Научная статья на тему 'Существование и единственность решения вариационных задач Дирихле нелинейных вырожденных дифференциальных уравнений'

Существование и единственность решения вариационных задач Дирихле нелинейных вырожденных дифференциальных уравнений Текст научной статьи по специальности «Математика»

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Аннотация научной статьи по математике, автор научной работы — Исхоков С. А.

The paper is devoted to exostance and uniqueness of a generalized solution to a nonlinear differential equation of rhe form Lu≡∑|k|≤(-1)k(akx|u(k)x|(k))=F, x∈Ω (*) whose coefficients have a power degeneracy in the boundary ∂Ω of a bounded domain Ω⊂Rn. The variational method, based on elements of rhe embedding theorems of different metrics for weight function spaces are proved and then in sections 2 and 3 solvability og variational Dirichlet oriblems for the equation (*) are investigated

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Existance and uniqueness of solutions for variational dirichlet problems of a nonlinear degenerate differenrial equation

The paper is devoted to exostance and uniqueness of a generalized solution to a nonlinear differential equation of rhe form Lu≡∑|k|≤(-1)k(akx|u(k)x|(k))=F, x∈Ω (*) whose coefficients have a power degeneracy in the boundary ∂Ω of a bounded domain Ω⊂Rn. The variational method, based on elements of rhe embedding theorems of different metrics for weight function spaces are proved and then in sections 2 and 3 solvability og variational Dirichlet oriblems for the equation (*) are investigated

Текст научной работы на тему «Существование и единственность решения вариационных задач Дирихле нелинейных вырожденных дифференциальных уравнений»

UDC 517.918

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR VARIATIONAL DIRICHLET PROBLEMS OF A NONLINEAR DEGENERATE DIFFERENTIAL EQUATION S, A. Iskhokov

1. Embedding theorems of different metrics for weighted function spaces

Let 0 be a bounded n-dimensional domain in the space Rn of points x = (x\,x2,... ,xn) with (n — l)-dimensional boundary dfi. Denote by p(x) the regularized distance from a point x e fi to dfi (the definition see, for example, in [1]) and by Uk the Sobolev generalized derivative d H kn °f the function u(x), where k = (ki,... , kn) is a multi-index and |k| = ki + ... + kn.

Suppose that r is a nonnegative integer, a and p are real numbers, and p ^ 1. The symbol Lpa(Q) denotes the class of measurable real-valued functions u(x) defined on fi which have all the generalized Sobolev derivatives Uk (x) of order r(|k| = r) with the finite seminorm

\u;Lla(Q )\\ = { E / P?a(x) U k (x) |p dx} ^

When r = 0, the class Lrp, a(fi) coincides with the weighted space Lp,a(fi), which has the norm

i/p

\u,LPMn )\\ = I pP^ x Hx) lp dx'

© 2008 Iskhokov S. A.

We denote by Wpa(Q) and Vpa(Q) the spaces with the respective norms IbWyfi)\\ = {\\u;Lrpa(n)f + \\u;Lp(fiF}1/p , (1.1)

||u;VpУfi)|| = {||u;Lp,a(fi)||p + \u,Lp,a-^mP) 'P ■ (1.2)

The space Wp,a(fi) is a Banach space with the norm (1.1) for any p > 1, a G R and a natural number r. For a negative integer r and p > 1, the space Vp o(ii) is defined by the equality Vpa(Q) = (V—-a(fi))*, where q = p/{p - 1).

Let C^(fi) denote the set of infinitely diflterentiable functions in fi

o

with compact support. The symbol Wp,a(0) denotes the closure of C^(fi) with respect to the norm of the space Wpa(Q).

Henceforth the symbol Bm(OQ) stands for Besov space of functions defined on the boundary 8Q (for its definition, see, for example, [2,3]).

Basic properties of spaces Wp,a(Q) and Vpa(ii) were examined by S. M. Nikol'skii, P. I. Lizorkin and N. V. Miroshin in [2,4-6]. In particular, the following theorems were proven.

Theorem 1.1. Let OQ be a dosed differentiable manifold. Then:

1) for all a G (—<x>,+ <x), p G [1, ro), and a nonnegative integer r, Vpa(Q) is a Banach space with the norm (1.2) and the set C^(fi) is dense inV;,a(n);

2) ifa+ 1/p G{ ,2,... ,r}, then Vpa(Q) = W p,a(fl) (up to the norm equivalence).

Theorem 1.2. Let r G N,p > 1,

— — < a < r — (1.3)

pp

and so be a natural number for which

r — a--<so < r — a---hi. (1-4)

pp

Let Oil G CSo+1+E, where £ G (0,1). Then any function u G Wp,a(il) has the s0 boundary functions

dns

= M X e Brp-a-s-1 /p( dQ), s = 0,l,...,so - 1, (1.5)

an

d su

and

||VS; Brp-a-s-1 m )|| < C\\u,wr a(Q )||, s = 0,1,... ,s0 - 1,

where the positive constant C is independent of u(x). Here n is the normal vector to OQ.

Conversely, for given functions

Mx) e Bp-a-s-/p{3U) (s = 0,1,... - 1) there exists a function u e Wp, aa(Q) satisfying (1.5) and

so-i

\u,wra(tt)|| < C £ ||Vs;B/p(on)||,

s=0

where the constant C > 0 is independent of functions ^s (x) (s = 0,1,... , s0 -1 ).

Theorem 1.3. 1. If dQ is a closed differentiable manifold and a < -1 /p or a > r - I/p, then Wrp, a(Q) = WrPia(Q).

2. If Oil e Cwhere the integer so is the same as in Theorem 1.2 and £ e (0,1), then under the condition (1.3) the equality

° ( d su

wrp,a(n) = G w^(O) : —

holds.

= 0, s = 0,1,... , s - 1

on

Further in this section we prove embedding theorems of different metrics for weighted spaces aa (fi), V^ (fi), that will be applied in the next sections to investigating unique solvability of variational Dirichlet problems.

For normed spaces Bi, B2 with norms B ||, B || an embedding Bi —► B2 means that all elements of the space Bi can be regarded as elements of the space B2 and, moreover, Hf; B21| < C||f; Bi || for any f G B where the constant C > 0 does not depend on f.

Theorem 1.4. Let dQ be a dosed differentiable manifold, r be a natural number, and s be a integer satisfying 0 < s < r. Then for real numbers p, q, a, as such that

1 < p ^ q < oo, a.s >--, a — s ^ as H--------(1-6)

q q P

we have the embedding Wpa(Q) —► W^^il).

Proof. It is enough to prove Theorem 1.4 for the case n = 1 and Q = (0, a), where a > 0. Then the assertion of the theorem can be extended to the case n > 1 and a bounded domain iic Rn by the approach developed in the book [2].

Proposition 1.5 [7,8]. Let w0(t) and wi(t) be positive measurable functions defined in (0, œ), 1 < p < q < œ. Then the following Hardy inequality

o / o \1 \ 1 /q / o W/P

J ¡J v(t) dT I w0(t) dt I < C ¡J t)u!(t) dt

holds for nonnegative functions p(t) if and only if the weight functions w0(t), wi(t) satisfy the condition

1 /q / o w/p'

W (t) dT | = C < œ

sup I / wo (t) dt 1 l w P t dt 0<t<o \ J I \ J

where p = p/(p — 1).

It follows from proposition 1.5 that the inequality

1/q /a W /p

' X'

■M

f (y) dy

dxI « (/dx) (!,,

holds if/?</x+l + i — i, —| < /x. Henceforth the shorthand notation A(u) < B(u) stands for the inequality A(u) < cB(u), where c > 0 is a constant independent of u(x).

First we consider the case s = 1. So a\ > —| and + l + | — K

Let u(x) be a function defined on (0, a) which has all generalized derivatives of order < r. Since (see [2])

| ur- >(a) I << \\u;Lp( Ja) \\ + \\xYur (x)-,Lp( Ja) \\, Ja = (0,a),

for arbitrary p > 1 and arbitrary 7 e R, and

u{r-V {x)= u{r-1) (a)+ / u

(y)

by applying (1.7) we have

||xai(x); Lq( Ja) y « \\u;Lp( Ja) \ + \\xau{r) (x)-,Lp( Ja) y q \ 1/q

dx) « \\u; Lp J W + Wx^r) (x);Lp( J J

raiq

u r y dy

Thus

Wra J — W^-] J . (1.8)

Let s ^ 1 and a, as satisfy (1.6). Denote a.j = as - (s - j) for j = 1, 2,... ,s - 1; applying embedding (1.8), we obtain

w— J.

W— Ja) — W— (Ja) — ••• The theorem is proven.

rs 0 < s < r. Then under the conditions

nn n n

1 ^ P ^ 1 < 00, s---1--JiO, a-sH----^ as,

p q p q

we have the embedding

Vp^m — Vqr-Ss((î) .

x

Proof. The embedding Vpra(fl) —► Vr-S (0), in the case p = q, was proven in [5]. We consider the case p < q. Let £ e 0 be an arbitrary-point, and J(£) be the ball around it of radius p(£)/2. Then by the Troisi's theorem (see [4,9]) for any real number t and qi >1 the relation

J(pt-n/qi (0\\u-,Lp( J(m\)qi d£ ~ j(/(x)\u(x)\)qi dx (1.9) n n

holds, where the sign ~ means that there exists a two-sided estimate of the left-hand side of the relation in terms of the right-hand side with positive constants.

It follows from the results of [1] that

£ fc);Lq(mn < £ y^'>.lp(myLp(m)y (i.io)

for 1 < p < q < +oo, 0 < to < r and r — to — ^ + ^ > 0, where I (0) = {z e Rn : | z | < 1/2}■

Let £ be a fixed point in fi. For v (z) = u (£+ \zp (£)'), by applying (1.10) we have

k\=m I{0)

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q \ i/q dz

<

I

p \ i/p dz

l\=r Vo) V J

p \ 1 /p dz\ . (1.11)

By performing the change of variables z = (y — £) in integrals of inequality (1.11) and multiplying both sides of obtained inequality by pa-r+(n/p)-(n/q) £ we get af^er some transformations and integrating with respect to £ eii

E f Pqa-rq+iqn/p)-n{£)pqm{off x(v,0uk(y)|qdy) d£

\k\=mQ Q

< E J Pqa-n( £)(/ y Ip dyj q'P d£

f x(y,0 Iu(y) Ip dy^J q'P d£, (1.12)

n

fl(a-r)-v,i

l

k

m

r

where x(y, 0 the characteristic function for the set J(£). By the Troisi's inequality (1.9) the left-hand side of the inequality (1.12) can be estimated from below by-

const Y, Jpaq+qm(y)p-rq+{qn/p)(y)\Uk(y) \q dy.

\k\= mQ

To estimate from above the integrals on the right-hand side of the inequality (1.12) we first apply the following generalized Minkowski inequality

r /r \q/p {fir \p/q i q/p

J U \F(x,y)\p dyj dx J \F(x,y)\q dxj dy\

X Y Y X

and then use the inequality (1.9). We get as a result

| {pa-r+m+(n/p)-n/q x\uk x\)q dA11 / < \\uVPra(Q)\\. \k\= ™fi

(1.13)

Now, if we choose m = r — s, then under the condition as ^ a — s + ^ — ^ the inequality (1.13) implies the embedding Vp a(ii) —► V^-^^}). The theorem is proven.

2. Homogeneous variational Dirichlet problem

Let r be a natural number and pk > 2 for all k : k < r. In a bounded domain fi c Rn with (n — l)-dimensional boundary dfi of class CC we consider the nonlinear differential equation

Lu = J2 —!)\k\ (ak J) \Uk (x)\pk - Uk (x))k = F, x G Q, (2.1)

\k\^r

with real-valued coefficients ak(x).

A function U(x) is said to be a generalized solution to the equation (2.1) if it satisfies the integral identity

]T ak(x)\Uk (x) \p— Uk {x)vik) (x) dx = (F, v) (2.2)

Q \k\^r

for any v e C^ (fi). Henceforth (F, v) stands for the value of the functional F on the function vX- If F is an ordinary function, then (F, v) is the inner product of FX and v(x) in L2(0).

Suppose that the coefficients o,k(x) (|k| ^ r) satisfy the following inequalities

ClpPkia-r+\k\+(n/p))-n X < ok (x) < cippk*a-r+\k\+{n/p))-n X , (2.3)

where ci and c2 are positive constants independent of x e 0. The numbers pk (|k| < r) have to satisfy the conditions

pk = p for |k| = 0, |k| = r; n n

r-\k\----> 0, 2 < p < pk for k : 0 < |A;| < r. (2.4)

p p k

The equation (2.1) was studied in [10] in the case when its right-hand side F is an ordinary function and 2 < pk < p (Vk : |k| < r). Here we F

metrics proven in section 1, study unique solvability of variational Dirichlet problems for equation (2.1) when 2 < p < pk (Vk : |k| < r), that is, in our

pk

Dirichlet problem for class of degenerate nonlinear differential equations was also studied in [11].

In this section we study solvability of the following homogeneous variational Dirichlet problem

Problem Do- For a given functional F e Vq--a (fi), q = p/ (p — 1), find the generalized solution to the differential equation (2.1) belonging the space Vpr^iil).

Consider the functional associated with the equation (2.1)

E(u)= V — /ak X h{k) X)\Pk dx. pk

\kKr fi

Under the conditions (2.4) with the aid of the Theorem 1.6 it is easily-established that

Jp-n(x)(pa-r+'k\+(n/p(XukX|)Pk dx << \\u-,v;a{ii)\\P n

for any multi-index k : |k| < r and any function u G Vp a(Q). So it follows from (2.3) that functional E(u) is finite for u G Vp a(ii). Let F be a given functional from V""a(fi) and let $ (u) = E (u) — (F, u).

Theorem 2.1. Under the conditions (2.3) and (2.4) for any given functional F G V""a (fi) there exists a unique function U G Vpa (fi) such

that

inf$ (u) = $(U) = E(U) — {F,U), where the in£mum is taken over all u(x) G Vpa (fi). Proof. By Theorem 1.6 it follows from (2.3) that

)ir << E{u). (2.5)

Let e be an sufficiently small arbitrary positive number. Applying the

Young inequality

em a iw e - \h\M2

\ab\ < -+-U_ (£ > o, 1/Ml + 2/M2 = 1), (2.6)

pi

we have

HF,u)l < \\F;V-ra(nt)\\ \\u;Vpria(n)\\

ep „ e "q ,, „ « - N VP,a (°) + — h ^,7-a (°) • (2-7)

p ii f, ii q ii h, ii

From (2.5) and (2.7) we obtain

> E(u) -\(F,U)\2 -—11^^(0)11. u

A0 = inf $t(u), where the infimum is taken over all u(x) G Vpa(ii). Applying the Clarkson inequality it is easy to show that

F

^ /u — v\ T / u + v\

Consider a sequence {um}^i=1 C Vpa(Q) such that $ (um) = A0 + £m, £m > 0, and £m ^<x> as m Applying inequality (2.8) we find

E < A0 + \ {£m+s + em) - $ (Mm+S2+Mm) (2.9)

for any natural number s. By virtue of (2.5) it follows that $ ium+s + um\ ^ Aq + 1 ^^ + ^

So

Um $ ( Um+S + Um ] = An.

Hence, by letting m ^<x> in the inequality (2.9) we obtain

Um E (= o. m—V 2 J

It implies that (see (2.5)) the sequence {um}^=1 is a Cauchy sequence in Vpra(Q). Since Vpra (fi) is a complete space, there exists unique function U e Vp 0 (fi) such that \\um — U : Vpa(Q)\\ ^ 0 as m

By virtue of Theorem 1.6 and condition (2.3) we have

|E(um) — E(U)J PPHa — r+ W + (n/p)) — n(x)|

\k\^rQ

x ^ X^" — |U(k xr1 dx

« E \\pa-r+{n/p)-{n/Pk)x(u(m)X — U(kx);Lpk(Q)\\

\k\^r

x {\\pa-r+{n/p)-{n/Pk)Xum X; Lpjn)\ \

+ \\p«-r+\k\+ (n/p)- n/pk) xU k X; Lpk(n) \ \ }Pk/P

« \\um — U;Vpria(il)\\ £ {\\um-, Vpam\\ + \\U-,Vpom\\}Pk/P.

>,aV"/|| / , XW^m, <■ p,a\

\k\^r

Thus lim E (um) = E (U) and, consequently,

m—

um U .

Let us prove uniqueness of the function U(x)

another function U G Vpa (fi) such that A0 = $ (U 'U - U

Suppose there exists Then by virtue of

(2.8) we get

E

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< A0 - $

U + Ui

Since (see (2.5))

and

it implies that

Hence

E fc^'.iO

A0 < $

UU

E №'.=0.

U - UvXaW)|l =°

Ux

Theorem 2.2. Let conditions (2.3), (2.4) be satisfied. Then the function U(x) from Theorem 2.1 is the unique solution to the problem D0 and the inequality

||U;V;ia(0)f < M llF;V-la(n)llq is valid, where constant M > 0 does not depend on F.

(2.10)

Ux

$ (u) on the space V^ ^ (fi). So (see, for example, [12]) the Gateaux derivative d$ (u, v) of the functional $ (u, v) at the point u in the direction v exists for any u,v G Vra (fi) and (U,v) = 0 (Vv G Vpa(ii)), that implies identity (2.2). Thus U(x) is a solution to the problem D0.

Conversely, if U(x) is a solution to the problem Do, then U(x) satisfies the identity (2.2) and, consequently, d$(U,v) = 0 (Vv G Vp a(ii)). By-virtue of results of [12, Chapter IV] the second Gateaux derivative d2$ (u, vi,v2) of the functional $ (u) exists and d2$ (U,v,v) > 0 for all

v e Vpa (fi). Therefore (see [12, Chapter IV]), the functional $ (u) has a U

Since U X is a solution to the problem D0 and belongs to Vra (fi), by substituting v(x) = U(x) in (2.2) we get E(U) = (F,U); that implies inequality (2.10) by virtue of (2.5). The theorem is proven.

Under some additional requirements on the smoothness of the boundary dfi and on the power of the degeneracy a we can weaken the condition (2.3) for lower coefficients ak(x) (|&| < r) of the equation (2.1).

Theorem 2.3. Let

a+- {1,2,... ,r-l}, —— < a. < r — — (2.11)

p p p

and let the boundary dQ of the domain fi C Rn belong to class Cs°+1) s

the following conditions is satisfied:

(A) 2s0 > r.

(B) dfi is not the zero set of a polynomial of degree at most r — 1. Then assertions of Theorems 2.1 and 2.2 are valid if the conditions

(2.3) for |fc| < r are replaced by

0 < a,k(xx) < C2pPfc({n/p))-n X , (2.12)

where c2 > 0 does not depend on x e 0.

Proof. We note that according to Theorem 2 from [13] under conditions of Theorem 2.3 for u e Vpa(Q) the following Poincare type inequality

E \\pa-r+\k\u{k); Lp(fi)\\ < c\\u-,Lrpta(n)\ (2.13)

\k\<r

holds, where the constant c > 0 does not depend on u(x). Using the inequality (2.13), it is proven that the inequality (2.5) holds when the leading coefficients ak(X (N = r) satisfy condition (2.3) and the lower coefficients ak(X (|^| < r) satisfy the condition (2.12). The remaining part of the proof IS clS those of Theorems 2.1 and 2.2.

3. Nonhomogeneous boundary conditions

If conditions (2.11) are satisfied and dfi e where s0 is the

natural number from (1.4) and e e (0,1), then by virtue of Theorems 1.1 and 1.3

Vpr,am = wp^m

(up to the norm equivalence) and every function u(x,t) e V^^ii) satisfies boundary conditions

dsu dns

= 0, s = 0,1,... ,so — 1-

dQ

D

Problem Do- For given functional F e V--a (fi) find a generalized

r

p,a

conditions

d sU

-r -a '

solution U(x) e WIa(fl) to the equation (2.1) satisfying the boundary

= 0, s = 0,l,...,so - 1. (3.1)

dfi

dn

In this section we study the solvability of a variational problem with nonhomogeneous boundary conditions of the form (3.1).

Problem D. For a given functional F e W--a(Q) =f (W^^^l))* and a given set of boundary functions

(x) e Bi-a-s-1 /p (dQ), s = 0,1,... ,s0 - 1,

find a generalized solution U(x) e Wpa (Q) to differential equation (2.1) mnd

dsU

TT

' P,a

that satisfies the boundary conditions

dns

= ^s, s = 0,1,..., so - 1. (3.2)

dfi

Suppose that coefficients o,k(x) of the differential equation (2.1) satisfy the condition

clflP*(a+(1/p/p^ (x) < ok (x) < c2pPk{-a+{1/p)/pk)) (x), (3.3)

= ^s, s = 0,1,... , So - 1.

an

where ci, c2 are positive constants independent of x.

Let 1 be a given set of boundary functions G BTp a s 1/p (dd).

By virtue of Theorem 1.2 there exists a function ^ (x) G a (fi) such that

dns

We denote by W^(fi) the set of functions u(x) G Wpa(fi) such that

u(x) — V(x) G WPaM-

By virtue of Theorem 1.4 applying the same technique as in proof of Theorem 2.1, one can prove the following theorem.

Theorem 3.1. Oil G where e G (0,1). Let pk = p for

|k| = 0, |k| = r, and 2 < p < pk for 0 < |k| < r. Then under the conditions (2.10) and (3.3) for every given functional F G W""a (fi) and every given set {^KLq of boundary functions ^s G

BP"a"s"1/p (dtt), there exists a

unique function U G Wy, (fi) such that

inf $ (u) = $ (U) = E (U) — {F, U), (3.4)

u G W

Ux D

and it satisfies the inequality

\\U;WPa(0)\\p << ^ \\U; WpyO)\\Pfc"^^ W^JO)\\

\k\<r

+ \\F;Wrjn)f + \\*; W^il)\\p. (3.5)

Moreover, if

Pk <p+l (Vk: k <r), (3.6)

Ux

\\U;WPa(0)\\p << \\*; WprJQ)f0 + \\F-Wl,_a(il)\\q , (3.7) where = max {p, p/(p + 1 — pk)} if > Wpa(fi)\\ > 1 and = p if

\\*; W,jn)\\ <1. '

Proof. The first part of the proof is as that of Theorem 2.2. We prove the estimates (3.5), (3.7).

Ux

(2.2). Since UX X e Wp,a (fi), by substituting vX = UX) X in (2.2) we obtain

E(U) — B[U= {F,U — *), (3.8)

where

B[U-,^} = ]T akX U k X \P— U(k (x)$ (k)( X dx.

\ k \ <rQ

By the embedding Theorem 1.4 from (3.3), it follows that

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\\u; wpa(n)\\P << E(u) (Vu e W^l)). (3.9)

Applying condition (3.3) and the Holder inequality with exponents Ai = pk/(pk — 1 ),A2=pk we obtain

\ k \ <rQ

{Pk-1 )/Pk

< E { /(/+^XI^(fc)XI)Pfc dx

I k | ^

X {/dx

Q

«J2 )\P||Ф; wpjn)\\p. (3.10)

I k I <r

The last inequality is valid by virtue of the embedding Theorem 1.4. Now, using inequality

\(F-U -Ф>\ < \\F;W-La\\ {\\U;WPaa(n)\\ + \\Ф; Wpjfi)\\}

and (3.9) and (3.10) from (3.8), we find that

\\U;wpa(fi)\\p << E \\U;wp,a(n)\ГЦФ;wpa(fi)\\

I k I <r

+ \\F-,Wq-r-a\\ \\U;Wpjn)\\ + \\ \\Ф;wpjtt)\\.

Furthermore, by applying the Young inequality (2.6) we obtain inequality

Uwpyo)ir 'll*; wrAn)ll

< £ ||u w;ia(n)||p + K(e) ||*; w;ia(n)f2 .

e

obtain (3.7) from (3.5). Theorem 3.2 is proven.

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2. Nikol'skii S. M. Approximation of functions of several variables and imbedding theorems. New York: Springer-Verl., 1975.

3. Triebel H. Interpolation theory, function spaces, differential operators. Berlin: VEB Deutscher Verl. Wiss., 1977; Amsterdam: North-Holland, 1978.

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8. Kufner A., Kheinig G. P. The Hardy inequality for higher-order derivatives (Russian) // Trudy Mat. Inst. Steklov. 1990. V. 192. P. 105-113. Translation in Proc. Steklov Inst. Math. 1992. N 3. P. 113-121.

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(3.5).

In the case when condition (3.6) is satisfied

^=p^pk — l) > ^ and it follows from (2.6) that

REFERENCES

11. Iskhokov S. A. On analogue of the first boundary value problem for a nonlinear equation with nonpower degeneracy and generalized right-hand side // Dokl. RAN. 2005. V. 405, N 1. P. 20-22. Translation in Doklady Math. 2005. V. 72. N 3. P. 843-845.

12. Fmik S., Kufner A. Nonlinear differential equations // Stud. Appl. Mechanics. Amsterdam; New York: Elsevier Sci. Publ. Co., 1980. V. 2.

13. Lizorkin P. I. On the theory of degenerate elliptic equations (Russian) // Trudy Mat. Inst. Steklov. 1985. V. 172. P. 235-251. Translation in Proc. Steklov Inst. Math. 1987. N 3. P. 257-274.

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