q1
σ2 =
.
p1 q1 − (p − 1)(q − 1)
σ1 =
(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) with p = q = 2:
−∆u ≥ a(x)v q1
(x ∈ RN \ S),
−∆v ≥ b(x)up1
(x ∈ RN \ S),
u(x), v(x) ≥ 0
N
(2.5)
(x ∈ R \ S),
the quantities defined in (2.4) take the form
(α − β + (γ − δ − 2)q1 ) − 2p1 q1
,
p1 q1 − 1
(γ − δ + (α − β − 2)p1 ) − 2p1 q1
σ2 =
.
p1 q1 − 1
σ1 =
(2.6)
Further we consider the system
−∆p u ≥ a(x)|Dv|q1
(x ∈ RN \ S),
−∆q v ≥ b(x)|Du|p1
(x ∈ RN \ 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 (p1 q1 − (p − 1)(q − 1)) − p1 q1 .
Then system (2.7) has no nontrivial (non-constant) solution.
We also consider the systems of higher-order differential inequalities
(−∆)k u ≥ a(x)|Dv|q
(x ∈ RN \ S),
(−∆)l v ≥ b(x)|Du|p
(x ∈ RN \ S)
(2.8)
and
(−∆)k u ≥ a(x)v q
(x ∈ RN \ S),
(−∆)l v ≥ b(x)up
(x ∈ RN \ S),
u ≥ 0,
v≥0
(x ∈ RN ),
where k, l ∈ N, p, q > 1.
For system (2.8), there holds the following theorem.
Theorem 2.5. Let
n
max |α| + β + (|γ| + δ + 2l − 1 + (2k − 1)p)q,
(2.9)
4
E. GALAKHOV, O. SALIEVA, L. UVAROVA
EJDE-2014/216
o
|γ| + δ + (|α| + β + 2k − 1 + (2l − 1)q)p ≥ N (pq − 1).
Then system (2.8) has no nontrivial (non-constant) solution.
For system (2.9), one has
Theorem 2.6. Let
n
o
max |α| + β + (|γ| + δ + 2l + 2kp)q, |γ| + δ + (|α| + β + 2k + 2lq)p ≥ 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∞ (RN ; R+ ) be a family of test functions to be specified below.
Multiplying the first inequality (2.3) by uλε ϕR and the second one by vελ ϕR ,
where uε = u + ε, vε = v + ε, ε > 0 and max{1 − p, 1 − q} < λ < 0, we obtain
Z
Z
Z
q1 λ
p λ−1
a(x)v uε ϕR dx ≤ cλ |Du| uε ϕR dx + |Du|p−1 |DϕR |uλε dx,
(3.1)
Z
Z
Z
b(x)up1 vελ ϕR dx ≤ cλ |Dv|q vελ−1 ϕR dx + |Dv|q−1 |DϕR |vελ dx.
(3.2)
Application of Young’s inequality to the first terms on the right-hand sides of the
obtained relations results in
Z
Z
Z
c|λ|
|DϕR |p λ+p−1
q1 λ
p λ−1
a(x)v uε ϕR dx +
|Du| uε ϕR dx ≤ cλ
dx, (3.3)
p−1 uε
2
ϕR
Z
Z
Z
c|λ|
|DϕR |q λ+q−1
b(x)up1 vελ ϕR dx +
|Dv|q vελ−1 ϕR dx ≤ dλ
vε
dx, (3.4)
2
ϕq−1
R
where the constants cλ and dλ depend only on p, q, and λ. Further, multiplying
each differential inequality (2.3) by ϕR and integrating by parts, we arrive at
Z
Z
p−1
Z |Dϕ |p
1/p
p
R
(1−λ)(p−1)
a(x)v q1 ϕR dx ≤
|Du|p uλ−1
ϕ
dx
dx
,
R
ε
p−1 uε
ϕR
(3.5)
Z
Z
Z
q−1
q
1/q
|DϕR | (1−λ)(q−1)
q
b(x)up1 ϕR dx ≤
|Dv|q vελ−1 ϕR dx
dx
. (3.6)
q−1 vε
ϕR
Combining (3.3)–(3.6) and taking ε → 0, we obtain a priori estimates
Z
Z |Dϕ |p
Z |Dϕ |p
p−1
1/p
p
R
R
λ+p−1
(1−λ)(p−1)
a(x)v q1 ϕR dx ≤ Dλ
u
dx
dx
,
p−1
p−1 u
ϕR
ϕR
(3.7)
Z
Z
Z |Dϕ |q
q−1
1/q
|DϕR |q (1−λ)(q−1)
q
R
λ+q−1
b(x)up1 ϕR dx ≤ Eλ
dx
dx
,
q−1 v
q−1 v
ϕR
ϕR
(3.8)
where Dλ , Eλ > 0 depend only on p, q, and λ.
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BLOW-UP OF SOLUTIONS
5
Applying the Hölder inequality with exponent r to the first integral on the righthand side of (3.7), we obtain
Z |Dϕ |p
p−1
p
R
λ+p−1
u
dx
ϕp−1
R
(3.9)
0
Z
Z
p−10
p−1
0
|DϕR |pr
pr
pr
− rr
(λ+p−1)r
b
(x) pr0 −1 dx
≤
b(x)u
,
ϕR dx
ϕR
where 1r + r10 = 1.
Choosing the exponent r so that (λ + p − 1)r = p1 , from (3.7) and (3.9) we have
Z
0
p−1
Z
Z
p−10
r0
|DϕR |pr
pr
pr
b(x)up1 ϕR dx
a(x)v q1 ϕR dx ≤ Dλ
b− r (x) pr0 −1 dx
ϕR
(3.10)
Z |Dϕ |p
1/p
R
(1−λ)(p−1)
×
u
dx
.
ϕp−1
R
Applying the Hölder inequality with exponent y > 1 to the last integral on the
right-hand side of (3.10), we obtain
Z
1/y
Z
|DϕR |p (1−λ)(p−1)
b(x)u(1−λ)(p−1)y ϕR dx
u
dx
≤
p−1
ϕR
(3.11)
0
Z
1/y0
y0
|DϕR |py
b− y (x) py0 −1 dx
×
,
ϕR
where y1 + y10 = 1.
Choosing y in (3.11) so that (1 − λ)(p − 1)y = p1 and taking into account (3.10),
we reach the estimate
Z
0
Z
p−1
Z
p−10
r0
|DϕR |pr
pr
pr
a(x)v q1 ϕR dx ≤ Dλ
b(x)up1 ϕR dx
b− r (x) pr0 −1 dx
ϕR
0
1 Z
10
Z
py
0
|DϕR |py
py
− yy
p1
b(x)u ϕR dx
b
(x) py0 −1 dx
;
×
ϕR
i.e.,
Z
a(x)v q1 ϕR dx
≤ Dλ
×
Z
Z
p1
b(x)u ϕR dx
0
b
− yy
(x)
1 Z
p−1
pr + py
|DϕR |py
py 0 −1
ϕR
0
0
b
− rr
(x)
|DϕR |pr
pr 0 −1
ϕR
0
p−10
pr
dx
(3.12)
10
py
dx
,
where the exponents r and y are chosen so that
1
1
+
= 1, (1 − λ)(p − 1)y = p1 ,
y y0
(3.13)
1
1
+ 0 = 1, (λ + p − 1)r = p1 .
r
r
Note that such choice of r and y is possible due to our hypotheses on p and p1
provided that λ < 0 is small enough in absolute value. Similarly, choosing s and z
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E. GALAKHOV, O. SALIEVA, L. UVAROVA
EJDE-2014/216
such that
1
1
+ 0 = 1, (1 − λ)(q − 1)z = q1 ,
z
z
(3.14)
1
1
+ 0 = 1, (λ + q − 1)s = q1 ,
s s
and estimating the right-hand side of (3.8) by the Hölder inequality, we obtain
Z
b(x)up1 ϕR dx
≤ Eλ
Z
Z
×
q1
a(x)v ϕR dx
|DϕR |qz
0
− zz
a
1 Z
q−1
qs + qz
(x)
0
qz 0 −1
ϕR
0
0
− ss
a
(x)
|DϕR |qs
qs0 −1
ϕR
q−10
qs
dx
(3.15)
10
qz
dx
.
Combining (3.12) and (3.15), we finally arrive at
Z
1−mn
a(x)v q1 ϕR dx
≤ Dλ EλN
×
and
Z
≤
Z
Z
r0
b− r (x)
×
Z
Z
s0
ϕR
0
ϕR
dx
n(q−1)
Z
0
qs
z0
a− z (x)
|DϕR |qz
0
qz 0 −1
ϕR
n0
qz
dx
(3.16)
m0
py
dx
(3.17)
0
p−10 Z
10
y0
|DϕR |py
pr
py
dx
b− y (x) py0 −1 dx
ϕR
1−mn
− rr
a− s (x)
qs0 −1
pr 0 −1
0
b
|DϕR |qs
|DϕR |pr
b(x)up1 ϕR dx
Eλ Dλm
0
s0
a− s (x)
(x)
|DϕR |pr
pr 0 −1
ϕR
0
|DϕR |qs
0
−1
ϕqs
R
0
dx
Z
m(p−1)
0
pr
0
b
− yy
(x)
|DϕR |py
py 0 −1
ϕR
0
0
q−10 Z
10
z0
|DϕR |qz
qs
qz
dx
,
a− z (x) qz0 −1 dx
ϕR
p−1
1
q−1
1
+ , m :=
+ .
(3.18)
pr
py
qs
qz
Simple calculations taking into account (3.13) and (3.14) give explicit values of m
and n, namely,
q−1
p−1
, n=
.
(3.19)
m=
q1
p1
Our assumptions imply that the exponent on the left-hand side of (3.16), (3.17) is
such that
p1 q1 − (p − 1)(q − 1)
1 − mn =
> 0.
p1 q1
Thus from (3.17) and our assumptions on a and b we have
Z
a(x)v q1 dx ≤ CRN +σ1 ,
(S R \S 1/R )∩BR (0)
Z
(3.20)
b(x)up1 dx ≤ CRN +σ2 .
n :=
(S R \S 1/R )∩BR (0)
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BLOW-UP OF SOLUTIONS
7
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 function ϕR ∈ C01 (RN ; [0, 1]) and integrating by parts, we obtain
Z
Z
(|Du|p−2 Du, DϕR ) dx,
a(x)|Dv|q1 ϕR (x) dx ≤
RN
RN
Z
Z
p1
(|Dv|q−2 Dv, DϕR ) dx,
b(x)|Du| ϕR (x) dx ≤
RN
RN
which because of relations
(|Du|p−2 Du, DϕR ) ≤ |Du|p−1 |DϕR |,
(|Dv|q−2 Dv, DϕR ) ≤ |Dv|q−1 |DϕR |
and the Hölder inequality, results in
Z
a(x)|Dv|q1 ϕR (x) dx
RN
≤
p−1
p1
b(x)|Du|p1 ϕR (x) dx
Z
(4.1)
RN
Z
×
RN
Z
p1
p−1
p1
1 −p+1
1− p
b− p1 −p+1 (x)|DϕR | p1 −p+1 ϕR
p1 −p+1
p1
(x) dx
,
b(x)|Du|p1 ϕR (x) dx
RN
Z
≤
a(x)|Dv|q1 ϕR (x) dx
q−1
q
1
(4.2)
RN
×
Z
q1
q−1
RN
q1
1 −q+1
1− q
a− q1 −q+1 (x)|DϕR | q1 −q+1 ϕR
q1 −q+1
q1
(x) dx
.
Substituting (4.1) into (4.2) and vice versa, we obtain
Z
1− (p−1)(q−1)
p1 q1
a(x)|Dv|q1 ϕR (x) dx
RN
≤
Z
q−1
RN
Z
×
q1
1− q
a− q1 −q+1 (x)|DϕR | q1 −q+1 ϕR
p1
p−1
RN
q1
1 −q+1
1− p
b− p1 −p+1 (x)|DϕR | p1 −p+1 ϕR
1 −q+1)
(p−1)(q
p1 q1
(x) dx
p1
1 −p+1
p1 −p+1
p1
(x) dx
,
1− (p−1)(q−1)
p1 q1
b(x)|Du|p1 ϕR (x) dx
Z
RN
≤
Z
q−1
RN
×
q1
Z
RN
p−1
q1
1 −q+1
1− q
a− q1 −q+1 (x)|DϕR | q1 −q+1 ϕR
p1
1− p
b− p1 −p+1 (x)|DϕR | p1 −p+1 ϕR
(x) dx
p1
1 −p+1
q1 −q+1
q
1
1 −p+1)
(q−1)(p
p1 q 1
(x) dx
;
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E. GALAKHOV, O. SALIEVA, L. UVAROVA
EJDE-2014/216
i. e.,
Z
a(x)|Dv|q1 ϕR (x) dx
RN
≤
Z
RN
×
Z
q1
1 −q+1
1− q
p1
p−1
RN
Z
q1
q−1
a− q1 −q+1 (x)|DϕR | q1 −q+1 ϕR
1− p
b− p1 −p+1 (x)|DϕR | p1 −p+1 ϕR
(x) dx
p1
1 −p+1
1 −q+1)
p(p−1)(q
q −(p−1)(q−1)
1 1
(x) dx
p
(4.3)
q1 (p1 −p+1)
1 q1 −(p−1)(q−1)
,
b(x)|Du|p1 ϕR (x) dx
RN
≤
Z
q−1
RN
×
q1
q1
1 −q+1
1− q
a− q1 −q+1 (x)|DϕR | q1 −q+1 ϕR
Z
p−1
RN
p1
1− p
b− p1 −p+1 (x)|DϕR | p1 −p+1 ϕR
(x) dx
p1
1 −p+1
p
(x) dx
p1 (q1 −q+1)
1 q1 −(p−1)(q−1)
(4.4)
1 −p+1)
p(q−1)(p
q −(p−1)(q−1)
1 1
.
Choosing test function ϕR ∈ C01 (RN ; [0, 1]) so that
(
1 (|x| ≤ R),
ϕR (x) =
0 (|x| ≥ 2R),
and
|DϕR (x)| ≤ cR−1
(x ∈ RN ),
(4.5)
we obtain
Z
N−
(p−1)((|α|+β)(q−1)+q1 (|γ|+δ+1))+p1 q1
p1 q1 −(p−1)(q−1)
,
N−
(q−1)((|γ|+δ)(p−1)+p1 (|α|+β+1))+p1 q1
p1 q1 −(p−1)(q−1)
.
a(x)|Dv|q1 dx ≤ cR
BR (0)
Z
b(x)|Du|p1 dx ≤ cR
BR (0)
Taking R → ∞, we complete the proof similarly to Theorem 2.2.
5. Proof of Theorem 2.5
Multiplying inequalities (2.8) by the test function ϕR ∈ C02k−1 (RN ; [0, 1]) and
integrating by parts, we obtain
Z
Z
q
a(x)|Dv| ϕR (x) dx ≤
(Du, D((−∆)k−1 ϕR )) dx,
RN
RN
Z
Z
b(x)|Du|p ϕR (x) dx ≤
(Dv, D((−∆)l−1 ϕR )) dx,
RN
RN
which by relations
(Du, D((−∆)k−1 ϕR )) ≤ |Du| · |D((−∆)k−1 ϕR )|,
(Dv, D((−∆)l−1 ϕR )) ≤ |Dv| · |D((−∆)l−1 ϕR )|,
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9
and the Hölder inequality, results in
Z
a(x)|Dv|q ϕR (x) dx
RN
Z
1/p
≤
b(x)|Du|p ϕR (x) dx
(5.1)
RN
Z
×
−
p
1
1
b− p−1 (x)|D((−∆)k−1 ϕR )| p−1 ϕR p−1 (x) dx
RN
p−1
p
,
Z
b(x)|Du|p ϕR (x) dx
Z
1/q
≤
a(x)|Dv|q ϕR (x) dx
RN
(5.2)
RN
q−1
q
1
− 1
q
.
a− q−1 (x)|D((−∆)l−1 ϕR )| q−1 ϕR q−1 (x) dx
Z
×
RN
Substituting (5.1) into (5.2) and vice versa, we obtain
1
1− pq
Z
a(x)|Dv|q ϕR (x) dx
RN
Z
≤
RN
×
q−1
q
1
− 1
pq
a− q−1 (x)|D((−∆)l−1 ϕR )| q−1 ϕR q−1 (x) dx
Z
RN
p−1
p
1
− 1
p
b− p−1 (x)|D((−∆)k−1 ϕR )| p−1 ϕR p−1 (x) dx
,
1
1− pq
b(x)|Du|p ϕR (x) dx
Z
RN
≤
Z
RN
×
q−1
q
1
− 1
q
a− q−1 (x)|D((−∆)l−1 ϕR )| q−1 ϕR q−1 (x) dx
Z
RN
p−1
p
1
− 1
pq
b− p−1 (x)|D((−∆)k−1 ϕR )| p−1 ϕR p−1 (x) dx
;
i.e.,
Z
a(x)|Dv|q ϕR (x) dx
RN
Z
≤
RN
×
−
q
1
Z
b
1
− p−1
k−1
(x)|D((−∆)
ϕR )|
p
p−1
RN
Z
1
a− q−1 (x)|D((−∆)l−1 ϕR )| q−1 ϕR q−1 (x) dx
q−1
pq−1
− 1
ϕR p−1 (x) dx
(p−1)q
pq−1
(5.3)
,
b(x)|Du|p ϕR (x) dx
RN
≤
Z
1
− q−1
a
RN
×
Z
RN
(x)|D((−∆)
l−1
ϕR )|
q
q−1
− 1
ϕR q−1 (x) dx
p(q−1)
pq−1
p−1
pq−1
p
1
− 1
.
b− p−1 (x)|D((−∆)k−1 ϕR )| p−1 ϕR p−1 (x) dx
(5.4)
10
E. GALAKHOV, O. SALIEVA, L. UVAROVA
EJDE-2014/216
Denote K = max{k, l}. Choosing the test function ϕR ∈ C02K−1 (RN ; [0, 1]) so
that
(
1 (|x| ≤ R),
ϕR (x) =
0 (|x| ≥ 2R),
and
|Dµ ϕR (x)| ≤ cR−1
(x ∈ RN ; 0 ≤ |µ| ≤ 2K − 1),
(5.5)
we have
Z
a(x)|Dv|q dx ≤ cRN −
|α|+β+(|γ|+δ+2l−1+(2k−1)p)q
pq−1
b(x)|Du|p dx ≤ cRN −
|γ|+δ+(|α|+β+2k−1+(2l−1)q)p
pq−1
,
BR (0)
Z
.
BR (0)
Taking R → ∞, 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 function ϕR ∈ C02K (RN ; [0, 1]), where
K = max{k, l}, and integrating by parts, we obtain
Z
Z
q
a(x)v ϕR (x) dx ≤
u · (−∆)k ϕR dx,
RN
RN
Z
Z
b(x)up ϕR (x) dx ≤
v · (−∆)l ϕR dx.
RN
RN
Taking into account that
u · (−∆)k ϕR ≤ u · |(−∆)k ϕR )|,
v · (−∆)l ϕR ≤ v · |(−∆)l ϕR )|
and using the Hölder inequality, we arrive at
Z
Z
1/p
a(x)v q ϕR (x) dx ≤
b(x)up ϕR (x) dx
RN
RN
×
Z
b
1
− p−1
k
(x)|(−∆) ϕR |
p
p−1
RN
Z
b(x)up ϕR (x) dx ≤
Z
RN
p−1
p
− 1
ϕR p−1 (x) dx
(6.1)
,
1/q
a(x)v q ϕR (x) dx
RN
×
Z
q
1
RN
−
1
a− q−1 (x)|(−∆)l ϕR | q−1 ϕR q−1 (x) dx
q−1
q
(6.2)
.
Substituting (6.1) into (6.2) and vice versa, we obtain
1
Z
1− pq
Z
q−1
q
1
− 1
pq
q
a(x)v ϕR (x) dx
≤
a− q−1 (x)|(−∆)l ϕR | q−1 ϕR q−1 (x) dx
RN
RN
×
Z
RN
Z
1
1− pq
b(x)u ϕR (x) dx
≤
p
Z
RN
p−1
p
1
− 1
p
b− p−1 (x)|(−∆)k ϕR | p−1 ϕR p−1 (x) dx
,
1
− q−1
a
l
(x)|(−∆) ϕR |
RN
×
Z
RN
1
q
q−1
− 1
ϕR q−1 (x) dx
p
−
1
q−1
q
b− p−1 (x)|(−∆)k ϕR | p−1 ϕR p−1 (x) dx
p−1
pq
;
EJDE-2014/216
BLOW-UP OF SOLUTIONS
11
i.e.,
Z
a(x)v q ϕR (x) dx
RN
Z
≤
RN
×
Z
RN
Z
q
1
−
1
a− q−1 (x)|(−∆)l ϕR | q−1 ϕR q−1 (x) dx
q−1
pq−1
(6.3)
(p−1)q
p
1
− 1
pq−1
b− p−1 (x)|(−∆)k ϕR | p−1 ϕR p−1 (x) dx
,
b(x)up ϕR (x) dx
RN
≤
Z
RN
×
p(q−1)
q
1
− 1
pq−1
a− q−1 (x)|(−∆)l ϕR | q−1 ϕR q−1 (x) dx
Z
RN
(6.4)
p−1
pq−1
p
1
− 1
.
b− p−1 (x)|(−∆)k ϕR | p−1 ϕR p−1 (x) dx
Choosing the test function ϕR ∈ C02K (RN ; [0, 1]) so that
(
1 (|x| ≤ R),
ϕR (x) =
0 (|x| ≥ 2R)
and estimate (5.5) holds, we obtain
Z
|α|+β+(|γ|+δ+2l+2kp)q
pq−1
,
a(x)|Dv|q dx ≤ cRN −
BR (0)
Z
|γ|+δ+(|α|+β+2k+2lq)p
pq−1
b(x)|Du|p dx ≤ cRN −
.
BR (0)
Taking R → ∞, we complete the proof similarly to Theorems 2.2, 2.4 and 2.5.
Acknowledgements. This research was supported by grants of Russian Foundation 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 Federation 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: galakhov@rambler.ru
Olga Salieva
Moscow State Technological University “Stankin”, Vadkovsky lane 3a, Moscow, 125994,
Russia
E-mail address: olga.a.salieva@gmail.com
Liudmila Uvarova
Moscow State Technological University “Stankin”, Vadkovsky lane 3a, Moscow, 125994,
Russia
E-mail address: uvar11@yandex.ru