Strongly nonlinear problems with gradient dependent lower order nonlinearity

,Yo;oniineor Analysis. Theory. Pnnted in Great Bnram.

,Uethodr

&

Appkatiom. Vol.

II.

No.

1. pp. 5-15.

1987.

0362-546&437 P3.M) c .oO Pergamon Journals Ltd.

STRONGLY NONLINEAR PROBLEMS WITH GRADIENT DEPENDENT LOWER ORDER NONLINEARITY Dipartimento

IMatematico-Statistico, (Receiued 7 Augwr

T. DEL VECCHIO Universitk di Napoli, Via Mezzocannone 8, 80131 Napoli, Italy 1985; received for publication 20 January 1986)

Key won& and phrases: Strongly nonlinear problems,

elliptic equations.

1. INTRODUCTION

IN THIS paper the existence of solutions to some strongly nonlinear elliptic problems (equations and unilateral problems) is investigated. The differential operator is A(u) + g(x, u, Du). Here A denotes a nonlinear operator, with coefficients functions subject to some polynomial growth restriction. No growth assumption is imposed on the function g(x, s, 5) with respect to s, but only a sign condition. Let us remark that no further regularity is assumed on g and on the coefficients of the operator, beside Caratheodory conditions. The peculiarity of these problems is that it is still possible to work within the framework of reflexive Sobolev spaces. However, the abstract mappings induced are in general neither everywhere defined nor bounded. Previous existence results in this domain are obtained in [l, 4,5,9, 121, where g = g(x, u). However, in some of these papers the problems are studied in Wm+‘,m > 1. Finally, in [2] the function g depends on f too, but the principal part of the operator is linear with respect to Du. In order to prove the existence of solutions, let us consider a sequence of approximating problems, with solutions u&. A major role in this method is therefore played by suitable compactness properties of the sequence uE in some Sobolev space. The Frehse’s compactness result [8] is the essential tool in this approach. 2. THE

EXISTENCE

THEOREM

Let R be a bounded domain of R” and a(x, s, 5) a function: (x, s, s’) E R x R x R”+ where the components

a(x, s, s’) E RN,

a,(~, s, j) verify the following conditions:

ai(x, s, E) are Caratheodory

functions, that is, measurable

in x for each fixed (s, E) E RiVC1 and continuous in (s, 5) for almost allx E Q;

(2.1)

there exists a constant p such that: I@, s , E>l s P(l.$-’

+ ls’lp-‘>, 5

p’

1;

(2.2)

T. DEL VECCHIO

6

there exists a constant cr such that: a(x, s, j)E 2 cY/jlp;

(2.3)

for all (s, c), (s, q) E RN+’ with 5 # q andx E R:

[4x, s, t> - 4x9s7 r1)1[5- 171 > 0.

(2.4)

From (2.2) we have 0,(x, u, Du) E LJ”(S2) if u E W’*p(Q) (where (l/p) + (l/p’) = 1). Then we can put for u, u E W’*P(Q) a(u, 0) =

a(x, u, Du)Du dr. i s-2

The operator A : WA.J’(S2)-+ W- t-P’(Q) defined by (Au, u) = a(u, u), is bounded and pseudomonotone (cf. Ill]). Furthermore, let g(x, s, 5) : S2 x R x RN + R be a Caratheodory function such that: g(x,s, 5); 3 0

a.e. in R, for each (s, 5) E RN+l,

Ig(x,~,5)l~~(x,~)I~ly

=.inQ,

(2.5)

for each (s, E) E RN+l, where y < p and sup b(x, s) = h,(x) E L p’(p-~)(R) JSJ6( We set

for each t > 0.

g(x,s9 5)

gE(X, s7E)=1 + Elg(X, s, E)/. Since g, E L”(R),

(2.6) (2.7)

there exists a solution uE of the equation uE E W$P(Q) : A(d)

+ g,(x, uE, Du’)

=f

(cf. [3, 10, 111). We shall prove that the sequence uE is weakly compact and, if u is a weak cluster point of uE, u is a solution of the equation u E WA-“(Q) : A(u) + g(x, u, Du) =

f.

We have the following result: if u’ -

LEMMA 2.1. Under the above hypotheses, and Jn gE(x, uE, DuE)uEc Cl, then

ug(x, u, Du) E L’(R)

and

u in W’+‘(S2)-weak DuE+ Du in measure

g,(x, uE, Dd)+g(x,

u, Du)

in L’(Q)-strong. Proof. Since uE- u weakly in W’~p(Q), also the sequence uE converges to u in measure. But g(x, s, 5) is a Caratheodory function then g(x, uE, DuE) --, g(x, u, Du) a.e. in R. By Fatou’s lemma

IR

g(x, u, Du)u dx S lim inf E

g,(x, uE, DuE)u” dx d Cl

(2.8)

Strongly nonlinear problems

that is ~g(x. U, Du) E L’(R). For any b > 0 and for any measurable IE

set E we have:

/g,(x, UE, DUE)1 dJ

(where Fg = {x E Q : IuE(x)I 2 a}, G6 = {x E R; Id( < S} and where C2 is a constant such that IIDu~]I~P(~)s C,). Given q > 0, let 6 = 2Ct/r,7. Then, for meas sufficiently small, J-&&x, nE, Dn’)] dx < 17

VE.

By Vitali’s theorem we have gE(x, cP, Du’) + g(x, u, Du) in L’(Q) strong. We shall prove the following existence theorem. THEOREM 2.2. For anyfE

W- l,P’ (Q) there exists u E W$p ($2) such that

g(x, ~1,Du), ug(x, u, Du) E L’(Q) and (Au, 0) + J-n&, u, Du)u d.x = (f, u> for all u E W&P(Q) f~ L” (d) and u = u. Before proving theorem 2.2 we recall a result of Frehse: PROPOSITION 2.3. [S] Let uE E W,$P(52) such that J-njD~Elp-2 Du” D@dx s K~~c$~~~=~~,VE

for all $I E W$” (C-2)r\ L” (Q) and uE + u

in W&P(n)-weak.

(2.11)

Then DK”

From this proposition COROLLARY

+

Du in measure - .

we can deduce some interesting

consequences.

2.4. Let (Us) be a sequence

of the solutions of the problems: -div(]Du”]P-*DUE) =fE

UEE W$P(s2), If fE--fo in W-‘up’ u E WkP(S2) such that

(2.12)

fE E w-‘.p’(Q).

weak (E+ 0) and ]]f,]]r.~(n) s C3, then there exists a function Du’-+

Du in measure

8

T. DEL VECCHIO

and u is the solution of the equation -div(IDujP-‘Dc4) Proof. The sequence (u’) is bounded subsequence ~6 such that:

in WA+‘(Q), then there exists u E W$P(Q) in W$p(Q)

11”’-u By the proposition

(2.13)

=f,,.

and a

weak.

2.3 we have Du” + Du

in measure

and also IDU~‘IP-~DU~‘-+

We can conclude, by Vitali’s theorem,

inmeasure.

IDuIP-*Du

that, for all s
I Du” IP-2DcP’ + IDulp-‘Du

in L’(B)

- strong.

By the uniqueness of the weak limit, we have I DUE’ IP-2DUE’ -

In conclusion,

in LP’(S2) weak.

1DuIP-*Du

we can pass to the limit in iR

IDU~~-*DU~DU&

=
U)

vu E wp(Q)

and we have IR

ID+‘DuDu

Vu E Wk”(Q)

d_x= (f,,, u)

that is u verifies (2.13). By the uniqueness of the solution II, the whole sequence converges. COROLLARY

2.5.Let (u’) be a sequence of the solutions of the problems

-div(l DuEIP-2DuE) = g, + h uE E W&S2), If

s llg’llLv2)

h E W-‘*P’(S2),gE

E W+“(Q).

CJ and uE- u in WA+‘(R) weak, then u”+ u in Wby’(Q)strongly

Vr
Proof. Given o > 0, let fi be a function such that

6 E C;(Q)

and

I/h - hllw-~.p~cQ, c u.

Let fiE be the solution of the problems -div(lDriEIP-2DiiE) E” E W$P(R).

= g, + ti

(2.14)

Strongly nonlinear problems

9

We have

(ItiE- UEIIPW;,lp(Q) G CjO.

(2.15)

0

By the previous corollary there exists a function li E W$‘(Q)

such that

Dii’ + Dti in measure

(and, by Vitali’s theorem, Dii’-, Finally by the inequality

Dii

in L’(R)

Vr < p).

lluE - 4 w;J(n)

we have the assertion. Corollary 2.5 is again true in the case that the differential operator -div(a(x, u, Do)) is of the type of Leray-Lions [lo]. In fact we can prove, by modifying Frehse’s proof [8], the following proposition.

W-‘*p’(n)

PROPOSITION 2.6. Let befE

and (u”) a sequence such that:

(2.16)

for all @ E Wkp(Q)

n L”(Q).

Then Du’+

Du

in measure.

Now we can prove theorem 2.2. Let us consider the equation. -div(a(x,

uE, DuE)) + g,(x, uE, Due) =f

UCE Wp(Q),

fE

where g, is defined by (2.7). Since g, E L”(R), there exists a solution uE E W$‘(Q) such that: IR

a(x, uE, Du’)Du

dx +

(2.17)

w-‘qs2)

IR

uE of (2.17) (cf. [ll]),

that is a function

g,(x, uE, Du’)u dx = (f, u)

(2.18)

for all u E W$P(R). We shall prove that exist u E Wb@(R) and a subsequence of (P) denoted again by (u’) such

T. DEL VECCHIO

10

that UE--

in W$” (Q) weak

u

Du’ + Du

in measure

(2.19)

gE(x, uE, DuE)uE s C, so that from lemma 2.1 we have: I-

JR

f

g,(x, uE, DuE)uE dx G C,

(2.20)

in L’ (R) strong.

(2.21)

g,(x, uE, DuE)uE dx = (f, uE).

(2.22)

ug(x, u, Du) dx s lim inf E

JR

and gsx, u”, DUE) ---, g(x, u, Du) Using uE as a test function in (2.17)‘we have a(x, uE, Du”jDu’ du +

IR

From (2.2) and (2.5) we have also: 41UE IIPw;,P(n) s

Ml w-‘.P’(Q) IIUEIIwp(Q)

So, there exists u E W$p(Q) and a subsequence llE-

u

denoted again by uE such that:

in WkP(Q)

- weak.

(2.23)

Furthermore g,(x, UE, DuE)uE dx s +

I g,(x,

IR

a(x, uE, Dd)Du’

uE, Du’)u’ du = (f,

CLX

uc> s Cslifll w-w(n)

(2.24)

R

If 4 E W$J’(Q) f~ L”(Q)

fR

we have

a(x, uE, Du’)D$

s bIIr.=(n) s

IR

dx = -

gE(x, uE, Du”)@ dx + (f, r$)

Igdx, uE, DUB)1 dx + (f, $G

aI@IIL’(R)+ (f, G>. (see(2.9)).

Then, from the proposition

(2.25)

2.26, DuE+ Du

Thus (2.19) is proved.

iR

in measure.

(2.26)

11

Strongly nonlinear problems

Let K > 0, we define the function TK(u):

TK(U)(X) =

K

if u(x) > K

u(x)

if -K s u(x) s K

1 -K

if u(x) < -K

If we use as the test function in (2.17) (u’ - 7’,(u)) we have: (Ad,

uE - TK(U)) +

g&1

UE, DUE)(UE - G(U))

Lx

= (f, UE - TK(U)).

(2.27)

then (AuE, uE - u) = (Ad, =-

I

gc(x,

uE - TK(u)) + (AU’, TK(u) - u)

Du”)(u’

uE,

- T&l))

dx + (f, UE-

u>

R

+

(f, u - TK(u)) + (Au’, TK(u) - 4.

(2.28)

By (2.20), (2.21) and by observing that TK(u) E L”(R) we have: lim sup g,(x, UC,DuE)(uE - MU)) E IR s -

I

g(x, u,

k

Du)(u - z-K(U))dx.

(2.29)

s-2

Then lim sup(Au’, uE - u) s -

g(x, u, W(u

E

- m4)

dJl

(2.30) For K+

+m we have

limsup(Au’, By (2.31) and since A is pseudomokotone, AuE-

Au

we have: in W-‘@‘(R)

u>+ f

(2.32)

weak.

Then we can pass to the limit in (2.18) for all u E WA+‘(Q) n L”(R)

(Au,

(2.31)

uE - u) s 0.

g(x, u, Dub dx = (f, 4.

and we have (2.33)

JR

To prove that

(Au, 4 +

1 g(x, u, Dub R

d-x= (j-74

(2.34)

T. DEL VECCHIO

12

it is sufficient to pass to the limit, for K-+ +x, in (2.35) This is allowed by observing that g(x, u, Du)TK(u) --;, g(x, u, Du)u I&, u, Du)&&)I

=z I&,&

a.e. in S2 (2.36)

Du)/ lI(/ E L’(Q)

and by using the Lebesgue theorem. Remark 1. We can prove that

ug(x, u, Du) E L’(R) and that u is an admissible function for the equation Au + g(x, II, Du) = f

that is (Au, u) + sng(x, u, Du)u CLX= (f, u) by using a result of Brezis-Browder [4]. In fact we have proved that g(x, u, Du) E L’(R) and we know that g(x, u, Du)u(x) 3 0; then (by [4, theorem 11): ug(x, u, Du) E L’(R) and Jog@, u, Du)u du = (9, u) = (-Au +f, u). Remark 2. The hypothesis

of coercivity of A is stronger than that of Leray-Lions [lo], so we can use Frehe’s theorem [8]. We observe that, from the [6. lemma 3’1 and (2.31), we have also the strong convergence of uE to u in WA-P(,).

Remark 3. If R is an unbounded domain of R”, the proof of theorem 2.2 is the same if we use a proper definition of g,(x, S, g) and if we use Frehse’s result for any subdomain C of S2 with compact closure in Q. We may therefore assume that DuE converges strongly to Du in Lq(C) Vq
PROBLEAM

Now we study a variational inequality related to equation (2.33). We define K to be the closed convex set K = {u E WAJ’(Q) : u s 0). THEOREM

3.1. For everyfe

there exists a function u E W$P(Q)

P’(Q)

ug(x, u, Du) E L’(R), (Au -f,

such that

u E K and

u - u) + 1 g(x, u, Du)(u - u) dx 2 0 R

for all u E K n L^(S2). Proof. Let us consider the problem Au’ + f

[(d)+]P-’

+ gE(x, uE, Du”) =f

(3.1)

13

Strongly nonlinear problems

where (u’)+ = max{tP(x), 0) and g, is defined by (2.7). If CPis a solution of the problem (3.1), we have alI uE IIWkP(Q) s

I I

a(x,uEDu")Du" dx +

R

+

g&t UE7DuE)uE

(3.2)

dx = (f, rt’);

cl

that is

s CI,. WII WpyQ) Sothere

exists u E IA@’

and a subsequence, UC-

u

again denoted by (uE), such that:

in W$p(Q) weak.

On the other hand: gE(x,UE, Du')u"dx

S

+

a(x,u”,

i

dx

DuE)DuEdx + -

g&7 UE,DuE)uEdx=(f,u')s Cl1

(3.3)

R

that is the sequence g,(x, u&,Dd) is bounded in L’(Q) Furthermore

+I 1 [(u”)+]P-‘u”

(see (2.9)).

gE(x,uE,DuE)uE cirsCll,

dx + j

(3.4)

R

& R

that is (Us)+ + 0 in ZY(R)-strong. Then u s 0. Finally the subsequence (l/.s)[(cP)+j~- 1 is bounded in L l(Q). If we use as the test function (u’)+ in (3.1), we have: 1 ;I,

W>+lP d-xs ,i- f(4’ R

dr

that is

p s IlfllLP~p, /I$z I/CUE>+ & II LW)

IILW)

hence

(u’>+p-1 < llfllL@(Q) II&P-l IILW)

’

T. DEL VECCHIO

11

By observing that (3.5)

we can conclude that (l/~)[(u~)+]P-~ is bounded in LP’(S2) and so in L’(R). Then (2.15) is verified and DLL’--* Du in measure. If we use in (3.1) as the test function (u’ - TK(u)) we have: (Au”, CL’- TK(u)) +

+

i

R;

g,(x, uE, Du’)(u”

- G(u))

dr

[(UC)+lp-l(uE - MU)) dx = (f, UE- z-K(U)),

(3.6)

then (Ad,

uE - u) +

s (Ad, +

iR

iR

g&,

u”, DuE)(uE - TK(u)) dr

uE - TK(u)) + (Au’,

g,(x, uE, Du”)(zP

TK(u) - u)

- TK(u)) dx + fl

[(uE)+]P-‘(d R

= (Ad,

- TK(U)) CLx

TK(u) - u) + (f, uE - I-K(U)).

(3.7)

In conclusion lim sup(AuE, uE - u) 6 E +

For K+

-

IR

g(x, u, Du)(u - ~-K(U)) h

GZIIm4 - 4v’+yR)

(3.8)

9

+= we have lim sup(AuE, uE - u) c 0

(3.9)

E

and, by using the pseudomonotonicity

of A,

lim inf(AuE, uE - z) 2 (Au, u - z), e

for all 2 E Wbp (Q).

(3.10)

Now we use as the test function in (3.1) uE - u, with u E K n L=(Q): (Au”, uE - u) +

IR

g,(x, uE, DuE)(uB - u) dx

s (AU’, uE - u) +

J

* [(u’)+]p-‘(IL”

gE(x, uE, DuE)(uE - u) dx I $2 - u) dx = (f, UE-

4.

(3.11)

15

Strongly nonlinear problems

By (3. ll), by Fatou’s lemma and by strong convergence in L’( R ) of gE(X,u’, LW) to g(x, U, Du) we have:

Remark 4. The result of the existence of the solutions to the strongly nonlinear problems with g = g(x, u), have been obtained in [l, 5, 91. Acknowledgemenrs-This

unilateral

work has been performed as a part of National Research Project 1M.P.I. (40%).

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