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Eurasian
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2015, Volume 6, Number 4
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KORDAN NAURYZKHANOVICH OSPANOV
(to the 60th birthday)
On 25 September 2015 Kordan Nauryzhanovich Ospanov, professor of the Department "Fundamental Mathematics" of the L.N. Gumilyov Eurasian National University, Doctor of Physical and Mathematical Sciences (2000), a member of the Editorial Board of our journal, celebrated his 60th birthday.
He was born on September 25, 1955, in the village Zhanata- lap of the Zhanaarka district of the Karaganda region. In 1976 he graduated from the Kazakh State University, and in 1981 he completed his postgraduate studies at the Abay Kazakh Pedagogical Institute.
Scientific works of K.N. Ospanov are devoted to application of methods of functional analysis to the theory of differential equations. On the basis of a local approach to the resolvent representation he has found weak conditions for the solvability of the singular generalized Cauchy-Riemann system and established coercive estimates for its solution. He has obtained a criterion of the spectrum discreteness for the resolvent of the system and the exact in order estimates of singular values and Kolmogorov widths. He has original research results on the coercive solvability of the quasilinear singular generalized Cauchy-Riemann system and degenerate Beltrami-type system. He has established important smoothness and approximation properties of non strongly elliptic systems. K.N. Ospanov has found separability conditions in Banach spaces for singular linear and quasi-linear second- order differential operators with growing intermediate coefficients and established a criterion for the compactness of its resolvent and finiteness of the resolvent type.
His results have contributed to a significant development of the theory of two- dimensional singular elliptic systems, degenerate differential equations and non strongly elliptic boundary value problems.
K.N. Ospanov has published more than 140 scientific papers. The list of his most important publications one may see on the
http://mmf.enu.kz/images/stories/photo/pasport/fm/ospanov
K.N. Ospanov is an Honoured Worker of Education of the Republic of Kazakhstan, and he was awarded the state grant "The best university teacher".
The Editorial Board of the Eurasian Mathematical Journal is happy to congratulate Kordan Nauryzkhanovich Ospanov on occasion of his 60th birthday, wishes him good health and further productive work in mathematics and mathematical education.
EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879
Volume 6, Number 4 (2015), 59 – 76
ON MONOTONICITY OF SOLUTIONS OF DIRICHLET PROBLEM FOR SOME QUASILINEAR ELLIPTIC EQUATIONS IN HALF-SPACES
O.A. Salieva
Communicated by V.I. Burenkov
Key words: quasilinear equations, monotonicity, nonexistence of solutions.
AMS Mathematics Subject Classification: 35J60, 35J70, 35J92.
Abstract. We prove the monotonicity of nonnegative bounded solutions to the Dirich- let problem for a quasilinear elliptic equation of the form−∆pu=f(u)g(xn)withp≥3 in a half-space, where xn is the normal coordinate of the argument. This assertion im- plies new nonexistence results for the case f(u) = uq with the appropriate values of q.
1 Introduction
Let n≥2. Denote Rn+ ={x= (x1, . . . , xn) : xn>0}, ∂Rn+={x= (x1, . . . , xn) : xn = 0}.
Consider the following Dirichlet problem:
−∆pu=f(u)g(xn) (x∈Rn+),
u= 0 (x∈∂Rn+), (1.1)
where ∆pu := div(|Du|p−2Du) is the p-Laplacian, and f is a nonnegative function satisfying the following condition:
f ∈ C1(R+∪ {0})∩C2(R+) and for any M > 0 there exist a = a(M) > 0 and A =A(M)>0such that
asq≤f(s)≤Asq and |f0(s)| ≤Asq−1 in [0, M] (F1) for some q > p−1.
A typical example of a functionf satisfying (F1) is f(u) = uq with q > p−1.
As forg(xn), we suppose that it is a nonnegative non-decreasing continuous function such that
g(txn)≤ctαg(xn) (G1)
for all (at least sufficiently small) t > 0 and xn > 0 with some constants c > 0 and α > −p independent of t. Further, in Theorem 1.2, we require in addition that there exists the limit
g0 := lim
xn→0g(xn)∈(0,+∞). (G2)
60 O.A. Salieva
As an example one can take g(xn) = xαn
1 +xαn with α ≥0.
We study the problems of existence and monotonicity of nonnegative bounded so- lutions of problem (1.1) in the case p≥3.
For 1 < p < 3 and g(xn) ≡ 1 the problem was studied by a number of authors.
In particular, the semilinear case (i.e., p= 2) was considered in the pioneering papers of H. Berestycki, L. Caffarelli, and L. Nirenberg [2], [3] for n = 2 and in those of E.N. Dancer (see [8]) for general n. For p 6= 2, the operator ∆p becomes nonlinear and either singular (for 1 < p < 2) or degenerate (for p > 2), which leads to various additional difficulties. The monotonicity of nonnegative bounded solutions to (1.1) with general n and g(xn) ≡ 1 was shown by A. Farina, L. Montoro, and B. Sciunzi in [11] for 2n+ 2
n+ 2 < p <2 and in [12] for 2 < p <3 (see also the development of this approach in [10]). We also mention the paper of H. Zou [21] where a nonexistence result for problem (1.1) is obtained in the functional class W1,p(Rn+)∩Lqloc(Rn+)under more restrictive assumptions on f. A detailed treatment of the case q < p−1 was given recently by E.N. Dancer, Y. Du, and M. Efendiev in [9]. For problems with singular coefficients, nonexistence results were obtained by M.F. Bidaut-V´eron and S.I. Pohozaev in [4] and by E.I. Galakhov and the author in [14].
The case p ≥ 3 and g(xn) ≡ 1 was considered in another recent paper of E.I.
Galakhov and the author [15]. Here we extend the methods of this paper to more general functions g.
The main statements of the present paper are as follows.
Theorem 1.1. Let p > 2. Assume that f satisfies (F1) and g satisfies (G1). Then any nonnegative nontrivial solution u(x0, xn) ∈ W1,p(Rn+)∩L∞(Rn+) of problem (1.1) is strictly monotonically increasing in xn on R+ for each fixed x0 ∈Rn−1.
Theorem 1.2. Problem (1.1) with f(u) = uq and g satisfying (G1) and (G2) has no nontrivial nonnegative bounded solutions if p > 2 and either
n−1≤ p(p+ 3)
p−1 , q > p−1 or
n−1> p(p+ 3)
p−1 , p−1< q < qc(n−1, p), where
qc(n, p) = [(p−1)n−p]2+p2(p−2)−p2(p−1)n+ 2p2p
(p−1)(n−1) (n−p)((p−1)n−p(p+ 3)) . Note that for g(xn)≡1 these results coincide with those of [15].
The method of proof is based on certain rescaling techniques and on the weak comparison principle in unbounded narrow domains, similarly to [13], [11], [12]. This allows us to apply the moving plane method due to A.D. Alexandrov and J. Serrin [1], [18].
We note that for p >2 the comparison principle can be derived from the weighted Poincar´e inequality proven in [5], but one must cope with the possible blow-up of the
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 61 constant in this inequality as the functions under comparison (i.e., the solution u and its reflection with respect to a moving plane) approach zero. In [12], this blow-up is balanced by the power-like decay of f, but such an argument works only for2< p <3.
Therefore we modify the approach used in [12] as follows:
1. First, we use a scaling argument to show that u is monotonic in some layer near the boundary. In fact, the boundary maximum principle due to J.L. V´azquez [20]
implies that for eachx0 = (x1, . . . , xn−1)∈Rn−1 there exists a constantδ(x0)>0 such that the function u(x0,·) is strictly increasing in xn for all xn ∈ (0, δ(x0)).
We show that such δ >0 can be chosen independently of x0. Hence the moving plane process can get started.
2. In order to be able to continue this process for any xn, we formulate the com- parison principle for the solution uand its reflectionuλ with respect to a moving plane xn = λ rather than for generic u and v. This allows us to use the specific properties of uλ.
3. In the proof of the comparison principle, we estimate (u− uλ)+ on a narrow unbounded set Aλ,δ such that u(x)≤ uλ(x) on its boundary, as in [11] and [12].
By another scaling argument, we show that u and uλ are uniformly bounded away from zero on the set supp(u−uλ)+∩Aλ,δ. Therefore we can apply tou−uλ
the weighted Poincar´e inequality with a uniformly bounded constant in any set from a sufficiently fine covering of Aλ,δ, which implies that (u−uλ)+ ≡ 0, that is, u(x) ≤ uλ(x) in Aλ,δ, and hence the moving plane process can be continued.
Thus we obtain the monotonicity of u(x0, xn) inxn ∈R+ for any x0 ∈Rn−1. Since monotonic solutions of (1.1) are necessarily stable, this implies nonexistence of positive bounded solutions to (1.1) withf(u) =uq for anyq∈(p−1, qcr(n, p)), where qcr(n, p)is the critical exponent for stable solutions of (1.1) obtained in [6]. By passing to the limit asxn→ ∞, we can extend this result to the caseqcr(n, p)≤q < qcr(n−1, p).
The rest of the paper consists of three sections. Section 2 contains some known results, which we will use in the sequel. In Section 3, we formulate our preliminary results, namely, lower estimates on δ and u based on scaling arguments. In Section 4, we prove Theorems 1.1 and 1.2.
2 Some known results
In the sequel we will make use of the following results in [12]. Let Ω be an open set in Rn. For a nonnegative weight functionρ ∈L1(Ω), we denote
L2ρ(Ω) ={u: Ω→R, Z
Ω
ρu2dx <∞},
Hρ1,2(Ω) ={u: Ω→R, kuk2H1,2 ρ (Ω)=
Z
Ω
ρ(u2+|∇u|2)dx <∞}, H0,ρ1,2(Ω) - the closure ofC0∞(Ω) with respect to the norm ofHρ1,2(Ω).
62 O.A. Salieva
Theorem 2.1. Let C∗ >0. Assume that ρ is a weight function such that Z
Ω
dy
ρt|x−y|γ ≤C∗, (2.1)
with t = p−1p−2r, p−1p−2 < r <1, and n−2t < γ < n−2 if n ≥3 (or γ = 0 if n = 2).
Then, for any w∈H0,ρ1,2(Ω)∩W01,1(Ω),
kwkLq(Ω)≤Cρk∇wkL2ρ(Ω) (2.2) for any 1≤q <2∗(t), where
1 2∗(t) = 1
2− 1 n + 1
t 1
2 − γ 2n
, (2.3)
Cρ= ˆC(C∗)2t1(Cη)1−2t1, (2.4) Cˆ = nω1
n, where ωn is the volume of the unit ball in Rn, and Cη =
1−η
α n−η
1−η ω1−
α
n n|Ω|αn−η
with
η= 1− 1 2t − 1
q and
α=n−
n−1− γ 2t
2t 2t−1. Proof. See Theorem 5.1, [12], with Ω1 = Ω and Ω2 =∅.
Corollary 2.1. Assume that w∈H0,ρ1,2(Ω). If the weight ρ fulfils (2.1), then kwkL2(Ω) ≤KpCρ|∇wkL2ρ(Ω),
and there exists 0< θ <1 such that
Kp =C|Ω|(p−1)nθ (2.5)
and C > 0 depends only on the numerical parameters.
Proof. See Corollary 5.3, [12], with Ω1 = Ω and Ω2 =∅.
Proposition 2.1. Let u∈C1(Rn+) be a solution to problem (1.1) with p >1. Assume that f satisfies (F1) and that |∇u| is uniformly bounded on Rn+. Let Ω0 ⊂⊂ Rn+ and 0< δ <dist(Ω0, ∂Rn+) be such that f(u(x))>0 in Ω0δ, where
Ω0δ ={x∈Rn+: d(x,Ω0)< δ} ⊂⊂Rn+.
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 63
Consider a finite covering Ω0 ⊂ SS
i=1
Bδ(xi) with xi ∈ Ω0 and S =S(δ). Then there exists C(n)>0 such that
Z
Ω0
dy
|∇u(y)|τ|x−y|γ ≤C(n) S(δ) a2δ2( inf
x∈Ω0δu(x))2q (2.6) where γ is as in Theorem 2.1, a=a(M) with M = sup
x∈Ω0δ
u(x) as in (F1), and max{p−2,0} ≤τ ≤p−1.
Proof. See Proposition 4.2 and Corollary 4.3, [12], and references therein.
Remark 6. In particular, if in Proposition 2.1 Ω0 is a cube of diameter d, then (2.6) holds with
S(δ) = d
δ
+ 1 n
(see Remark 4.4, [12]).
3 Preliminary results
For our auxiliary results, we will need the condition (F) sup
x∈Rn+
f(u(x)) up−1(x) <∞ or a stronger one
(F0) lim
u→0+
f(u) up−1 = 0.
Forx0 ∈Rn−1, we denote z(x0) = inf
xn>0 : ∂u(x0, xn)
∂xn
= 0
. (3.1)
Remark 7. One necessarily has z(x0) > 0 for any x0 ∈ Rn−1 due to the boundary maximum principle for quasi-linear operators [20].
Lemma 3.1. Let p > 1. Suppose that f satisfies (F1), g satisfies (G1), u ∈ W1,p(Rn+)∩ L∞(Rn+) is a nonnegative nontrivial solution of problem (1.1), and the pair (f, u) satisfies (F).
Then there exists κ >0 such that u(x0, xn)≥u(x0, yn) for all xn, yn such that 0≤yn≤xn<κ.
64 O.A. Salieva
Proof. If inf
x0∈Rn−1
z(x0)>0, the assertion obviously holds, sinceu= 0on∂Rn+ andu≥0 in Rn+. Hence we suppose that
inf
x0∈Rn−1
z(x0) = 0. (3.2)
By Remark 7, z(0)>0. Define
Z0 ={x0 ∈Rn−1 : z(x0)≥z(0)},
Zk ={x0 ∈Rn−1 : 2−kz(0)≤z(x0)≤21−kz(0)} (k ∈N). (3.3) Evidently, Rn−1 =
∞
S
k=0
Zk. Some of the sets Zk may be empty.
We claim that there exists a sequence{x0m} of pointsx0m ∈Rn−1 with the following properties.
1. For each m ∈ N, there exists km ∈ N such that x0m ∈ Zkm and km → ∞ as m → ∞.
2. Definerm = dist x0m,
∞
S
j=km+1
Zj
!
. Then
m→∞lim sup 2kmrm =∞. (3.4)
In fact, a sequence with property 1 does exist, since otherwise the number of non- empty sets Zk would be finite, say K, and hence
inf
x0∈Rn−1
z(x0)≥2−Kz(0), which would contradict (3.2).
Now assume that each sequence with property 1 does not satisfy 2, that is, for each such sequence one has
m→∞lim sup 2kmrm <∞. (3.5) We construct a sequence {x0m} such that x01 = 0 and x0m+1 ∈
∞
S
j=km+1
Zj is chosen so that
|x0m+1−x0m| ≤2dist x0m,
∞
[
j=km+1
Zj
!
= 2rm.
We show that the series
∞
X
m=0
|x0m+1−x0m| necessarily diverges. In fact, otherwise one would have
|x0m| ≤ |x00|+
m−1
X
j=0
|x0j+1−x0j| ≤0 +
∞
X
j=0
|x0j+1−x0j| ≤
≤2
∞
X
j=0
rj ≤2c
∞
X
j=0
2−kj <∞ (m∈N),
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 65 where c:= lim
j→∞sup 2kjrj <∞ due to (3.5).
Thus the sequence {x0m} is bounded and therefore converges to a point x0∗ ∈ Rn−1 up to a subsequence. Denote zm =z(x0m). Sinceu∈C1,β(Rn+)and ∂u(x0m, zm)
∂xn = 0 by (3.2), for some constant c >0 we have
∂u(x0m,0)
∂xn = ∂u(x0m,0)
∂xn −∂u(x0m, zm)
∂xn ≤czmβ →0as m→ ∞ and by continuity
∂u(x0∗,0)
∂xn = 0,
which is impossible due to the boundary maximum principle.
Hence claim (3.4) holds, i. e. there exists a sequence {x0m} with elements in Rn−1 such that
x0m ∈Zkm, Brm/2(x0m)⊂
km
[
j=0
Zj, that is,
z(x0)≥2−kmz(0)≥z(x0m)/2 =zm/2for all x0 ∈Brm/2(x0m). (3.6) Now for each m∈N we introduce a rescaled function
um(x) = 1
u(x0m, zm) ·u(˜xm), where x˜m = (zmx1+xm1 , . . . , zmxn−1+xmn−1, zmxn).
By (1.1) and (F), one has
−∆pum(x) = − zmp
up−1(x0m, zm)∆pu(˜xm) =dmup−1m (x)g(xn), (3.7) where
0≤dm =zpmf(u(˜xm))g(zmxn)
up−1(˜xm)g(xn) ≤zmp+α· sup
x∈Rn+
f(u(x))
up−1(x). (3.8) Evidently, one hasum(0, . . . ,0,1) = 1and ∂um(0, . . . ,0,1)
∂xn = 0. Due to (3.1) and (3.6), u is monotonic in the cylinder
Zm :=n
(x0, xn) : 0≤ |x0−x0m| ≤ rm
2 ,0≤xn≤ zm 2
o ,
and therefore the rescaled functions um are monotonic in the respective cylinders Zm0 :=
(x0, xn) : 0≤ |x0| ≤ rm
2zm, 0≤xn ≤ 1 2
. Note that by (3.3) and the definition zm =z(x0m) with x0m ∈Zkm we have
2−kmz(0)≤zm ≤21−kmz(0),
66 O.A. Salieva
which by (3.4) implies
m→∞lim rm
2zm ≥ lim
m→∞
2km−1rm
z(0) =∞. (3.9)
Being a solution of the equation
−∆pum(x) = dm(x)up−1m (x)g(xn) (3.10) obtained in (3.7), um satisfies the Harnack inequality (see, e.g., Theorem 7.2, [17]), i.e., for some constant c1 >0, each δ > 0, and each compact set X ⊂ Rn+ containing (0, . . . ,0,1)we get
sup
x∈Xδ
um(x)≤c1 inf
x∈Xδum(x)≤c1,
where Xδ ={x = (x1, . . . , xn) ∈ X : xn ≥δ}. Due to the monotonicity of um(x0, xn) in xn∈[0,1/2] inZm0 , for X ⊂Zm0 this implies
sup
x∈X
um(x)≤c1.
By standard estimates (see [19]) for the solution um of (3.10) with dm satisfying (3.8) we have
kumkC1,β(X) ≤c2(X) with some β >0and c2(X)>0.
Due to (3.9), the sequence of cylinders Zm0 covers the whole layer Rn−1×[0,1/2].
Hence, by the Arzela–Ascoli Theorem, there exists a subsequence of {um} converging to some function
u0 ∈Cloc1,β0(Rn+) with β0 >0. Moreover, by the fact that lim
m→∞zm = 0, (3.7)–(3.8), and the boundedness of u it follows that u0 satisfies
−∆pu0 = 0 (x∈Rn+),
u0 = 0 (x∈∂Rn+). (3.11)
Hence u0(x0, xn) =c2xn with some c2 ∈R by Theorem 3.1 [16]. Furthermore, one has c2 = 1, since u0(0, . . . ,0,1) = lim
k→∞um(0, . . . ,0,1) = 1. On the other hand,
∂u0(0, . . . ,0,1)
∂xn = lim
m→∞
∂um(0, . . . ,0,1)
∂xn = 0,
which contradicts u0(x0, xn) = xn. Thus the assumption (3.2) cannot hold. This completes the proof.
Now defineuλ(x0, xn) =u(x0,2λ−xn) for λ >0,x0 ∈Rn−1, and xn∈[0, λ].
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 67 Lemma 3.2. Let p >1. Suppose that f satisfies (F0), g satisfies (G1) and (G2), and u∈W1,p(Rn+)∩L∞(Rn+) is a nonnegative nontrivial solution of problem (1.1). Denote
Aλ ={(x0, xn) : uλ(x0, xn)< u(x0, xn), 0≤xn ≤λ}.
Then for each u such that u(x0, xn) monotonically increases in xn ∈ [0, y] for each fixed x0 ∈Rn−1 and some y∈(0, λ) one has
cλ := inf
x∈Aλu(x)>0. (3.12)
Proof. Suppose, to the contrary, that inf
x∈Aλu(x) = 0. Then there exists a sequence {(x0k, yk)} such that
k→∞lim u(x0k, yk) = 0 (3.13) and (x0k, yk)∈Aλ, that is, for some z(x0k)∈[0, λ) one has
uλ(x0k, z(x0k)) =u(2λ−x0k, z(x0k))< u(x0k, z(x0k)).
Since u(x0k, xn) monotonically increases in xn ∈ [0, y], for some t(x0k) ∈ [y, z(x0k)) one has ∂u(x0k, xn)
∂xn
xn=t(x0k)
= 0, and the sequence{t(x0k)} converges to some t0 ∈[y, λ] up to a subsequence. Now introduce the family of functions
uk(x0, xn) = u(x0 +x0k, xn) u(x0k, xn) . Note that
∂uk(0, xn)
∂xn
xn=t(x0k)
= 0. (3.14)
Similarly to (3.7), due to (1.1) one has
−∆puk(x) = − 1
up−1(x0k, xn)∆pu(x0+x0k, xn) =
= 1
up−1(x0k, xn)up−1(x0+x0k, xn)· f(u(x0+x0k, xn))g(xn)
up−1(x0+x0k, xn) =dkup−1k (x)g(xn),
(3.15)
where dk:= f(u(x0+x0k, xn))
up−1(x0+x0k, xn) tends to zero as k → ∞ due to (3.13) and (F0).
Repeating the argument of the proof of Lemma 3.1, we obtain that there exists u0(x) := lim
k→∞uk(x) = cxn with c >0. But (3.14) implies
∂u0(0, . . . ,0, xn)
∂xn
xn=t0
= 0.
This contradiction proves the claim.
To prove the central results of the present paper, we will also need the following statement on passing to the limit.
68 O.A. Salieva
Lemma 3.3. Let a function u ∈ Cloc1,β(Rn+;R+) with β > 0 be a weak solution of the equation
−∆pu(x) =uq(x)g(xn) (x= (x0, xn)∈Rn+ =Rn−1×R+), (3.16) where p, q > 1 and g satisfies (G1), (G2). Suppose also that for any x0 ∈ Rn−1 there exists a positive bounded limit
U(x0) := lim
xn→∞u(x0, xn), (3.17)
where the convergence in (3.17) is uniform inC1(G)for any compact subset G⊂Rn−1. Then U is a weak solution of the equation
−∆pU(x0) =g0Uq(x0) (x0 ∈Rn−1), (3.18) where g0 is the limit in hypothesis (G2).
Proof. The locally uniform convergence of (3.17) in C1 means, in particular, that
∂u(x0, xn)
∂xi → ∂U(x0)
∂xi (i= 1, . . . , n−1),
∂u(x0, xn)
∂xn →0
(3.19)
uniformly on any compact subset G ⊂ Rn−1. Introduce test functions ϕ ∈ D(Rn−1) and ψ ∈ D(R) with suppψ ⊂(0,1) and
1
R
0
ψ(xn)dxn = 1. Multiplying equation (3.16) by ψ(xn−k)ϕ(x0) (k∈N) and integrating by parts, we get
Z
R
Z
Rn−1
uq(x0, xn)g(xn)ψ(xn−k)ϕ(x0)dxndx0 =
= Z
R
Z
Rn−1
|Du(x0, xn)|p−2∂u(x0, xn)
∂xn · ∂ψ(xn−k)
∂xn ϕ(x0)dxndx0+ +
Z
R
Z
Rn−1
|Du(x0, xn)|p−2
n
X
i=2
∂u(x0, xn)
∂xi ·∂ϕ(x0)
∂xi ψ(xn−k)dxndx0.
(3.20)
Change of variables and passing to the limit as k → ∞ result in Z
R
Z
Rn−1
uq(x0, xn)g(xn)ψ(xn−k)ϕ(x0)dxndx0 = Z
R
Z
Rn−1
uq(x0, s+k)g(s+k)ψ(s)ϕ(x0)ds dx0 →
→g0 Z
R
ψ(s)ds Z
Rn−1
Uq(x0)ϕ(x0)dx0 =g0 Z
Rn−1
Uq(x0)ϕ(x0)dx0,
(3.21)
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 69 Z
R
Z
Rn−1
|Du(x0, xn)|p−2∂u(x0, xn)
∂xn · ∂ψ(xn−k)
∂xn ϕ(x0)dxndx0 =
= Z
R
Z
Rn−1
|Du(s+k, x0)|p−2∂u(s+k, x0)
∂s · ∂ψ(s)
∂s ϕ(x0)ds dx0 →
→ Z
R
Z
Rn−1
|DU(x0)|p−2∂U(x0)
∂s · ∂ψ(s)
∂s ϕ(x0)ds dx0 = 0.
(3.22)
Similarly Z
R
Z
Rn−1
|Du(x0, xn)|p−2
n−1
X
i=1
∂u(x0, xn)
∂xi · ∂ϕ(x0)
∂xi ψ(xn−k)dxndx0 =
= Z
R
Z
Rn−1
|Du(s+k, x0)|p−2
n−1
X
i=1
∂u(s+k, x0)
∂xi · ∂ϕ(x0)
∂xi ψ(s)ds dx0 →
→ Z
R
ψ(s)ds Z
Rn−1
|DU(x0)|p−2
n−1
X
i=1
∂U(x0)
∂xi · ∂ϕ(x0)
∂xi dx0 =
= Z
Rn−1
|DU(x0)|p−2
n−1
X
i=1
∂U(x0)
∂xi ·∂ϕ(x0)
∂xi dx0.
(3.23)
Combining (3.20)–(3.23), we get g0
Z
Rn−1
Uq(x0)ϕ(x0)dx0 = Z
Rn−1
|DU(x0)|p−2(DU, Dϕ)dx0 (3.24)
for any ϕ ∈C01(Rn−1). Thus U is a positive weak solution of equation (3.18).
4 Proof of Theorems 1.1 and 1.2
Proof of Theorem 1.1. Now denote
Λ ={λ≥0 : u(x)≤uλ(x)for all x0 ∈Rn−1, xn ∈[0, λ]}
and λ0 = sup Λ. By Lemma 3.1, (0,κ/2) ⊂ Λ and hence λ0 ≥ κ/2 > 0. To prove Theorem 1.1, we must show that λ0 = ∞, that is, Λ = R+. Since Λ is closed by the continuity of u and non-empty, it suffices to show that it is relatively open in the topology of R+.
Suppose the converse. Similarly to the proof of Theorem 1.1 in [12], we multiply (1.1) and the corresponding equation for uλ by the test function
Ψ = (u−uλ)α+ϕ2RχRn−1×[δ;+∞),
where α >0 and δ >0 will be specified later. Here ϕR∈ C0∞(Rn−1) is a nonnegative function such that
ϕR ≡1 inBR(0)⊂Rn−1, ϕR ≡0 inRn−1\B2R(0),
|DϕR| ≤cR−1 inB2R(0)\BR(0) with some c > 0.
(4.1)
70 O.A. Salieva
After subtraction, for each λ >0 we obtain in a standard way (see [12]) αc1
Z
C(2R)
(|Du|p−2 +|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α+1+ ϕ2Rdx≤
≤c2 Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+| · |Dϕ2R|(u−uλ)α+dx+
+ Z
C(2R)
(f(u)g(xn)−f(uλ)g(xn,λ))(u−uλ)α+ϕ2Rdx:=I1+I2,
(4.2)
whereC(2R) ={(x0, xn) : |x0| ≤2R, δ ≤xn≤λ}, xn,λ= 2λ−xn ≥xnfor 0< xn≤λ.
Setting
I1 :=c2 Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+| · |Dϕ2R|(u−uλ)α+dx (4.3)
and
I2 :=
Z
C(2R)
(f(u)−f(uλ))(u−uλ)α+ϕ2Rdx, (4.4)
we can rewrite inequality (4.2) as αc1
Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α+1+ ϕ2Rdx ≤I1+I2. (4.5)
We proceed in two steps:
Step 1: Estimation of I1.
We use the argument from [12] (Section 6, step 1), simplifying it by means of Lemma 3.1. Namely, by (4.3), using the parametric Young inequality, we obtain
I1 ≤I1a+I1b, (4.6)
where
I1a :=δ0Cˆ Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2ϕ2R(u−uλ)α−1+ dx (4.7)
and
I1b := ˆCδ0 Z
C(2R)
(|Du|p−2+|Duλ|p−2)|DϕR|2(u−uλ)α+1+ dx. (4.8) For sufficiently smallδ0 >0, I1a can be estimated from above by the left-hand side of (4.5). It remains to estimate I2b.
Note that by Proposition 4.2 [11], for any ε > 0, if λ − y < ε, one can find δ = δ(ε) > 0 such that supp(u−uλ)+ ⊂ Aλ,δ, where Aλ,δ = Rn−1 ×[y−δ, λ] and δ(ε)→ 0+ as ε →0+. Here the case supp(u−uλ)+∩(Rn−1×[0, δ]) 6=∅ is excluded
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 71 due to the factor χRn−1×[δ;+∞) in the test function, where δ =δ(ε)and ε >0 is still to be fixed. Thus, if we denote the width of supp(u−uλ)+ in the direction xn bydλ:
dλ := sup
(x0,z1)∈supp (u−uλ)+
sup
(x0,z2)∈supp (u−uλ)+
|z1−z2|, we have dλ ≤d(ε) :=ε+δ(ε), whered(ε)→0 asε→0.
Now consider a family {Qi}Ni=1 of N =N(R) disjoint cubes Qi with edge d(ε) and with the xn coordinate of the center, say yC, such thatyC = y−δ(ε)+λ2 and
C(2R)⊂[
i=1
Qi. (4.9)
It follows that the diameter of each cube Qi is diamQi =√
nd(ε), i= 1, . . . , N. (4.10) By (4.1) and (4.8), since∇u,∇uλ ∈L∞(Rn+), we get
Ib1 ≤
N
X
i=1
C δ0R2
Z
C(2R)∩Qi
[(u−uλ)+]α+12 2
dx. (4.11)
Similarly to [12], due to Proposition 2.1 we can use in each cube Qi the weighted Poincar´e inequality of Corollary 2.1 with ρ≡ |Du|p−2 and take advantage of the con- stant Cˆ that turns to be independent of the index i in (4.9). Thus we obtain
Ib1 ≤
N
X
i=1
Ci∗ C δ0R2
Z
Qi
|Du|p−2[(u−uλ)+]α−1|D(u−v)+|2dx. (4.12)
We estimate the constant Ci∗ by Proposition 2.3 and Remark 2.4. Denote C := sup
x∈Rn+
|u(x)−u(z)|
|x−z| ,
which is finite due to the fact that u ∈ L∞(Rn+) and standard elliptic estimates. By Lemma 3.2, we have
inf
x∈supp(u−uλ)+
u(x) =cλ >0, (4.13)
where cλ is the constant defined in (3.12), whence one has
x∈(Qinfi)δ
u(x)≥cλ/2>0 (i= 1, . . . , N). (4.14) Moreover, by Lemma 3.1 we havey≥κ >0and thusxn >κ−δ > 0insupp(u−uλ)+, whence
dist(Qi,{u= 0})≥κ−δ >0 (i= 1, . . . , N). (4.15)
72 O.A. Salieva
Combining Proposition 2.1 and Remark 6 with (4.14) and (4.15), we deduce that the constants Ci∗ in (4.12) are in fact independent of i and (unlike the argument in [12]) of R. Hence,
I1 ≤δ0Cˆ Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2ϕ2R(u−uλ)α−1+ dx+
+C(dQ, δ0) R2
Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α−1+ dx,
(4.16)
where δ0 >0 can be chosen arbitrarily small and
C1(dQ, δ0)→0as dQ →0.
Step 2. Estimation of I2 in (4.5).
We can assume that p ≥ 3 and therefore q > p−1≥ 2, since the case 2< p < 3 was considered in [12]. Note that
f(u)g(xn)−f(uλ)g(xn,λ) = (f(u)−f(uλ))g(xn) +f(uλ(g(xn)−g(xn,λ))
≤c(λ)(f(u)−f(uλ))
with some constant c(λ) > 0 by the assumptions on the continuity and monotonicity of g(xn). By the Taylor expansion of f(·), we obtain
f(u) =f(uλ) +f0(ξ)(u−uλ) with uλ(x)≤ξ(x)≤u(x) forx∈supp(u−uλ)+.
Forx= (x0, y)such that x0 ∈Rn−1 and 0< δ <minn
κ, cλ
2C −(λ−y)o
with κ defined in Lemma 3.1, the Lipschitz continuity of u and (4.13) imply inf
x∈supp(u−uλ)+
uλ(x)≥ cλ 2 >0 and consequently
ξ0 := inf
x∈supp(u−uλ)+
min{u(x), uλ(x)}> cλ
2 >0. (4.17)
On the other hand,
ξ1 := sup
x∈supp(u−uλ)+
max{u(x), uλ(x)} ≤ kukq−1L∞(Rn) <∞. (4.18)
On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations 73 Therefore, similarly to the previous step (see (4.11) and (4.12), we can apply to I2 the weighted Poincar´e inequality given in Corollary 2.1:
I2 ≤Ckξ1kq−1
N
X
i=1
Z
C(2R)∩Qi
[(u−uλ)
α+1 2
+ ]2dx≤
≤C3(dQ)C4
N
X
i=1
Z
C(2R)∩Qi
|Du|p−2|D(u−uλ)+|2(u−uλ)α−1+ dx≤
≤C3(dQ)C4(y−δ, ξ0)
N
X
i=1
Z
C(2R)∩Qi
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α−1+ dx, (4.19) where C3(dQ)→0 asdQ→0, and
C4(y−δ, ξ)≤C(y−δ)[n−2p+(p−1)r−γ](p−1)rp−2
sup
x∈Qi
ξs(x),
with p−2p−1 < r < 1, s = s(p, q, r) = [2p−2−(p−1)r−2q](p−1)rp−2 , and γ = 0 if n = 2 or any γ > n−2 if n ≥3. If s ≥0, we have sup
x∈Qi
ξs(x) ≤ξs1 with ξ1 defined in (4.18), while for s < 0, we obtain sup
x∈Qi
ξs(x)≤ξ0s, where ξ0 is defined in (4.17). In both cases the quantity C4(y−δ, ξ) is uniformly bounded by some C5(y) for small δ >0.
Hence, combining (4.2)–(4.19), we get αc1
Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α+1+ ϕ2dx≤
≤δ0c3 Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2ϕ2(u−uλ)α−1+ dx+
+C1(dQ, δ0) R
Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α−1+ dx+ C2(dQ, δ0)
R +
+C3(dQ)C5(y) Z
C(2R)
(|Du|p−2+|Duλ|p−2)|D(u−uλ)+|2(u−uλ)α−1+ dx.
(4.20) Choosingδ0 >0 andε >0to be sufficiently small (recall that 0≤dQ≤ε+δε, and hence dQ →0 as ε → 0), we can pass to the limit as R → ∞ and complete the proof similarly to that of Theorem 1.1 in [12].
Proof of Theorem 1.2. Since monotonic solutions of (1.1) are stable (see [12], Theorem 1.4), Theorem 1.1 and Proposition 2.3 [6] imply the assertion of Theorem 1.2 forp−1<
q < qc(n, p). For qc(n, p) ≤ q < qc(n −1, p), by Lemma 3.3 one can reduce the dimension of the problem by one, passing to the limit as xn → +∞ and using once more Proposition 2.3 [6], which leads to the same result.
