On nonexistence of solutions to some nonlinear inequalities with transformed argument

On nonexistence of solutions to some nonlinear inequalities with transformed argument

Olga Salieva

Moscow State Technological University “Stankin”, 3a Vadkovsky lane, Moscow, 127055, Russia Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya ul., Moscow, 117198, Russia

Received 25 May 2016, appeared 23 January 2017 Communicated by László Simon

Abstract. We obtain results on nonexistence of nontrivial solutions for several classes of nonlinear partial differential inequalities and systems of such inequalities with trans- formed argument.

Keywords: nonlinear inequalities, nonexistence of solutions, transformed argument.

2010 Mathematics Subject Classification: 35J60, 35K55.

1 Introduction

In recent years conditions for nonexistence of solutions to nonlinear partial differential equa- tions and inequalities attract the attention of many mathematicians. This problem is not only of interest of its own, but also has important mathematical and physical applications. In par- ticular, Liouville type theorems of nonexistence of nontrivial positive solutions to nonlinear equations in the whole space or half-space can be used for obtaining a priori estimates of solutions to respective problems in bounded domains [1,4].

In [5–7] (see also references therein) sufficient conditions for nonexistence of solutions were obtained for different classes of nonlinear elliptic and parabolic inequalities using the nonlinear capacity method developed by S. Pohozaev [8]. On the other hand, there ex- ists an elaborated theory of partial differential equations with transformed argument due to A. Skubachevskii [9]. But the problem of sufficient conditions for nonexistence of solutions to respective inequalities with transformed argument remained open. Some special cases of such problems were treated in [2,3].

In this paper we obtain sufficient conditions for nonexistence of solutions to several classes of elliptic and parabolic inequalities with transformed argument and for systems of elliptic inequalities of this type.

The structure of the paper is as follows. In §2, we prove nonexistence theorems for semi- linear elliptic inequalities of higher order; in §3, for quasilinear elliptic inequalities; in §4, for

BEmail: olga.a.salieva@gmail.com

systems of quasilinear elliptic inequalities; and in §5, for nonlinear parabolic inequalities with a shifted time argument.

The lettercwith different subscripts or without them denotes positive constants that may depend on the parameters of the inequalities and systems under consideration.

2 Semilinear elliptic inequalities

Letk∈N. Consider a semilinear elliptic inequality

(−_∆)^ku(x)≥ |u(g(x))|^q (x∈_Rⁿ), (2.1) whereg∈ C¹(_Rⁿ;Rⁿ)is a mapping such that

(g1) there exists a constantc>0 such that|J_g⁻¹(_x)| ≥c>_{0 for all x}∈_Rⁿ; (g2) |g(x)| ≥ |x|for allx ∈_Rⁿ.

Example 2.1. The dilatation transform g(x) = γx with any γ ∈ _Rsuch that |γ| > 1 satisfies assumptions (g1) withc=|γ|⁻ⁿand (g2).

Example 2.2. The rotation transform g(x) = Ax, where A is a n×n unitary matrix (and therefore|g(x)|=|x|for all x∈_Rⁿ), satisfies assumptions (g1) withc=1 and (g2).

In some situations assumption (g2) can be replaced by a weaker one:

(g’2) there exist constantsc₀ >0 andρ>0 such that|g(x)| ≥c₀|x|for allx ∈_Rⁿ\B_ρ(0). Remark 2.3. We assume without loss of generality that c₀ ≤1.

Example 2.4. The contraction transform g(x) = γx with 0 < |γ| ≤ 1 satisfies assumptions (g1) withc= |γ|⁻ⁿ and (g’2) withc₀=|γ|and any ρ>0.

Example 2.5. So does the shift transform g(x) = x−x₀ for a fixed x₀ ∈ _Rⁿ with c = 1, c₀=1/2 andρ=2|x₀|.

Lemma 2.6. There exists a nonincreasing function ϕ(s)≥0in C^2k[_0,∞), satisfying conditions ϕ(s) =

(1 (0≤s≤1),

0 (s≥2), (2.2)

and

Z ₂

1

|ϕ⁰(s)|^q0

ϕ^q0−1(s)^ds<_{∞, (2.3)}

Z ₂

1

|_∆^kϕ(s)|^q0

ϕ^q0−1(s) ^ds<_∞. (2.4)

Here and below q⁰ = _q−^q₁.

Proof. Take ϕ(s)_{equal to} (₂−s)^λ with a sufficiently largeλ > 0 in a left neighborhood of 2 (see [7]).

Theorem 2.7. Let either n ≤ 2k and q > 1, or n > 2k and 1 < q ≤ _n−ⁿ_2k. Suppose that g satisfies assumptions (g1) and (g2). Then inequality(2.1)has no nontrivial solutions u∈ L_q,loc(_Rⁿ).

Proof. Assume for contradiction that a nontrivial solutions of (2.1) does exist. Let 0< R<_∞ (in particular, the case R=1 is possible). The function

ϕ_R(x) =ϕ |x|

R

,

where ϕ(s)is from Lemma2.6, will be used as atest functionfor inequality (2.1). Multiplying both sides of (2.1) by the test function ϕ_R and integrating by parts 2k times, we get

Z

Rⁿ|u(x)| · |_∆^kϕ_R(x)|dx≥

Z

Rⁿ|u(g(x))|^qϕ_R(x)dx. (2.5) Using (g1), (g2), and the monotonicity ofϕ_R, one can estimate the right-hand side of (2.5) from below as

Z

Rⁿ|u(g(x))|^qϕ_R(x)dx=

Z

Rⁿ|u(x)|^qϕ_R(g⁻¹(x))|J_g⁻¹(x)|dx≥c Z

Rⁿ|u(x)|^qϕ_R(x)dx. (2.6) On the other hand, applying the parametric Young inequality to the left-hand side of (2.5), we get

Z

Rⁿ|u(x)| ·_∆^kϕ_R(x) dx

≤ ^c 2

Z

Rⁿ|u(x)|^qϕ_R(x)dx+c₁ Z

Rⁿ

∆^kϕ_R(x)

q⁰

ϕ^1−q

0 R (x)dx

= ^c 2

Z

Rⁿ|u(x)|^qϕ_R(x)dx+c₁R^n−2kq0 Z

Rⁿ

∆^kϕ₁(x)^q

0

ϕ^1−q

0 1 (x)dx

= ^c 2

Z

Rⁿ|u(x)|^qϕ_R(x)dx+c₂R^n−2kq0

(2.7)

with some constantsc₁,c₂>0. Combining (2.5)–(2.7), we have c

2 Z

Rⁿ|u(x)|^qϕ_R(x)dx≤ c₂R^n−2kq0.

Restricting the integration domain in the left-hand side of the inequality, we obtain c

2 Z

BR(0)

|u(x)|^qdx≤ c₂R^n−2kq0.

Taking R→∞, we get a contradiction forn−2kq⁰ < 0, which proves the theorem in all cases except the critical one (wheren−2kq⁰ =_0).

In the critical case we get

Z

Rⁿ|u(x)|^qdx< _∞

and hence _Z

supp∆^kϕR

|u(x)|^qdx≤

Z

B2R(0)\BR(0)

|u(x)|^qdx →0 asR→_∞.

But (2.5), (2.6) and the Hölder inequality imply c

Z

B_R(₀)

|u(x)|^qdx≤ Z

supp∆^kϕ_R

|u(x)|^qdx ¹_q

· Z

supp∆^kϕ_R

∆^kϕ_R(x)^q

0

ϕ^1−q

0 R (x)dx

_q¹₀

(2.8)

and therefore Z

BR(0)

|u(x)|^qdx ≤c _Z

supp∆^kϕR

|u(x)|^qdx ¹_q

→0 asR→_∞

since the second factor on the right-hand side of (2.8) can be estimated from above byc₂R^n−2kq0 as before, wheren−2kq⁰ =0. Thus for a nontrivialuwe obtain a contradiction in this case as well. This completes the proof.

Theorem 2.8. Let either n ≤ 2k and q > 1, or n> 2k and1 < q≤ _n−ⁿ_2k. Suppose that g satisfies assumptions (g1) and (g’2). Then inequality(2.1)has no nontrivial solutions u∈ L_q,loc(_Rⁿ)such that

lρ:= lim

R→_∞

R

B2R(0)|u(x)|^qdx R

B_c0R(₀)\B_ρ(₀)|u(x)|^qdx < _∞ (2.9) (in particular, u∈Lq(_Rⁿ)).

Proof. Similarly to estimate (2.6), forR> ρwe get Z

Rⁿ|u(g(x))|^qϕ_R(x)dx =

Z

Rⁿ|u(x)|^qϕ_R(g⁻¹(x))|J⁻_g¹(x)|dx

≥c Z

Rⁿ\Bρ(0)

|u(x)|^qϕ_R x

c₀

dx≥c Z

Bc0R(0)\Bρ(0)

|u(x)|^qdx.

(2.10)

Then (2.5) and (2.7)–(2.10) imply Z

Bc0R(0)\Bρ(0)

|u(x)|^qdx≤c₁ Z

B2R(0)

|u(x)|^qdx+c₂R^n−2kq0,

where c₁,c₂ > 0, and the constant c₁ can be chosen arbitrarily small. Hence by assumption (2.9) forc₁ < _2l¹

ρ+1 and sufficiently largeRwe have Z

B_c0R(0)\Bρ(0)

|u(x)|^qdx≤2c₂R^n−2kq0,

i.e., the conclusion of Theorem2.7for any subcriticalqremains valid in this case as well. The critical case can be treated similarly to the previous theorem.

Further we consider the inequality

(−_∆)^ku(x)≥ |Du(g(x))|^q (x∈_Rⁿ). (2.11) Theorem 2.9. Let either n≤2k−1and q> 1, or n> 2k−1and1<q≤ _n−2kⁿ₊₁. Suppose that g satisfies assumptions (g1) and (g2). Then inequality(2.11)has no nontrivial solutions u∈W_q,loc¹ (_Rⁿ). Proof. Multiplying both sides of (2.11) by the test function ϕ_R and integrating by parts 2k−1 times, we get

Z

Rⁿ(Du(x),D(_∆^k−1ϕ_R(x)))dx≥

Z

Rⁿ|Du(g(x))|^qϕ_R(x)dx, which implies

Z

Rⁿ|Du(x)| · |D(_∆^k−1ϕ_R(x))|dx≥

Z

Rⁿ|Du(g(x))|^qϕ_R(x)dx. (2.12)

Using (g1) and (g2), we can estimate the right-hand side of (2.12) from below as Z

Rⁿ|Du(g(x))|^qϕR(x)dx=

Z

Rⁿ|Du(x)|^qϕR(g⁻¹(x))|J_g⁻¹(x)|dx

≥c Z

Rⁿ|Du(x)|^qϕ_R(x)dx.

(2.13)

On the other hand, applying the parametric Young inequality to the left-hand side of (2.12), we get

Z

Rⁿ|Du(x)| ·D(_∆^k−1ϕ_R(x)) dx

≤ ^c 2

Z

Rⁿ|Du(x)|^qϕ_R(x)dx+c₁ Z

Rⁿ

D(_∆^k−1ϕ_R(x))^q

0

ϕ^1−q

0 R (x)dx

≤ ^c 2

Z

Rⁿ|Du(x)|^qϕ_R(x)dx+c₂R^{n−(2k−1)q0}

(2.14)

with some constantsc₁,c₂>0. Combining (2.12)–(2.14), we have c

2 Z

Rⁿ|Du(x)|^qϕ_R(x)dx≤c₂R^{n−(2k−1)q0}.

Restricting the integration domain in the left-hand side of the inequality, we obtain c

2 Z

BR(0)

|Du(x)|^qdx≤c2R^{n−(2k−1)q0}.

Taking R →∞, we get a contradiction for n−(2k−1)q⁰ <0. The critical case can be treated similarly to the previous theorems.

Theorem 2.10. Let either n≤2k−1and q>1, or n>2k−1and1<q≤ _n−2kⁿ₊₁. Suppose that g satisfies assumptions (g1) and (g’2). Then inequality(2.11)has no nontrivial solutions u∈W_q,loc¹ (_Rⁿ) such that

m_ρ:= lim

R→_∞

R

B2R(0)|Du(x)|^qdx R

Bc0R(0)\Bρ(0)|Du(x)|^qdx <_∞ (2.15) (in particular, u∈W_q¹(_Rⁿ)).

Proof. It is similar to that of Theorem2.8.

3 Quasilinear elliptic inequalities

Further consider the inequality

−_∆pu(x)≥u^q(g(x)) (x∈ _Rⁿ), (3.1) where g∈C¹(_Rⁿ_;Rⁿ)satisfies assumptions (g1) and (g’2).

Theorem 3.1. Let p > 1 and p−1 < q ≤ ⁿ⁽_n^p₋⁻_p¹⁾. Suppose that g satisfies assumptions (g1) and (g’2). Then inequality(3.1)has no nontrivial nonnegative solutions u∈W_p,loc¹ (_Rⁿ)∩L_q,loc(_Rⁿ).

Proof. We use the same test functions ϕ_R as in the previous section. Chooseλso that 1−p<

λ <0. Multiplying both sides of (3.1) by u^λ(x)ϕ_R(x), integrating by parts, and applying the parametric Young inequality withη>0, we get

λ Z

Rⁿu^λ−1(x)|Du(x)|^pϕ_R(x)dx+

Z

Rⁿu^λ(x)|Du(x)|^p−1|Dϕ_R(x)|dx

≥

Z

Rⁿu^q(g(x))u^λ(x)ϕ_R(x)dx

≥c_η Z

Rⁿu^q+λ(g(x))ϕ_R(x)dx−η Z

Rⁿu^q+λ(x)ϕ_R(x)dx.

(3.2)

Since−_∆pu≥0,usatisfies the weak Harnack inequality Z

B2R(0)u^q+λ(x)dx≤cRⁿ inf

x∈B_R(0)u^q+λ(x) and by iteration

Z

B_2R(₀)u^q+λ(x)dx ≤cRⁿ inf

x∈Bc0R(0)u^q+λ(x)≤c Z

B_c0R(₀)u^q+λ(x)dx, possibly with a differentc>0 (see [10]). Therefore

Z

Rⁿu^q+λ(x)ϕ_R(g⁻¹(x))|J_g⁻¹(x)|dx ≥c Z

Bc0R(0)u^q+λ(x)dx

≥c Z

B2R(0)u^q+λ(x)dx ≥c Z

Rⁿu^q+λ(x)ϕR(x)dx.

(3.3)

For a sufficiently smallη>0 (note thatc_η →+_∞asη→+0), we can estimate the right-hand side of (3.2) from below, since due to (g1), (g’2), and (3.3)

c_η Z

Rⁿu^q+λ(g(x))ϕ_R(x)dx−η Z

Rⁿu^q+λ(x)ϕ_R(x)dx

=cη

Z

Rⁿu^q+λ(x)ϕ_R(g⁻¹(x))|J_g⁻¹(x)|dx−η Z

Rⁿu^q+λ(x)ϕ_R(x)dx

≥(c_ηc−η)

Z

Rⁿu^q+λ(x)ϕ_R(x)dx≥c₁ Z

Bc0R(0)\Bρ(0)u^q+λ(x)dx

≥c₂ Z

B2R(0)\Bρ(0)u^q+λ(x)dx

(3.4)

with some constantsc₁,c₂>_0.

On the other hand, applying the parametric Young inequality with exponents _p−^p₁ and p to the integrand at the left-hand side of (3.2) represented as

(pεϕ_R)^p

−1 p u

(λ−1)(p−1)

p ·(pεϕ_R)¹

−p p |Dϕ_R|

with 0< ε<|λ|, and then applying it again with exponents _λ+^q+_p^λ₋₁ and _q−^q+_p+^λ₁ (note that these exponents are greater than 1 for a sufficiently small|λ|due to the assumption q> p−1) to u^λ+p−1(x)|Dϕ_R(x)|^pϕ¹_R^−p(x)represented as

c₂(q+λ) 2(λ+p−1)^ϕR

^λ+_q+^p−_λ¹

u^λ+p−1·

c₂(q+λ) 2(λ+p−1)^ϕR

−^λ+_q+^p−_λ¹

|Dϕ_R|^pϕ¹_R^−p,

we get λ

Z

Rⁿu^λ−1(x)|Du(x)|^pϕ_R(x)dx+

Z

Rⁿu^λ(x)|Du(x)|^p−1|Dϕ_R(x)|dx

≤(λ+ε)

Z

Rⁿu^λ−1(x)|Du(x)|^pϕ_R(x)dx+c₃(ε)

Z

Rⁿu^λ+p−1(x)|Dϕ_R(x)|^pϕ¹_R^−p(x)dx

≤(λ+ε)

Z

Rⁿu^λ−1(x)|Du(x)|^pϕ_R(x)dx+ ^c2 2

Z

Rⁿu^q+λ(x)ϕ_R(x)dx +c₄(ε)

Z

Rⁿ|Dϕ_R(x)|^{pq(−q+p+λ1)}ϕ

1−^p_q⁽₋^q+_p+^λ₁⁾

R (x)dx ≤(λ+ε)

Z

Rⁿu^λ−1(x)|Du(x)|^pϕ_R(x)dx +^c2

2 Z

Rⁿu^q+λ(x)ϕ_R(x)dx+c₅(ε)R^n−p

(q+λ) q−p+1

(3.5)

with some constantsε,c3(ε),c₄(ε),c5(ε)>0. Combining (3.2)–(3.5), we have c₂

2 Z

B_2R(₀)\B_ρ(₀)u^q+λ(x)ϕ_R(x)dx≤c₅(ε)R^n−p

(q+λ) q−p+₁.

Choosing λ sufficiently close to 0 and taking R → ∞, we obtain a contradiction for n−

pq

q−p+1 < 0, i.e., p−1 < q < ⁿ⁽_n^p₋⁻_p¹⁾. The critical case can be treated similarly to the previous theorems.

Further we consider the inequality

−_∆pu(x)≥ |Du(g(x))|^q (x∈ _Rⁿ). (3.6) Theorem 3.2. Let p−1 < q ≤ ⁿ⁽_n^p₋⁻₁¹⁾. Suppose that g satisfies assumptions (g1) and (g2). Then inequality(3.6)has no nontrivial nonnegative solutions u∈W¹_p,loc(_Rⁿ)∩W_q,loc¹ (_Rⁿ).

Proof. Multiplying both parts of (3.6) by the test function ϕ_R and integrating by parts, we get Z

Rⁿ|Du(x)|^p−2(Du(x),Dϕ_R(x)))dx≥

Z

Rⁿ|Du(g(x))|^qϕ_R(x)dx, which implies

Z

Rⁿ|Du(x)|^p−1· |Dϕ_R(x)|dx ≥

Z

Rⁿ|Du(g(x))|^qϕ_R(x)dx. (3.7) Using (g1) and (g2), one can estimate the right-hand side of (3.7) from below as

Z

Rⁿ|Du(g(x))|^qϕ_R(x)dx=

Z

Rⁿ|Du(x)|^qϕ_R(g⁻¹(x))|J_g⁻¹(x)|dx

≥c₀ Z

Rⁿ|Du(x)|^qϕ_R(x)dx.

(3.8)

On the other hand, applying the Hölder inequality to the left-hand side of (3.7), we obtain Z

Rⁿ|Du(x)| · |Dϕ_R(x)|dx

≤ _Z

Rⁿ|Du(x)|^qϕ_R(x)dx

^p−_q¹ _Z

Rⁿ|Dϕ_R(x)|^q−qp+1ϕ

1−_q−^q_p+1 R (x)dx

^q−p_q⁺¹ .

(3.9)

Combining (3.7)–(3.9), we have Z

Rⁿ|Du(x)|^qϕ_R(x)dx≤c₁ Z

B2R(0)

|Dϕ_R(x)|^q−qp+1ϕ

1−_q−^q_p+1 R (x)dx

and hence _Z

BR(0)

|Du(x)|^qdx≤ c2Rⁿ⁻

q q−p+1

with some constantsc₁,c2 > 0. Taking R → ∞, we obtain a contradiction forn− _q−^q_p+1 < 0.

The critical case can be treated similarly to the previous theorems.

Remark 3.3. If g satisfies (g’2) instead of (g2), a version of Theorem 3.2 can be proven for a class of solutions that satisfy (2.15) (in particular, u ∈ W_p,loc¹ (_Rⁿ)∩W_q¹(_Rⁿ)) similarly to Theorems2.8 and2.10.

4 Systems of quasilinear elliptic inequalities

Now consider a system of quasilinear elliptic inequalities

(−_∆pu(x)≥v^q1(g₁(x)) (x∈_Rⁿ),

−_∆qv(x)≥u^p1(g₂(x)) (x∈_Rⁿ), (4.1) whereg₁,g₂∈C¹(_Rⁿ;Rⁿ)are mappings that satisfy (g1) and (g’2).

Introduce the quantities

σ₁ =n− ^qq1 q₁−q+₁^, σ2 =n− ^pp1

p₁−p+1,

σ=σ₁(p−1)(q₁−q+1) +σ₂q₁(p₁−p+1), τ=σ₁p₁(q₁−q+1) +σ₂(q−1)(p₁−p+1).

(4.2)

Then there holds the following.

Theorem 4.1. Let p,q,p₁,q₁>1, p−1< p₁, q−1<q₁, andmin(σ,τ)≤0. Suppose that g₁and g₂ satisfy assumptions (g1) and (g’2). Then system(4.1)has no nontrivial nonnegative solutions

(u,v)∈(W_p,loc¹ (_Rⁿ)∩L_p1,loc(_Rⁿ))×(W_q,loc¹ (_Rⁿ)∩L_q1,loc(_Rⁿ)).

Proof. Assume that there exists(u,v)– a nontrivial nonnegative solution of system (4.1). Let {ϕR}be the same family of test functions as in Sections 2 and 3.

Multiplying the first inequality (4.1) by u^λ_εϕ_R and the second one by v^λ_εϕ_R, where u_ε = u+ε,v_ε =v+ε,ε>0 and max{1−p, 1−q}< λ<0, we get

Z

v^q1(g₁(x))u^λ_ε(x)ϕR(x)dx ≤λ Z

|Du|^pu^λ_ε⁻¹ϕRdx+

Z

|Du|^p−1|DϕR|u^λ_ε dx, Z

u^p1(g₂(x))v^λ_ε(x)ϕ_R(x)dx ≤λ Z

|Dv|^qv^λ_ε⁻¹ϕ_Rdx +

Z

|Dv|^q−1|Dϕ_R|v^λ_ε dx, which can be rewritten as (note that|λ|= −λsinceλ<0)

Z

v^q1(g₁(x))u^λ_ε(x)ϕ_R(x)dx+|λ|

Z

|Du|^pu^λ_ε⁻¹ϕ_Rdx≤

Z

|Du|^p−1|Dϕ_R|u^λ_ε dx, (4.3) Z

u^p1(g₂(x))v^λ_ε(x)ϕ_R(x)dx+|λ|

Z

|Dv|^qv^λ_ε⁻¹ϕ_Rdx≤

Z

|Dv|^q−1|Dϕ_R|v^λ_ε dx. (4.4)

Here and below we omit the argument xif it is the only one in a certain integral, andRⁿif it is the integration domain. Application of the Young inequality to the right-hand sides of the obtained relations results in

Z

v^q1(g₁(x))u^λ_ε(x)ϕ_R(x)dx+ |λ| 2

Z

|Du|^pu^λ_ε⁻¹ϕ_Rdx≤ c_λ

Z |Dϕ_R|^p ϕ^p_R⁻¹

u^λ_ε^+p−1dx, (4.5) Z

u^p1(g₂(x))v^λ_ε(x)ϕ_R(x)dx+ |λ| 2

Z

|Dv|^qv^λ_ε⁻¹ϕ_Rdx≤ d_λ

Z |Dϕ_R|^q ϕ^q_R⁻¹

v^λ_ε^+q−1dx, (4.6) where constants c_λ andd_λ depend only on p,q, and λ. Further, multiplying each inequality (4.1) by ϕ_R and integrating by parts, we obtain

Z

v^q1(g₁(x))ϕ_R(x)dx≤ _Z

|Du|^pu^λ_ε⁻¹ϕ_Rdx

^p−_p¹ _Z

|Dϕ_R|^p ϕ_R^p−1

u⁽_ε^{1−λ)(p−1)}dx

!¹_p

, (4.7)

Z

u^p1(g₂(x))ϕ_R(x)dx≤ _Z

|Dv|^qv^λ_ε⁻¹ϕ_Rdx

^q−_q¹ _Z

|Dϕ_R|^q ϕ^q_R⁻¹

v⁽_ε^{1−λ)(q−1)}dx

!¹_q

. (4.8)

Note that the integrals on the left-hand sides of these inequalities can be estimated from below by R

B2R(0)v^q1(x)dx and R

B2R(0)u^p1(x)dx (with some positive multiplicative constants) respectively, similarly to the proofs of Theorems 2.7and3.1. Thus, combining (4.5)–(4.8) and takingε→0, we arrive to a priori estimates

Z

B2R(0)

v^q1dx ≤D_λ

Z |Dϕ_R|^p ϕ^p_R⁻¹

u^λ+p−1dx

!^p−_p¹

Z |Dϕ_R|^p ϕ^p_R⁻¹

u^{(1−λ)(p−1)}dx

!¹_p

, (4.9)

Z

B_2R(0)u^p1dx ≤E_λ

Z |Dϕ_R|^q ϕ^q_R⁻¹

v^λ+q−1dx

!^q−_q¹

Z |Dϕ_R|^q ϕ^q_R⁻¹

v^{(1−λ)(q−1)}dx

!¹_q

, (4.10)

where Dλ andEλ >0 depend only on p, q, andλ.

Apply the Hölder inequality with exponent r > 1 to the first integral on the right-hand side of (4.9):

Z |Dϕ_R|^p ϕ^p_R⁻¹

u^λ+p−1dx

!^p−_p¹

≤ _Z

u^(λ+p−1)rϕ_Rdx ^p−_pr¹

|Dϕ_R|^pr0 ϕ^pr

0−1 R

dx

!^p−1

pr0

, (4.11)

where 1 r + ¹

r⁰ =1.

Choosing the exponentr so that(λ+p−1)r= p₁, from (4.9) and (4.11) we get Z

B2R(0)v^q1dx

≤ D_{λ Z}

u^p1ϕ_Rdx

^p_pr⁻¹ _Z

|Dϕ_R|^pr0 ϕ^pr

0−1 R

dx

!^p−1

pr0

Z |Dϕ_R|^p ϕ^p_R⁻¹

u^{(1−λ)(p−1)}dx

!¹_p .

(4.12)

Applying the Hölder inequality with exponent y > 1 to the last integral on the right-hand side of (4.12), we obtain

Z |DϕR|^p ϕ_R^p−1

u^{(1−λ)(p−1)}dx≤ Z

u^{(1−λ)(p−1)y}ϕRdx

¹_y Z |DϕR|^py0 ϕ^py

0−1 R

dx

!_y¹₀

, (4.13)

where ¹_y+_y¹0 =1.

Choosingyin (4.13) so that(1−λ)(p−1)y= p₁and taking into account (4.12), we get the estimate

Z

B2R(0)v^q1dx≤ D_{λ Z}

B2R(0)u^p1dx

^p_pr⁻¹+_py¹ Z |Dϕ_R|^pr0 ϕ^pr

0−1 R

dx

!^p−1

pr0

Z |Dϕ_R|^py0 ϕ^py

0−1 R

dx

! ¹

py0

, (4.14) where the exponentsr andyare chosen so that









 1 y+ ¹

y⁰ =_1, (₁−λ)(p−₁)y= p₁, 1

r + ¹

r⁰ =1, (λ+p−1)r= p₁.

(4.15)

Note that this choice of r > 1 and y > 1 is possible due to our assumptions on p and p₁ provided thatλ < 0 is sufficiently small in absolute value. Similarly, choosings andz such

that 







 1 z + ¹

z⁰ =1, (1−λ)(q−1)z= q₁, 1

s + ¹

s⁰ =1, (λ+q−1)s=q₁,

(4.16)

and estimating the right-hand side of (4.10) by the Hölder inequality, we get Z

B2R(0)u^p1dx≤Eλ

Z

B2R(0)v^q1dx

^q_qs⁻¹+_qz¹ Z |Dϕ_R|^qs0 ϕ^qs

0−1 R

dx

!^q_qs⁻₀¹

Z |Dϕ_R|^qz0 ϕ^qz

0−1 R

dx

!_qz¹₀

. (4.17) Combining (4.14) and (4.17), we finally arrive at

Z

B2R(0)v^q1dx 1−µν

≤ D_λE_λ^ν

Z |DϕR|^qs0 ϕ^qs

0−1 R

dx

!^ν(q−1)

qs0

Z |DϕR|^qz0 ϕ^qz

0−1 R

dx

!^ν

qz0

×

Z |Dϕ_R|^pr0 ϕ^pr

0−1 R

dx

!^p−1

pr0

Z |Dϕ_R|^py0 ϕ^py

0−1 R

dx

! ¹

py0

(4.18)

and Z

B2R(0)u^p1dx 1−µν

≤E_λD^µ_λ

Z |DϕR|^pr0 ϕ^pr

0−1 R

dx

!^µ(p−1)

pr0

Z |DϕR|^py0 ϕ^py

0−1 R

dx

! ^µ

py0

×

Z |Dϕ_R|^qs0 ϕ^qs

0−1 R

dx

!^q−1

qs0

Z |Dϕ_R|^qz0 ϕ^qz

0−1 R

dx

! ¹

qz0

,

(4.19)

where

µ:= ^q−1 qs + ¹

qz, ν := ^p−1 pr + ¹

py. (4.20)

Simple calculations using (4.15) and (4.16) yield explicit values ofµandν, namely, µ= ^q−1

q₁ , ν= ^p−1

p₁ . (4.21)

Our assumptions imply that the exponents on the left-hand side of (4.18) and (4.19) are such that

1−µν= ^p1q1−(p−1)(q−1) p₁q₁ >0.

Thus from (4.19) we have Z

B_2R(0)v^q1dx≤CR^{p1q1−(pσ−1)(q−1)}, Z

B_2R(0)u^p1dx≤CR^{p1q1−(pτ−1)(q−1)}. (4.22) TakingR→_∞in (4.22), under the hypotheses of the theorem we arrive at a contradiction, which completes the proof.

Similarly one can consider a system

(−_∆pu(x)≥ |Dv(g(x))|^q1 (x∈ _Rⁿ),

−_∆qv(x)≥ |Du(g(x))|^p1 (x∈ _Rⁿ)_, ^(4.23) where p,q,p₁,q₁ >1 andp−1< p₁, q−1<q₁.

Introduce the quantities

σ =n− (p₁+p−1)q₁ p₁q₁−(p−1)(q−1)^, τ=n− (q₁+q−₁)p₁

p₁q₁−(p−1)(q−1)^.

(4.24)

Then one has

Theorem 4.2. Letmin(σ,τ)≤0. Then system(4.23)has no nontrivial solutions.

We leave the proof to the interested reader.

5 Nonlinear parabolic inequalities

Now let τ>0. Consider the semilinear parabolic inequality

∂u(x,t)

∂t + (−_∆)^ku(x,t)≥ |u(x,g(t))|^q (x∈ _Rⁿ;t ∈_R+) (5.1) with initial condition

u(x, 0) =u₀(x) (x∈_Rⁿ), (5.2) whereu₀∈ C(_Rⁿ)is a function that satisfies the condition

Z

Rⁿu₀(x)dx≥_{0, (5.3)}

andg:R+→_R+ is a continuous function such that (g3) t ≤g(t)andg⁰(t)≥1 for anyt ≥0.

Let 0<R,T <∞. We will use as a test function the product of two functions Φ(x,t) = ϕ

|x| R

·ϕ t

T

, where the function ϕ(s)is the one from Lemma2.6.