Existence of a Local Renormalized Solution of an Elliptic Equation with Variable Exponents in Rn

In R ⁿ = { x = ( x i ,X 2 , • • • ,x _n ) } , n ^ 2 , we consider the following second-order quasilinear elliptic equation with variable growth and a locally summable function f :

(0 2025 Kozhevnikova, L. M.

The concept of a renormalized solution is an important tool for studying large classes of degenerate elliptic equations with data in the form of a measure. The original definition is given in [1] for the Dirichlet problem

—diva(x, Vu) = ^, x G Q, u = 0, x G dQ,                                    (2)

and it was extended by M. F. Bidaut-Veron [2] to a local form for an equation with the p -Laplacian, absorption, and a Radon measure µ :

— A p u + | u | p ^{0 - 2} u = ^, p G (1 ,n ) , 0

— 1 < p ₀ .                 (3)

In particular, M. F. Bidaut-Veron proved the existence of a local renormalized solution in R ⁿ of the equation with ^ G L i , i_oc ( R ⁿ ). In the monograph [3], L. Veron generalized the concept of a local renormalized solution for an equation with power type nonlinearities of the form

— div a(x , V u ) + b (x , u, V u) = ^.

In the present work, the concept of a local renormalized solution is adapted to the equation (1) with variable growth exponents. We prove the existence of a local renormalized solution of the equation (1) in R ⁿ and establish its sign-definiteness.

• 1.    Lebesgue and Sobolev Spaces with Variable Exponents

In this section, we provide the necessary information from the theory of spaces with variable exponents. For Q C R ⁿ , denote

Li(Q) = {P G L^(Q) : 1 < P- < P+ < +M, where p- = vrai infxeQ p(x) and p+ = vrai supxeQ p(x). Taking p0 G L+(Q), we define the Lebesgue space Lp(^)(Q) as the set of measurable real-valued functions v on Q such that

Pp^,Q(v) = / |v(x)|P(x) dx < x,

Q with the Luxemburg norm ||v||lp0(q) = ||v||p0,Q = inf {k > 0 : Pp(^)(v/k) C 1} . For v G Lp(^)(Q), the following relations are valid:

IHiPo,Q ^{— 1 c} PpUqW ^c l|v|iP ( + ) ,Q + 1 .

Due to convexity, the following inequality holds:

| y + z | ^{p (x)} c 2 ^p + ^{- 1} (| y | ^{p (x)} + | z | ^{p (x)} ) , z,y G R, x G Q.                   (4)

For p _- >  1, we have the Young inequality

Ы C | y | ^{p (x)} + | z | ^p ‘ ^(x) , z,y G R,     p ’G ) = -p^-, x G Q.              (5)

P⁽J ^{— 1}

Let us define the Sobolev space with variable exponent

W^(Q) = {v G Lp^Q) : |Vv| G Lp^Q)} with the norm ||v||p0,Q = l|v||p0,Q + l|VvHp(•),Q• The space Wp^CQ) is defined as the completion of C“(Q) with respect to the norm || • Hw1 (q)- The spaces Lp(^)(Q), Wp^^Q), Wp^CQ) are separable, Banach, and reflexive for p- > 1 (see [4, Chapter 3, §3.2, §3.4, §8.1]).

For two bounded functions q ( ^ ) , r () G C ( Q ) we write q ( ^ ) < r (), provided inf _{x e} Q ( r (x) —

q (x)) >  0. Let C ⁺ ( Q ) = { p G C( Q) : 1 < p - C p ₊ <  +^ } , where p - = inf _xE q p(x), P + = s^uP x e Q p ^(x) .

Lemma 1 [5] . Let Q be bounded, pG ) ,q(9 G C ⁺ ( Q ) , p ₊ < n, qO <  p * () = nnp^ • Then the embedding W p ( . ) ( Q ) H L _q () (Q) is continuous and compact.

2.    Assumptions and Definition of the Renormalized Solution

Assumption P . We assume that the functions

a(x, s) = (a i(x, s), • • • , an(x, s)) : R2n H Rn, b(x, sg, s) : R2n+1 H R, from the equation (1), are Carath´eodory functions. Let there exist a nonnegative function Ф G Lp‘(),ioc(Rn) and positive numbers a, a such that for a. e. x G Rn and all s, t G Rn the following inequalities hold:

|a(x,s)| C a (|s|p(x)-i +Ф(x));(6)

(a(x, s) — a(x, t)) • (s — t) > 0, s = t;(7)

a(x, s) • s > a|s|p(x).

Here, s • t = £ⁿ ₌₁ S i t i , s = ( si,.. .,S n ), t = ( ti, • • • ,t n )•

Moreover, let there exist a nonnegative function Ф д G L y , i_oc ( R ⁿ ), a continuous nondecreasing function b : R ⁺ H R ⁺ , and a positive number b such that for a. e. x G R ⁿ and all s g G R, s G R ⁿ the following inequalities hold:

| b (x ,S 0 , s) | C ^a ( | s o | ) (ф o (x) + | s | ^{p (x)}) ;                                   (9)

b (x ,s o , s) s o >  b | s o | ^{P o (x)+1} , p () — 1 < poGG

Here, we assume that p,p o G C ⁺ ( R ⁿ ) and p ₊ < n .

Following [6, 7], we introduce the notation qoG) = pp^, qeG) = ^, qi() = qOq0—p(), q*» =       ₽‘(•)• О = . p‘(), where p+ = pp-y• Let the following additional assumption be satisfied

p ^{() — 1} < q o ⁽ • ⁾ •                                         ⁽¹¹⁾

Then we can define q 2 ( • ) = q o q j+y-^ •

Let us define the truncation T k ( r ) = max( — k, min( k,r )) . By Lip o ( R ) we denote the space of all Lipschitz functions on R whose derivative has compact support.

Definition 1. Let the domain Q be bounded. A measurable and a. e. bounded function u : Q H R is called a renormalized solution of the problem (1), (2) with f G L y (Q) if the following conditions are satisfied:

• a)    T k ( u ) G W p0 (Q) for any k >  0;

• b)    b (x , u, V u ) G L i (Q);

• c)    |V uK ^{) - 1} e L q₀(Q), 1 C qQ 20;

• d)    Hp^) ^{- 1} e L q₀(Q), 1 C qQ aG);

for any h e Lip g ( R ) and p e W r^^ Q), r() > q 2 ( • ), such that ph ( u ) e W p ( _ ) (Q), ph ( u ) =0 in R ⁿ \ Q, the following equality holds:

У ( b (x , u, V u) — f ) h ( u ) p dx + У a(x , V u ) • ( V uh ' ( u ) p + V ph ( u )) dx = 0.      (12)

R ⁿ

R ⁿ

Denote by L _{M j} i_oc ( R ⁿ ) , L i , i_oc ( R ⁿ ) , W p-^ i_oc ( R n ) the spaces of functions v defined in R n such that v e L ^ ( Q ) , L i (Q) W p\ ) ( Q ) for any bounded Q C R n , respectively. For brevity of notation, ^we write ^{( L} p^ ^{( Q )) n} = ^L p0 ^{( Q )}, ^{( L} p0, ioc ^{( R n )) n} = ^L p(^, loc ^{( R n )} -

Definition 1-loc. A measurable and a. e. bounded function u : R n ^ R is called a local renormalized solution of the equation (1) with f e L i , i_oc ( R ⁿ ) if the following conditions are satisfied:

a — loc) T k (u) e Wp^ lo^ Rn ) for any k> 0;

b — loc) b (x , u, V u) e L 1 , _loc ( R ⁿ );

c — loc)    V u p ^- e L q_Bioc ( R ⁿ ) , 1 C qG ) < q 2O ;

d ^{— loc)}    /V ^{1 e L} q ( • ) , loc ^{( R n ) , 1 C} q ⁽ ^ ) < qaO ;

for any h e Lip g ( R ) and p e Wp^ ( R n ), rQ > q 2 ( • ), with compact support the equality (12) holds.

Note that in [7] the definition of a renormalized solution of the Dirichlet problem for an equation with the pQ-growth in a bounded domain was formulated for the first time, and its existence was proven.

Let u be a local renormalized solution of the equation (1). For any k >  0, we have

VTk(u) = X{Rn: |u|

Using (4), we deduce from (6) that

|a(x,s)|p‘(x) C A|s|p(x) + Ф(x)(6')

with a nonnegative Ф e Li,i_oc(Rⁿ). It follows from (13), (6') that for any k > 0,

X{Rn: |u|

Evidently, q20 < p'G), and hence Ф e L_q0,i_oc(Rⁿ), q0 < q2(•)■ Therefore, in view of c — loc), (6), we get

a(x, Vu) e Lq_Woc(Rⁿ),  1 C qG) < q20.

Remark 1. Each integral in (12) is well defined. Let supp p = K. It is easy to prove that q2(•) > n, and hence the embeddings W^(K) C Wq-^(K) C W1(K) C C(K) hold. The first term in (12) is finite due to the condition b-loc), f e Li,i_oc(Rⁿ), and h(u)p e L^(K). Since supp h'C[—M, M] for some M > 0, the second term can be written as

У a(x, Vu) • (Vuh'(u)p + Vph(u)) dx

K

= У a(x, VTm(u)) • VTm(u)h'(u)pdx + J a(x, Vu) • Vph(u) dx.

K

K

Thanks to (13), (14), and h'(u)^ G L^(K), the first integral is defined and finite. Since a(x, Vu) G Lq^(K) for any q(-) < q₂(^ (see (15)) and h(u)Vp G L_r^(K) for any r0 > q₂(•), the term a(x, Vu) • V^h(u) is integrable on K.

The main result of the present work is the following

Theorem 1. Let f G Li,i_oc(Rⁿ) and Assumption P be satisfied. Then there exists a local renormalized solution u of the equation (1). If f ^ 0 for a.e. x G Rⁿ, then u ^ 0 for a. e. x G Rⁿ.

In [8], a nonlinear anisotropic elliptic equation of the form (1) (b(x, so, s) = b(x, so)) with variable exponent nonlinearities and a locally integrable function f is considered in Rⁿ. F. Mokhtari proved the existence of a local weak solution in Rⁿ.

3.    Preliminaries

The Lebesgue measure of a measurable set Q will be denoted by meas(Q). All constants appearing below are positive. We denote B(R) = {x G Rn : |x| < r}. Let Пл(^) = min(1, max(0, R + 1 — q)), q G R⁺.

Assertion 1. Let u be a local renormalized solution of the equation (1). Then for any R > 0 the following estimates hold:

J (|u| + 1)po(x)dx < Di,(16)

B(R)

У (|u| + 1)a-1|Vu|p(x)dx < D2(a), a< 0,

B(R)

meas ({B(R) : |u| > h}) < D\hp0, h ^ 1.

Moreover, | u|^p(•^{) 1}G Lq0,loc(Rn), 1 < q(^) < qs(), and

J |u|(p(x)-1)q(x)dx < D3;(19)

B(R)

V G Lq(^),ioc(Rn), 1 < q0 < q2(0, and

У |Vu|(p(x)-1)q(x)dx < D4,

B(R)

and also |Vu|^p(•^)-1G L_q(^),i_oc(Rⁿ), 1 < q0 < q₄(^), and (20) holds.

Here, the constants Di — D4 depend on N, where N(R +1) is the collection a, a, b, n, ^R, P, P0, IHlp’O^R+l), IlfII1,B(R+1), ^{which does not de}P^{end on} u.

• < Let p > 0, а < 0 and h(Q) = (1 — (|q| + 1)^a) sign q, h_p(q) = h(T_p(Q)), q G R. Then hp(Q) = |a|(|T_p(q)| + 1)^a—1X{|^|<_P}. Let h = h_p in (12). Using (6), for a nonnegative

^ € Wr^)(Rⁿ), r(-) > q'₂(•) with compact support we get

|a| j (\Tp(u)\ +1)^a-1a(x, Vu) •VT_p(u)^dx + J b(x,u, Vu)^h_p(u) dx Rⁿ                                          Rⁿ

= -Ja(x, Vu) • V^hp(u) dx + У ^fh_p(u) dx< а У \VT_p(u)\^p(x)1|V^| dx    (21)

Rⁿ                          Rⁿ                Rⁿ

+ а У   |Vu|^p(x)-1|V^| dx + а У Ф|V^| dx + J ^\f| dx.

{Rn: |u|>p}                             Rⁿ                Rⁿ

Then, using (5), we obtain

I = a J*\VTp(u)\^p(x)-1|V^ dx< H^ У |VTp(u)|^p(x)(|Tp(u)| + 1)^a-Mx Rⁿ                             Rⁿ

+ Ci(a, a,a) У |VHp^(x)(|Tp(u)| ₊ 1)^{(1-a)(p(x)-1)}^^-p(x)dx.

Rⁿ

Let -a* = infRn pyy-1 - 1 > 0. For a € (a*, 0) and Ta(x) = (_p(_x)p⁰₁⁽x(_1-a) > 1, applying (5) again, we deduce

I< ya у |VTp(u)|^p(x)(\Tp(u)\ + 1)^a-1vdx

Rⁿ

+ e J'(\Tp(u)\ + 1)p^0(xMx + C₂(e,a,a,a) j |VH^p(xK(x)^^1-p(x)T‘^(x)dx.

Rⁿ

Rⁿ

Combining (21), (22) and using (8), (10), we get the inequality a у /(|TP(u)| +1)a-1|VTp(u)|p(x)^dx + У |b(x,u, Vu)|^|hp (u)| dx

Rⁿ

Rⁿ

^ e У (|u| + 1)^p0^(x)^dx + C2 У |V^|^p(x)T‘^(xV^-p(x)T‘^(x)dx Rⁿ                       Rⁿ

+ а У   |Vu|^p(x)-1|V^| dx + a J Ф\V^\ dx + J ^\f| dx.

{Rn: |u|>p}                             Rn                Rn

In view of the condition c-loc), \Vu\p(•)-1 € Li,ioc(Rn), and hence lim ρ→∞

j Vupx 'V._?: dx = 0.

{Rn: |u|>p}

Passing to the limit as ρ → ∞ in (23) and noting (24), we establish a^ /(|^U| + 1)^a-1\Vu\^p(x)^dx + У |b(x,u, Vu)|^|h(u)| dx

Rⁿ

< e У (|u| + 1)^po(x)^dx + C₂У У^)

Rⁿ                      Rⁿ

Rⁿ

p^(x)Ta (x)               г                   г

^dx + а I Ф\V^\dx + / ^|f | dx.

Rⁿ

Rⁿ

Applying (10), we derive а у j(\u\ + 1)а-1 Vup"x^dx ■ b у |u|p0(xMh(u)| dx

Rn                                Rn

У                             f           p(x)Ta(x)             ГГ

• < e  (|u| + 1)po(x)^dx + C2                   ^dx + a   Ф|V^| dx + pf |

Rⁿ                       Rⁿ                          Rⁿ            Rⁿ

For a G (a*, 0], consider the function 0_a (x) = p(x)T‘ (x) = —,    ^po(x)p(x)        _:Rⁿ^ r+.

\ , ,                               ^a\ /        / ^a\ /    po(x)+1-p(x)+a(p(x)-1)

Denote 0a = maxxeRn 0a(x). Evidently, 0a is continuous, increasing for a G (a*, 0], and bounded form below by 0q. For any a G (a*, 0), find 0a and fix some 0 > 0a.

Let ^ = nR(|x|). Then it follows from (25) that a у j(\u\ +1)a-1 |Vu|p(x) nR dx + b У |u|po(x) nR |h(u)| dx < e j (\u\ + 1)po (x)nR dx

Rⁿ

Rⁿ

Rⁿ

+ C2 У (nR + 0^е|VnR|^в) dx + a0 У Ф|VnR|nR ¹dx + У nR|f | dx.

Rⁿ

Rⁿ

Rⁿ

Next, applying the inequality (5) and using the obvious inequality 0‘ < p‘(x), we establish

0 У Ф|VnR|n^{R 1}dx

Rⁿ

< I (0^e|VnR|^e+ Ф^е‘nR) dx< J (0^е|VnR|^e+ Ф^p‘^(x)nе + nR) dx. (27)

Rⁿ

Rⁿ

Combining (26), (27), we get

У (|u| + 1)^a-1|Vu|^p(x)

Rⁿ

< eC₃ У (|u| + 1)^po(x)

Rⁿ

nR dx + У ^|u|^po(x)n^e|h(u⁾l dx

Rⁿ

nR dx + C4 У (|VnR|^e+ nR(1 + \f | + Ф^p‘^(x))) dx.

For |u| > 1, we have |h(u)| = 1 — (|u| + 1)^a> 1 — 2^a> 0. Then, taking into account (4), we have the following chain of inequalities:

У (|u| + 1)p^o(x)nR dx< 2^po+^-1У(|u|^po(x)+ 1)nR dx< 2^po+    У    |u|p^o(x)nR dx

Rn                                       Rn                                    {Rn: |u|>1} nR dx < y-yy У |u|Po(x)nR|h(u)| dx + 2po+ У nR dx. Rn                           Rn

+ 2^po+   У

{Rn: |u|<1}

-

Combining (28), (29) and choosing a sufficiently small e > 0, we obtain

У (|u| + 1)^a-1|Vu|^p(x)nR dx +

У (|u| + 1)p^o(x)nR dx

Rⁿ

Rⁿ

< C5 У (|VnR|^e+ nR(1 + |f | + Ф^p‘^(x))) dx.

This yields (16) and (17) for a E (a*, 0). For a C a*, the inequality (17) also holds. The estimate (16) implies (18).

Let a E (1 — p-, 0) and v = (1 + |u|)^e, в = (a + p- — 1)/p- > 0. In view of ep(x) < p(x) — 1 < po(x), we use (16) to get

/

B(R)

v^p(x)dx C

j (1 + |u|)^po(x)dx C D1

B(R)

Then Vv = в(1 + |u|)e 1 Vusignu and, according to (17), the following inequalities are satisfied:

У |Vv|^p(x)dx C ep-

B(R)                      B(R)

j (1 + |u|)^(e-1)p(x)|Vu|^p(x)dx

c/

B(R)

(1 + |u|)^a-1|Vu|^p(x)dx C D2.

Combining (30), (31), we derive ||v||p0,_B(_R) C Ce- Hence, in view of Lemma 1, we obtain

Р*()

HvHs(^)(p(^)-1),B(R) ^CC7,   ^{1 C s}G < _p0 - 1 -

Therefore, we have for any a E (1 — p-, 0) that

У |u|^{es(x)(p(x)-1)}dx

B(R)

C У (|u|

B(R)

+ 1)^es^(x)(p(x)-1)dx C D₃-

The estimate (19) follows from (32) under the assumption 1 C q(x) < ^p Px^x)¹-!)^-), which is satisfied for any sufficiently small α.

Next, applying the inequality (5), for 1 C q(x) < p‘(x) and a < 0 we have:

У |Vu|(p(x)-1)q(x)

B(R)

dx C

/

B(R)

|Vu|^p(x)(|u| + 1)^a-1

Г               ^(1-a)q^(x)

dx +      (|u| + 1)p‘^(x)-q^(x).

B(R)

The first integral in (33) is estimated by (17), and the second one by (19), provided that P¹^^)< qo(x), which holds for 1 C q(x) < p—Oqo(x)’ and sufficiently small a < 0. Moreover, the second integral can be estimated using (16) under the assumption p^D < po(x), which holds for 1 C q(x) < pf-xp⁰^ and sufficiently small a < 0. Thus, the estimate (20) is established. >

Notice that in the case of a bounded domain Q, global estimates of the form (19), (20) for the entropy solution are established in [6, Proposition 3.2, 3.6, Corollary 3.5, 3.7].

Assertion 2. Let u be a local renormalized solution of the equation (1). Then for all k,R > 0, h ^ 0 the following estimate holds:

{B(R): |u|>k+h}

|b(x, u, Vu)| dx +

1 /

{B(R): h<|u|+h}

|Vu|^p(x)dx C D₅,

where the constant D^(N(R + 2)) independent of u.

< Consider the function

Tk,h(e) = Q - - hsign Q ' k sign q

for |-| < h, for h < |-| < k + h, for |-| > k + h.

Taking h(u) = Tkh(u), ^(x) = nR^x^ in (12), we get

У     nRa(x, Vu) •VudxR   J   b(x,u, Vu)Tkh(u)nR dx

{Rn: h<|u|+h}

{Rn: h<|u|}

+ У   a(x, Vu) •VnRTkh(u) dx=   J   Tkh(u)nRf dx.

{Rn: h^\u\}

Then, using (6), (10), we derive

{Rn: h<|u|}

У      a(x, Vu) -Vudx + k     J     |b(x,u, Vu)| dx

{B(R): h^\u\}

< k У     |f | dx + ka

{B(R+1): \u\^h}                ^

^ kllfll1,B(R+1) + ^ka

{B(R): \u\^k+h}

■ У      (|Vu|^p(x)_1+ Ф(x)) dx

{B(R+1): \ul^h}

У   (|Vu|^p(x)_1+ Ф(x)) dx.

B(R+1)

Combining the last inequality with (20) and applying (8), we obtain the estimate (34).

In particular, from (34) with h = 0 we have the estimate у    |b(x, u, Vu)| dx + k У     |Vu|p(x) dx < D5. >

{B(R): \u\^k}

{B(R): |u|}

Assertion 3. Let u be a local renormalized solution of the equation (1). Then for any h ^ 1, R > 0 the following inequality holds:

meas ({B(R) : |Vu| > h})< D₆(N(R + 2))h

γ0 ^,

p0_{_}p_

Yo =----FT

P0_ + 1

.

< We deduce from (36) that

|Vu|^p(x)dx< D₅k,

k > 0.

{B(R): \u\}

Let Ф(k,h) = meas {B(R) : |u| ^ k, |Vu|^p(x)^ h}, k,h> 0. It has been proved above (see (18)) that

Ф(k, 0) < D1k^_p0-.                                    (39)

Since h ^ Ф(k,h) is nonincreasing, for k,h > 0 we have

h

^Ф(0,h)< h¹ У ^Ф(0,-) ^dQ< 0

Ф(k, 0) + h

h

У^(Ф(0,Q) ^{- Ф(k,}^^{)) d}Q.

Note that

Ф(0, q) - Ф(к, q) = meas |b(R) : |u| < k, |Vu|^p(x)> q}.

Consequently, it follows from (38) that

∞

Ф(0, q) — Ф(k, q)^ dQ C D^k-                           (41)

Combining (39)–(41), we deduce

Ф(0,h) C Dik-p⁰- + D₅k/h-¹                                    - ^p0

Choosing k = hp0-+1, we derive Ф(0, h) < D^h p0-+1. Hence, in view of the inclusion {B(R) : |Vu|p(x) ^ h} D {B(R) : |Vu| ^ h1/p- }, we obtain the estimate meas ({B(R) : |Vu| > h1/p-}} C D^h-^/P-, h > 1, which implies (37). О

Lemma 2. Let either V = Lp(.)(Q) or V = Wp(,)(Q), and vj, j G N, v be functions from V , such that {vj}j∈N is bounded in V and vj → v a. e. in Q, j → ∞.                           (42)

Then vj ⇀ v weakly in V, j → ∞.

Lemma 3. Let vj, j G N, v G L^Q) be such that {vj}jeN is bounded in L^Q) and the convergence (42) takes place. Then vj ^ v weakly in L^Q) j ^ ж.

If, in addition, h G Lp(^)(Q), then vjh ^ vh strongly in Lp^Q), j ^ ж.

Below, we will use the Vitali theorem in the following form (see [9, Chapter 3, § 6, Theorem 15]).

Lemma 4. Let vj, j G N, v be measurable function in a domain Q, meas (Q) < ж, such that the convergence (42) takes place, s = 1 or s = p(), and the integrals j |vj(x)|s dx, j G N, Q uniformly absolutely equicontinuous. Then vj ^ v strongly in Ls(Q), j ^ ж.

Lemma 5 [10, Lemma 4.8]. Let the assumptions (6)-(8) hold in Q, vj G Wp()(Q), j∈N,

Vvj ^ Vv in Lp(^)(Q), j ^ ж, lim j→∞

У qj (x)

Q

dx = 0,

qj(x) = (a(x, Vvj) — a(x, Vv)} • V(vj — v).

Then, along a subsequence,

Vvj ^ Vv strongly in Lp()(Q), j ^ ro, a(x, Vvj) ^ a(x, Vv) strongly in L_p‘()(Q), j ^ ro.

Definition 2. Let Q be bounded. A measurable and a. e. bounded function u : Q ^ R is called a renormalized solution of the problem (1), (2) with f G Li(Q) if the assumptions a)-d) are satisfied and for any w G Wp()(Q) A L^(Q), w = 0 in Rn \ Q, such that

there exist k > 0, w⁺^,w “ G W^j(Rn), r() > q‘(•), ( w = w+“ a. e. for u> k, [ w = w "^ a. e. for u < —k,

the following equality holds:

У (b(x,u, Vu) — f }wdx + У a(x, Vu) ^Vwdx = 0.

Rⁿ

Rⁿ

Definition 2-loc. A measurable and a. e. bounded function u : Rn ^ R is called a local renormalized solution of the equation (1) with f G Li,i_oc(Rⁿ) if the assumptions a-loc)-d-loc) are satisfied and for any w G Wp()(Rⁿ) A L_M(Rⁿ) with compact support such that the assumptions (44) are satisfied, the equality (45) holds.

Remark 2. Each integral in (45) is well defined. Let suppw = K. The first term on the left-hand side is finite due to the assumption b — loc), f G Li,i_oc(Rⁿ), and w G L^(K). The second term can be written as

У a(x, Vu) ^Vwdx+ У a(x, Vu) ^Vwdx + J  a(x, Vu) •Vwdx.

{K: u<-k}

{K: u>k}

{K: |u|^k}

Thanks to (15) and (44), the product a(x, Vu) • Vw is integrable on {K : u < —k} and {K : u > k}. Finally, in view of (14) and w G W^^K), the product a(x, Vu) • Vw is integrable on {K : |u| C k}.

Theorem 2. Definitions 1-loc and 2-loc are equivalent.

The equivalence of Definitions 1 and 2 in the case of data in the form of a general measure is proven in [7], the equivalence of Definitions 1-loc and 2-loc is established similarly.

4. Proof of Theorem 1. Beginning

For any m∈N there exists a renormalized solution to the problem

—diva(x, Vum) + b(x,u^m, Vum) = f, x G B(m); um = 0, x G dB(m),

where f G Li(B(m)) (see, e. g., [11]).

Let us extend u^mby zero to Rⁿ. Obviously, u^mis a local renormalized solution of the equation (1). In this section, we will obtain some a priori estimates and convergence properties of the sequence {u^m}.

Let R > 0 be fixed and p G WrY^Rn), r() > q'₂(•), be with compact support supp^ C B(R). By Definition of 1-loc, for any h G Lipo(R), the solution um satisfies the equality

<(b(x,um, Vum) - f)h(u^m)^ + a(x, Vum) (Vu^mh‘(um)^ + V^h(u^m))> = 0, R ^ m. (47)

Moreover, for any w G Wp(^(Rⁿ) A L^(Rn) with compact support supp w C B(R) and such that (44) holds, the following equality is satisfied in view of Definition 2-loc:

I (b(x,u^m, Vum) - f }wdx +

Rⁿ

I a(x, Vum) • Vwdx = 0,

Rⁿ

R< m.

Step 1: a priori estimates.

Using the estimates (16)-(20) from Assertion 1, for any a < 0 and m ^ R + 1 we deduce

I (|u^m| + 1)^p0^(x)dx < Di,                               (49)

b(R)

J (|u^m| + 1)^a-1|Vu^m|^p(x)dx < D2, b(R)

meas ({B(R) : |u^m| > h})< D^h-p^0-, h > 1,

I |um|(_P(^x)^-₁)^q(^x)dx ^ D3, i ^ q₀₃₀.

B(R)

I |Vu^m|(^p(x)-1)q(x)dx < D4, 1 ^ q() < q₂() (or q₄(-)).

B(R)

Using the estimate (36) from Assertion 2, for any k > 0 and m ^ R + 2 we get

{B(R): \u^m\>k}

|b(x,u^m, Vu^m)|dx + ¹ k

I     |Vu^m|^p(x)dx ^ D5.

{B(R): |u^m|}

Hence, using (6‘), for m ^ R + 2 we obtain

Pp^B(R) (\VTk(um)|) < D5k,  k> 0, pp‘BB(R) (|a(x, VTk(um))\) < D7k,   k > 1.

From the inequality (54), due to the arbitrariness of k > 0, we establish the estimate

^b(x,um, Vum)^i,B(R) < D5.

Then, according to Assertion 3, for any h ^ 1 and m ^ R + 2 the following inequality holds (see (37)):

meas ({B(R) : |Vum| > h}) < D6h-Y0, yo = pp0—+ 1.

Hereinafter, the constants Di , Ci independent of m, k.

Step 2: a. e. convergence of subsequences of {um}, {∇um}. It follows from (51), (58) that meas ({B(R) : |um| ^ h}) ^ 0 uniformly with respect to m, h ^ от,      (59)

meas ({B(R) : |Vu^m| ^ h}) ^ 0 uniformly with respect to m, h ^ от.

Let us establish the following convergence along a subsequence:

u^m→ u a. e. in Rⁿ,  m → ∞.                       (60)

Let a E (1 — p-, 0). Consider a sequence v^m = (1 + |u^m|)^e, в = (a + p- — 1)/p- > 0. According to (49), (50), we have the estimates (see (30), (31))

J |v^m|^p(x)dx ^ D1, b(R)

У |Vv^m|^p(x)dx < D₂. b(R)

Let us also consider the sequences vm = (1 + (u^m) + )^e, vm = (1 + (u^m)^-)^e. It is clear that V^m| < (1 + |u^m|)^(a-i^)/p- |Vu^m|, V^m| < (1 + |u^m|)^(a-i^)/p- |Vu^m|. The estimates (61) imply the boundedness of vm, vm in Wp^^Bffi)) and, in view of the compactness of the embedding into L_p(^)(B(R)), we have the strong convergences vm ^ v, vm ^ v in L_p(^)(B(R)) and the a. e. convergence in B(R). Thus, the convergence (60) is proved in B(R). Next, taking R from N, we select a diagonal subsequence u^m(we denote it by the same indices) converging a. e. to u in Rⁿ.

It follows from (60) that for any k > 0,

Tk(um) ^ Tk(u) a. e. in Rⁿ, m ^ от.

The boundedness of {Tk(u^m)} in W^)(B(R)) for a fixed k > 0 follows from the estimate (55). Then one can select a weakly convergent subsequence Tk(um) ^ vk, m ^ от in Wp()(B(R)), where vk E Wp1()(B(R)). The convergence (60) implies that vk = Tk(u) E W^ )(B(R)). Thus, in view of the arbitrariness of R, we obtain the convergence

Tk(um) ■ Tk(u) in W R , m ^ от.

Next, the convergence of the subsequence ∇u^m (we denote it by the same indices)

∇um → ∇u a. e. in Rn,m

can be proved in much the same way as in [12, Step 4].

Step 3: strong convergences

|um|p(x)- 1     u         in LqO,loc(Rn),   1 < q(9

|Vum|p(x)- 1  ,\u 1 in LqWoc(Rn),  1 < qG)

a(x, Vum) ^ a(x, Vu) in Lq0,loc(Rn),   1 < q() < q2() (or q4()), m ^ от.

Applying Fatou’s lemma and the convergences (60), (63), from the estimates (52), (53) ^we derive ^{|u|p()-1 E}Lq^io^Rn), ¹^ q⁽⁾< q3⁽0, ^{|Vu|p()-1 E}Lq0,ioc^(Rn), ¹< q⁽⁾< q2⁽) ^(orq4⁽⁾⁾.

Applying the inequalities (4), (6) and Young’s inequality, for any measurable set

Q C B(R) and any e > 0 we establish j |a(x,Vum)|q(x) dx * aq+ 2q+-1 j (|Vum|(p(x)-1)q(x> + Фq(x)(x)) dx

QQ

* e / |Vum|^(p(x)-1)q(x)dx + C₈(e) meas (Q) + C₉   Ф^q(x)(x) dx,

B(R)                                               Q where 1 < q0 < q(-) < q20.

Taking into account the absolute continuity of the second integral on the right-hand side of (67), applying the estimate (53), for any e > 0 we find 5(e) such that for any Q with meas (Q) < 5(e) the following inequality holds:

J |a(x, Vu^m)|^q(x)dx * e  (Vm > R + 1).

Q

This implies that the sequence {|Vu^m|^(p(x)-1)q(x)}, {|a(x,Vum)|^q(x)} have uniformly absolutely equicontinuous integrals over B(R). By Lemma 4, we have the convergences (65), (66). The convergence (64) is established similarly, by using (52).

The estimate (35) for u^m and k = 1, h ^ 1 is written as

{B(R): h-1*|u^m|}

*

{B(R+1): lu^m[^h-1}

a(x, Vum) • Vumdx +

If|dx + a

/

{B(R+1): |u^m|>h-1}

/

{B(R): luml^h}

|b(x,u^m, Vu^m)| dx

(|Vu^m|^p(x)-1+ Ф(x)) dx,

m > R + 2.

Since |Vu^m|^p(x)-1converges strongly in L1(B(R + 1)) and in view of the absolute continuity of the integrals on the right-hand side of the last inequality, taking into account (59), for any e > 0 one can choose a sufficiently large h(e) > 1 such that for h ^ h the following estimate holds:

/

a(x, Vum) •Vum dx+

{B(R): h-1*|u^m|}

/

\b(x,u^m, Vum I dx <

{B(R): luml^h}

ε

2,

m ^ R + 3. (68)

The proof of the theorem will be continued in Section 6.

5.    Step 4. Strong Convergence of Truncations in W^/B (R))

In this step, the following strong convergence will be established:

VTk (um) ^VTk (u) in LpBloc(Rn), m ^to,(69)

a(x, VTk(um)) ^ a(x, VTk(u)) strongly in Lp‘0,ioc(Rn), m ^ to.(70)

It follows from (56) that for any k ^ 1 we have

||a(x, VTk(um))||p‘0,B(R) * Du(k),   m > R + 2-

For positive real numbers m, h, we denote by w(h,m) any value such that limsuplimsup |w(h, m)| = 0. h^+м m^+^

Moreover, by Wk(m) we denote any value such that, for a fixed k, limsupm _л. ^ |wk(m)| = 0. Let h, k, h — 1 > k > 0, фк(q) = qexp(Y2g²), Q G R, where y = b(k⁾• It is evident that

W⁽Q) = ^k⁽Q) - Y^|фк⁽Q)| ^Y7/8, Q ^G R.

Hence, for z^m = Tk(u^m) — Tk(u) we have

7/8 < ^k(zm) < max ^k(p) = Сц(к), m G N.

[-2k,2k]

In view of (60), we get

Фk(zm) ^ 0, ^k(zm) ^ 1 a. e. in Rn, m ^ от,

|фk(zm)| C фk(2k), 1 < ^k(zm) C ^k(2k), m G N.

Applying (73), (74), by Lemma 3 we establish the convergence

|фk(zm)| ““ 0 in LM(Rn),  m ^ от.

According to Lemma 3, for g G Lp0,loc(Rn) we have gфk(zm) ^ 0 in Lp0,^(Rn), m ^ от.

Taking w = фk(z^m)nR(|x|)nh-i(|u^m|), R > 0, as a test function in (48), we obtain

J a(x, VTh(u^m))V(nR(|x|)фk(z^m)nh-i(|um|)) dx

Rⁿ

+ I b(x,u^m, Vu^m)фk(z^m )nR(|x|)nh-i(|u^m|) dx

+ I fфk(z^m)nR(|x|)nh-i(|u^m|) dx = Ii^mh+ imh + imh = 0, m Y R + 1.

Rⁿ

Estimates for the integrals imh — imh. Thanks to (73), by Lebesgue’s theorem we obtain limhl C   I |fфk(zm)| dx = ^(m).                         (78)

B(R+1)

Evidently, z^mu^m Y 0 for |u^m| Y k. Therefore, in view of (10), we have

b(x, u^m, ^u^m)фk(z^m) Y 0 for |u^m| Y k.

Taking this into account and applying (8), (9), we estimate

• -Imh<     I     \b(x,um, Уи^т)\1Фк(zm)\nR(\x\) dx

{Rn: |um|}

• < b(k) I (\VTk(um)\^p(x)+ Фо(x)) |фк(z^m)lnR(lxl) dx< b(k) I Фо(х)|фк(z^m)\ dx (79) Rn                                                     B(R+1)

  --

• +bB / a(x, VTk (um)) •VTk (um№k (zm )|nR(|x|) dx = Im + I2m. a J

Rⁿ

In view of (75), we have

I2’i = ^-(k) /

B(R+1)

Фо(x)|фk(z^m)| dx = ш(т).

Now, using (78), (79), (80), from (77) we derive the inequalities im = im - im = J a(x, VTh(um) •Vz^(zm)nR(\x\)nh-i(\um\) dx

Rⁿ

- b^ [ a(x, VTk (u^m)) •VTk(u^m)\фk (z^m)\nR(\x\) dx ^a

Rⁿ

< w(m) +

/

a(x, VTh(u^m)) • Vu^mnR(\x\)\фk(z^m)\ dx

{Rn: h-i<|u^m|}

+ J   \a(x, VTh(u^m))\\VnR(\x\)\\фk(z^m)\dx = Цт) + I12^h + №, m > R + 1.

{Rn: |u^m|}

Estimate of the right-hand side in (81). Using (74), we have mh i2

{Rn: h-i<|u^m|}

a(x, VTh(u^m)) • Vu^mnR(\x\)\ф_k(z^m) dx

< \ фк (2k) \         J         a(x, Vu^m) •Vum dx.

{B(R+i): h-i^\u^ml}

Thanks to (68), we get

Imh< ^(h), m > R + 4.

Next, using (71) and (76) with g = 1, we obtain imh =     I     \ a(x, VTh(um)) \ \ VnR( \ x \ ) \ \ фк(zm)\ dx

{Rⁿ: |u^m|}

< Dii(h)Mkr   = Uh(m).    (83)

Combining (81)–(83), we establish the inequalities

Performing elementary transformations, we derive the equalities imh = j a(x, VTh(um)) •VT^um^k (zm)nR(|x|)nh-i(|um|)) dx

Rⁿ

• - I a(x, VTh (um)) VT (u^k (zm )nR(lxl)nh-i(luml)) dx

Rⁿ

• -    bB [ a(x, VTk (um)) •VTk (u^m^k (zm )lnR (|x|) dx ^a

Rⁿ

= I a(x, VTk (u^m)) •VTk (u^m)фф^т)nR (|x|)

Rⁿ

• -    I a(x, VTh(u^m)) •VTk (u^k (zm)nR (|x|)nh-i (|um|)) dx

Rⁿ

J a(x, VTk(u^m)) • Vz^m^(z^m)nR(\x\) +

Rⁿ

/

Rⁿ

a(x, VTk(um) • VTk(ффф^т)nR(\x\)

• -    J a(x, VTh (u^m)) •VTk (u)Фk (z^m)nR (|x|)nh-i(|um|) dx.

Rⁿ

Obviously, we have

Im = J a(x, VTk(um))

Rⁿ

Vz^m^(z^m)nR(\x\) dx

- b^ t a(x, VTk (um)) •VTk (ифф (zm )|nR(|x|) dx a

Rⁿ

+ у ^(a(x, °)

{Rn: \u^m\^k}

- nh-1 a(x, VTh(u^m))) • VTk(u)Фk(zm)nR dx = Am + Am + Im.

Using (71), we get

\Im< ^ J |a(x, V^ (u^m))\\VTk (u)фk (z^m)\dx< Dii(k)^VTk (u^k (z^m)^p^,B(R₊i). B(R+1)

In view of (76) with g = |VTk(u)| G L_p(.)(B(R + 1)), we obtain

Im = w(m).                                    (86)

Using (74), (71), we establish that

\Im| < Di2(h,k)||VTk(u)X{luml^k}^p^B(R₊i).                    (87)

By (60), we get

^VTk^(u)X{lum\^k}^{— VT}k^(u)X{\ul^k} =⁰ a.e. in Rn, m - ^ :

|VTk(u)|^p(x)X{|um|^k}< |VTk(u)|^p(x)G Li(B(R + 1)), m > R + 1.

Using Lebesgue’s lemma, we deduce that VTk(u)X{\um\^k} — 0 in Lp^B(R + 1)), m — .

Therefore, by (87), we obtain

Im = ^(m).                               (88)

Since im does not depend on h, it follows from (84), (85), (86), (88) that Im C ^k(m) + w(h).

From here, using the notation (43), we have

0 C У q^m(x)^k(z^m)nR(\x\) dx

Rⁿ

• = Im1 - J a(x, VTk(u)) • V(Tk(u^m) - Tk(u)^k(z^m)nR(\x\) dx

Rⁿ

C Uh(m) + ^(h) - I a(x, VTk(u))V(Tk(u^m) - Tk(u))^k(z^m)nR(^x\) dx

Rⁿ

= Mh(m) + w(h) - Im.

Thanks to (73), by Lemma 3 we establish the convergence nR(\x\)^k (zm)a(x, VTk (u)) ^ nR(\x\)a(x, VTk (u)) in Lp‘(^(Rn), m ^ to.

From here, applying (62), we deduce m I44

J ^k(z^m)a(x, VTk(u))V(Tk(u^m) - Tk(u))nR(\x\) dx = ^(m).

Rⁿ

Combining (89), (90), we arrive at f ^_k(z^m')q^m(x) dx C u_h(m) + u(h). Using (72), B(R)

passing to the limit in the last inequality first by m → ∞ and then by h → ∞, we establish that lim m→∞

q^m(x)

dx = 0.

B(R)

By Lemma 5 (v^m= Tk(um)), due to the arbitrariness of R > 0, we have the convergences (69) and (70).

6. Proof of Theorem 1. Ending

Step 5: limit function is a renormalized solution.

As in [11, Step 6], we established the convergence

b(x,u^m, Vu^m) ^ b(x,u, Vu) in Li,i_oc(Rⁿ),  m ^ to.               (91)

Let us prove that the limit function u satisfies Definition 2-loc. The assumptions a-loc)–d-loc) of Definition-loc are satisfied, which is proved in Steps 2, 3. Let us prove the equality (12). Let h G Lip₀(R) and ^ G Wr^^Rn), r0 > q’,(•) be with compact support. Since h is bounded and continuous, due to the convergence (60), by Lemma 3 we establish

h(um') ^ h(u) a. e. in Rn, m ^ to,(92)

h(um) —^ h(u) weakly in L^(Rn), m ^ to,(93)

V^h(um) ^ V^h(u) strongly in Lr(.)(Rn), r() > q'2(•), m ^ to.(94)

If supp h' C [-M, M] for M > 0, then for a. e. x G Rn we have

\h(u^m)\^p(x)+ \Vh(u^m)\^p(x)= \h(u^m)\^p(x)+ \Vu^mh'(u^m)\^p(x) C C12 + C13\VTm(u^m)\^p(x).

Then, applying the estimate (55), we obtain the boundedness of the sequence {h(um)} in W1^)(B(R)). From here and (92), by Lemma 2 we establish the convergence h(um) ^ h(u) weakly in W^) i_oc(Rⁿ), m ^ ^' Then we conclude the convergence

Vh(u^m)p ^ Vh(u)p weakly in L_p(.)(Rⁿ), m ^ от.

Applying (91), (93), we get

У(b(x,u^m, Vum)

Rⁿ

-

f )h(u^m)^dx = I

Rⁿ

(b(x, u, Vu) — f )h(u)p dx + w(m).

Taking into account the convergences (70), (95), we have

Уa(x,VTm(um) •Vh(u^m)^dx

Rⁿ

У a(x, VTm(u)) • Vh(u)^

Rⁿ

dx + ш

⁽m) = У

Rⁿ

a(x, Vu) • Vuh’(u)^ dx + ш(т).

Using the convergences (66), (94), we obtain

У a(x, Vum) • Vph(um) dx =

Rⁿ

J a(x, Vu) • V^h(u) dx + ш(т).

Rⁿ

Combining (47), (96)–(97), we get (12).

If f ^ 0 a. e. in Rn, then for each m G N the solution u^m of the problem (46) satisfies u^m ^ 0 a. e. in Rn. Applying the convergence (60), we establish that u ^ 0 a. e. in Rn. Theorem 1 is proved.