SOME RESULTS FOR SEMILINEAR DIFFERENTIAL-EQUATIONS AT RESONANCE

Some Results for Semilinear

Differential

Equations

at Resonance *

SONG-SUN LIN

Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu, Taiwan 300, Republic of China

Submitted by C. L. Dolph

1. INTRODUCTION

In this paper we investigate the existence of a solution of a so-called resonance problem, that is, for an equation of the type

Au = F(x, 24) in Q, _(1.1)

where the linear (differential) operator A is self-adjoint with a nontrivial kernel on L,(O) and the nonlinear map F(x, 0 satisfies some growth conditions for large values of ]r]. The starting of the problem is the well- known paper of Landesman and Lazer [ 11. They assume that F(x, <) = f(x) + h(x) with f(t) +f(+m> as t -+ *a, and ./X-co) <f(T) <f(+a>, and are able to prove necessary and sufficient conditions for (1.1) to be solvable. Since then, many works have been done on the programs. We refer to the extensive bibliographies of the paper by Brezis and Nirenberg ] 2 ] and the survey paper by Fucik [3].

This paper is stimulated by the work of Amann and Mancini 14 ]. In [4 ], they present a very general existence theorem for the case where the nonlinearity “does not cross an eigenvalue,” that is, F satisfies

j < F(x, t3 < ;z _ 6 ‘7’

or X+64y<2 _(I.21

- 1

for all large values of I<(, where 6 > 0 and 2 < 1 are two consecutive eigenvalues of A. In this paper, we assume F satisfies

(1.3)

* This work was partially supported by the National Science Council of the Republic of China.

574

0022-247X/83 $3.00

Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

the case could be regarded as the most general case, where nonlinearity does not cross eigenvalues. As in [4], we study (1.1) by the perturbation method, but different from [ 41, we consider the following perturbed equation of (1.1)

Au = FJX’ u) in 0, _(1.4)

where F,(x, c) = @(x, {) + r&, q = (6 - ~)/(a + E), E > 0, and 6 = I- j. In Section 2 we recall an abstract existence theorem of a nonresonance problem established by Amann and Mancini [4] by using the well-known existence theorem for coercive pseudomonotone mappings.

In Section 3 we obtain some a priori estimates of the projections of solutions u, of (1.4) on ker(A - j), ker(A - I), and (ker(A - 1) 0 ker(A - I)}‘. These estimates enable us to prove that either uJ]]uJ -+ 4 E ker(A -I) or uj/i]uj]] -+ w E ker(A - 1) if ]]uI]] + 03 asj-, co. Then, the existence theorems of a nonresonance problem are immediate by standard proof.

In Section 4 we consider the resonance case. We decompose F into two parts which are easier to handle. The approach results from De Figueiredo [S] and is simplified by Amann and Mancini [4]. By the estimates obtained in Section 3, we are able to obtain an existence theorem which generalizes most of the known results, where the nonlinearity does not cross eigenvalues.

2. NOTATION AND PRELIMINARIES

In this section, we recall a perturbation lemma and an existence theorem of a nonresonance problem given by Amann and Mancini [4].

Throughout the paper we denote by H a real Hilbert space and by

A: D(A)cH+H

a self-adjoint linear operator with dense domain D(A) and closed range R(A). Let N(A) be the kernel of A. Then R(A) = N(A)‘, which implies that

A-’ := [AID(A)nN(A)i]-l:N(A)l~N(A)l

is a continuous linear operator. We always assume that A -’ is compact. From these hypotheses, the spectrum a(A) of A is a pure point spectrum. More precisely, every ,l E a(A) - {0} is an eigenvalue of finite multiplicity, and a(A) - (0) has no finite cluster point. Hence c(A) is countable and can be enumerated in the following way:

. ..<~_.</1_,<~,:=0<~,<~,<....

Clearly, A,, E a(A) iff A is not invertible. In this case, & is an eigenvalue of finite or infinite multiplicity. In the case Ai E a(A) - (O), we denote Ni = ker(A -Ail) as finite dimensional. We note that the number of positive or negative eigenvalues can be infinite, finite, or zero.

Recall that a nonlinear operator is called bounded if it maps bounded sets into bounded sets. A map M: H + H is called monotone if

(M(u) - M(v), u - v) > 0 for all U, u E H.

We shall state two existence theorems given in [4] to the nonlinear operator equation

Au = B(u), _(2.1)

where we assume that B: H + H is continuous and bounded.

We first recall a perturbation lemma which is essentially well known and a complete proof is given in [4].

LEMMA 1. Suppose that either (i) N is finite dimensional, or (ii) B or -B is monotone.

Moreover, suppose that there exist a bounded map g: H--t H and a null sequence (ej) in R such that

(a) for every jE N, there exists a ui E D(A) such that Au,~ = B(uj) + Ej g(uj), and

Cal

suP lI”jll < O”. (2.2)

jcN

Then Eq. (2.1) is solvable.

By applying Lemma 1 and a well-known existence theorem for coercive pseudomonotone mappings (in the sense of Browder and Hess [6]), Amann and Mancini [4] prove the following existence theorem in the case that the nonlinearity “lies between two consecutive eigenvalues.”

LEMMA 2. Suppose that there exis,t twc consecutive eigenvalues 1 < 2 of A and positive constants y,, and y < (A - A)/2 such that

for all u E H. Moreover, let a := sign(l + 1) and suppose that either (i) N is finite dimensional, or

(ii) aB is monotone. Then Eq. (2.1) is solvable.

3. THE NONRESONANCE CASE

Throughout the remainder of the paper, we work on H = L,(R), where 0 is a finite measure space with measure m. Moreover, we suppose that

B(u)(x) := F(x, u(x)) for a.a. x E a,

where F: 0 x R + R is a Carathedory function, that is, F(x, r) is continuous

in <E R for a.a. x E J2 and measurable in x E 0 for every r E R.

Amann and Mancini [4] give a sufficient condition (N) for F, which guarantees that B satisfies the hypotheses of Lemma 2 and so resonance is excluded.

(N) There exist two consecutive eigenvalues i < 1 of A and numbers 6 > 0 and a,p>O such that

and

1% <)I< 0 ItI + a(x)

where a E L,(R).

for all r E R and a.a. x E 0,

To the situation where resonance may occur, they impose the following hypotheses:

(H) F(x, r) = J,c+f(x, r) for some ,lk E o(A), where I, # 0 if dim N = 00, and info(A) < ;1, < sup o(A), and

W + > (0 4% <I > -c(x) ItI - d(x),

(9 If(x, 0 G (A+ l

- A, - S) ((1 +&(x) for a.a. x E R and all

CER,

where f,, c, d E L*(Q) are nonnegative functions, and 6 > 0. In the case dim N < 03, the perturbed equation

Au = B(u) + EU _(3.1)

bounds on U, and so Au = B(u) is solvable, they impose a Landesman-Lazer type condition

(II,) (iii) i,(f+~‘-~-_)>Oforall~EN,-(O},

where f,(x) := lim infL, * &(x, <) and f*(x) := lim sup,, * oc f(x, 0, and 4’ :=max{d(x),O} f or a.a. x E 0 and & := 4’ - 4. The dual set of hypotheses (H,) has the form

(K) 0) @Xx, 0 G 0) Ill + d(x),

(ii) If(x, <)I < (A, - A,-, - 8) 1 <I + f,(x) for a.a. x E R and CE R, and

(iii) J,(f+#‘-f-#-)<Oforall#EN,-{O}, and the corresponding perturbed equation is

with 0 < E < .sO = 6/2.

Au = B(u) - EU (3.1’)

In the case dim N = co, to ensure aB is monotone, a := sign(k), they impose some kind of monotone conditions onf, namely, either

(M) (i) a((&, <) - f(x, v))/(( - V) + &) > 0 for a.a. x E R and all it+ II, or

(ii) there exists a number c > 0 such that

for a.a. x E Q and all r# II,

Then (2.1) is solvable if any one of the following sets of conditions is satisfied:

(1) (M(i)), a = 1 and (H,); (2) (M(i)), a = -1 and (HJ or

(3) (M(ii)) and either (H,), or (H-).

In this paper, we relax the restrictions (H, (ii)) and (H_(ii)) by assuming the following hypotheses:

W- 1) Q-(x, t) > -c(x) I <I - 4x1,

(H-2) I./(x, <)I < (&+i - 1,) lrl + Jo(x) for a.a. x E D and all <E R,

where fO, c, d E L,(Q) are nonnegative.

Condition (H-2) allows that the nonlinearity “can touch but not cross two eigenvalues” and then causes some difficulties to find solutions U, of (3.1) or

perturbed equation (3.1) or (3.1’) of (2.1), we consider the perturbed equations

Au = A/$ + nf(u) + qEU, (3.2)

where n = (6 - e)/(6 + E), E > 0, 6 = A,, , - Ak and f(u)(x) :=f(x, u(x)) for a.a. x E Q.

We first prove the following existence theorem of solution for (3.2).

LEMMA 3. Let hypotheses (H), (H-l), and (H-2) be satisJied. Then there

exists q, > 0 such that (3.2) has a solution u, for all 0 < E < E, $dim N < co or dim N = 03 and (M(i)) holds.

Proof: We note that (H-l) and (H-2) imply that

-f, (xl < f(x9 4 < x + fi (x> for a.a. x E B and < > 0,(3.3) and

at - f, (xl G f(x, a < fi (x) for a.a. x E S and r< 0, (3.4) where f, E L,(Q) is nonnegative (for example, f, = 6 + f, + c + d). Conversely, (3.3) and (3.4) imply (H-l) and (H-2) with c = f,, d = 0, and

fo=f*.

Let FE(x, 0 = A,{ + rf(x, c) + q&r. Then (3.3) and (3.4) imply

IfdimN=co,

a(F,(x, 4 -F&x, c))(r - 0

a{(1 - u) 1k + re) = & {-(A,+, + A,) + &} > 0.

Hence aB,(u), B,(u)(x) := F,(x, u(x)), is monotone. Therefore, Lemma 3 follows from Lemma 2.

In the remainder of the paper, we always assume that hypotheses (H), (H-l), and (H-2) are satisfied. If dim N = co, we also assume Ak+, # 0 and (M(i)) holds.

Before we can derive a priori bounds on solutions U, of (3.2), we need the following estimates:

LEMMA 4. supOCEGEO E ]( u,]] < co, where U, is n solution of (3.2).

Prooj By decomposing u into u = v + z with v E N, and z E Ni n D(A ), (3.2) takes the form

Lz = qf(u) + &-U, (3.5)

where L := A - I$.

Recall that for every u E D(A),

IlLu II2 > &k, u); for the proof see [4].

(3.6)

Then, following a device of BrCzis and Nirenberg [2], (H-l) implies

~;f(~~r)=I~lf(~~r)+clrl+~l-clrl-~

> I tl LO., 0 - 2~ I <I - 2d.

Moreover, (H-2) implies

U-(-v 0 > a-’ I.f(., 01’ - a-!A If(., <>I- 2~ ltl - 2d.

Therefore

and

Il.mIl G fJ II 24

II + Y,

(3.7)

cm>, u) 2 J- ’ Ilft~Il’ - Y(II u II + 113

(3.8) for all u E H, where y is a generic constant, independent of E but not necessarily the same in different formulas. Hence

11 LzlJ2 > d(Lz, z) = 6(Lz, u)

= &u(u)3 u) + 6rl& 11412

a rl ll.0Il’ + h& 11412

- r(llull + 1).

On the other hand,

lILzl12 G v2 IlfWl12 + 2V2&

Ilull Ilf(u)ll + V2E2

ll~l12.

(3.9) Therefore

By dividing the last inequality by E 11 u/l*, we have Y/E Ilull + Y/E 11412

2 (W t &I) Ilf(~)ll’/ll u II2 - 2rl Ilf(~>ll/ll f.4 II t (6 - WI* Suppose that there exists a sequence (uj) in D(A) such that

Ejll”jll+ C0 and cj+c’>O as j+co

and

L"j = Vj.ft"j) + VjEjfCUj)*

Let a = lim infj+, Ilf(“jIllll ujlle If a - co, a contradiction - is obvious. If a < co, then

02

&{(a-S)'t(atc')*}

>O,

a contradiction. Lemma 4 is proved. We further decompose u into

u=vtwty,

wherevEN,,wEN,+,, andyE (NR@N,+,)inD(A).

We note that for all y E (Nk @ Nk+ J’ f7 D(A),

lILYlIZ >/WY, Yh (3.10)

where /3 := A.,, 2 -I,. In fact, let Pi be the orthogonal projection of H onto eigenspace Nj, Nj = ker(A - Ajl). Then for all y E (Nk @ Nk+ ,)‘n D(A)

lILYlIz= C (nj-nk)2 IIpjYI12 j#k,k+ 1 > x (Aj-Ak)2 IIpjYI12 j>k+2 2 @k+2 -nk> j>T+2 (Aj-nk>IlPjyl12 > @k+2 - nk> 1 (dj-nk)(IPjy1)* t 2: (nj-ik)llpjYl12~ j>k+2 jck =(IEk+2-Ak) 2 (nj-‘k)IIPjY\12 j#k,k+ I

Since for all w E Nk+ , ,

Lw=(A-~k+,)W+(;lk+,-~k)W=~U’,

(3.5) takes the form

Ly + 6w = d(u) + ?jEU. (3.11)

The following estimates play a crucial role in deriving a priori bounds on solutions U, of (3.2).

LEMMA 5. If u is a solution of (3.1 l), then

IlLYll < Y(/Iw* + l), and

IV(u) - w G r(ll u II I’* + 11,

where y is a generic constant, independent of E. ProoJ By (3.10) and (3.11), we have

lILYlIZ >WYT Y>

=Mf(u>, u> +Pw 11412

-Pa llwl12~

Furthermore, by (3.7), (3.8), and Lemma 4, we obtain

IILYll*>P~-’ Ilf(4l’-P~ IIwl12 - Y(ll4l + 1). On the other hand, by (3.7), (3.1 l), and Lemma 4,

lILYlIZ 6 r* Ilf(~)ll’ + a*& Ilull Ilf(uIl + V2E2 11412 - a2 llwI12 ,< Il.f(~I12 - fz?* IIwl12 + Y(llUll + 1).

Therefore IlfWll’ - 6* 1/412 G V(ll~II + 1). (3.12) (3.13) Hence lILYlIZ < Y(llUll + 1).

Finally, (3.13) follows from the last estimate and Lemma 4.

Now suppose that (2.2) is false. Then there exist a null sequence (cl) in (0, E,,] and a sequence (uj) in D(A) such that

II ujll + co

as j+co, and

where uj=uj+wj+yj with ujENk, wjENk+,, and yjE(Nk@Nk+,)‘fl w >-

We note that there exists a constant c > 0 such that

IILYII a c IIYII

(3.15)

forallyE(N,@N,+,)‘nD(A).SincedimN,<az anddimN,+,<oo,we can assume (by passing to an appropriate subsequence if necessary) that

uj/ll ujll = uj/ll ujll + wj/ll ujll + Yj/lI”jll

-4+ vENkONk+, - (01,

and that

uj/ll”jll --) 4 + V

almost everwhere in 0. (3.16) Moreover, we can prove that the limit does not mix in Nk @ Nk+, , that is, either 4 = 0 or IJI = 0, if the following condition (E) is satisfied:

(E) For all #EN,--(O) and YEN,+,-{O), rn(x~QI $%f> v(x) + 01 > 0.

LEMMA 6. Let condition (E) be satisJied. rf II ujll -+ co as j-1 co, then

either

Proof. Let ~~/ll~~ll = #j and w~/IIu~[I = vj. Then, (3.13) implies

(3.17)

Suppose that w # 0, we shall prove 4 = 0. We first prove that

Iv > 01 = [w > 0, w + # > 01 and [w~ol=[w<o, w+#<Ol,

where

Iw > 01

= {x E Q I V(X)

>

O}, [W > 0, w + 4 > o] = (X E B I v(X) > o,

and V(X) + 4(x) > 0). Since

Suppose that m[ w > 0, w + 4 ,< 01 > 0. By using Egoroffs theorem, there exist a subset a’ of [w > 0, v + 4 < 0] with m(Q’) > 0, and numbers N > 0 and y > 0 such that for all j > N and a.a. x E Q’,

and

_u/jCx)

> Y, If uj(x) > 0, by (3.3),

6y/j(x) -fCx9 uj(x>>lll

ujll > h - 6uj(x)lll ujll -fI(x>lll ujll

> 6Y/2 -f~(x>lllujll

and if uj(x) < 0, by (3.4),

6Vj(x) -fCx, uj(x>>lll ujlI > h -fI(x>lll ujll

for all j > N and a.a. x E 0’. Hence

lim I

j+m

IO>O,~~+~GOI

16~j-f(uj>lllujlllZ 2 m(n’)(6Y/2)2 > O3

which contradicts (3.17). Similarly, we can prove [w < 01 = [w < 0, v + Q < 0] by (3.3) and (3.4).

We next prove that condition (E) implies that 4 = 0 if m[t,~ > 0, 4 ( O] = 0 and m[ v < 0, 4 > 0] = 0. In fact,

= I’

v4+ j

VA

l@>O.rn>Ol le<o.m<ol

if m[V > 0, $ < 0] =O and m[w > 0, 4 < 0] = 0. Hence

if (E) holds and 4 # 0, which contradicts j, WQ = 0.

Now suppose.that ##to. Then m(v>O, #<O]>O or m(yl<O, 4 > 0] > 0. Assume m[ w > 0, Q < 0] > 0. Since

Iv > O,# < 01 = Iv > 01 = [w > 0, I// + 4 > 01,

4 < 0] with m(Q’) > 0, and numbers N > 0 and y > 0 such that for all j > N and a.a. x E Q’,

uj(x) > O

and

_4jCx>

_{G --Y*}

BY (3.3),

ftx,

uj(x>>lll

ujll - 6V$(x)

G @jtx) + fI(x)lll ujll G + + fi(x)lll ujll,

for all j > N and a.a. x E Q’. Hence lim

1

i-00 IU>OI

If("j)/llujll

-

BWj12

2 m(n’)(6Y)2 > ‘3 which contradicts (3.17).

Similarly, if m[w < 0, 4 > 0] > 0, by (3.4) it leads to a contradiction to (3.17).

Hence, if v # 0, then 4 = 0. Lemma 6 is proved.

In application, condition (E) should not cause severe restriction. In the remainder of the paper, we always assume that condition (E) holds.

Let

I,(X) := lim infv and

I-i00 h,(x) := lim supv. I-*cc

We can now prove an existence theorem where resonance is excluded, by only considering the limiting functions 1, and k, .

THEOREM 1. _{Let hypotheses (H), (H-l), (H-2), and (E) be satisfied. Zf}

I

(Z+#+)’ + (IL#-)2 > 0 and

I [@-k+)w+]2+[(6-k~)w-]2>0,

0 R

for aZE 4 E Nk - {O) and w E Nk+, - {0}, then (2.1) is solvable.

Proof: Suppose that I/uj]l -+ co as j-+ co. Then it leads to a contradiction as follows: By Lemma 6, either Uj/llUjI] -+ 4 E N, - {O} or uj/l]uj]l +

V/EN/f+1 - PI* .

If Uj/]lUj]] + # E N, - {O}, by (3.13),

lim Ilf(“j)ll/ll ujll = 0.

j+m

On the other hand, by applying Fatou’s lemma,

lim

Ilf("j>ll'lllujll'

>!,l@$flf(x,

uj(x~)12111u,jl12

j-m

= I

,m>ol

liz_“ff(xY uj(x))2111

ujl12

+ I ,mio, liE$ff(x, uj(x>>2111

u,jl12

> I [e>o, (I+@+)* + j,,<o, ([-6)* > 03

a contradiction.

If uj/IIujII + v E N/c+, - {O), by

(3.13),

;it Ilf(“j)lll ujl/ - 6yill = O-

Again, by applying Fatou’s lemma,

)+z Ilf(“j)/ll ujll - 6Vjl12

> I

,rao, liE$f Iftx9 uj(x>>lll~,jll

-

61//,(X)1*

+ !;@<O,

lim inf I.@,

.i-m

uj(x))/ll ujJI - 6Vj(x)12

>

1 ,~I>ol liEkf(d -f(% uj(x>>Iuj(x))2 V'(X)

+ J~WOI j-c

lim inf (6 - f(x, uj(x))/uj(x))* v’(x)

a I R [(~-k+)y/+]Z+ [(d-k-)li/-]* >o,

a contradiction.

Hence ~~~~~~~~~

II 41 <

co, the theorem follows from Lemma 1.

COROLLARY. Suppose that constants do not being to Nk @ Nk+ , - (0). Then (2.1) is solvable if

(i) m[Z+ = 0] = 0 or m[EL = 0] = 0, and (ii) m[k+ = 6]= 0 or m[k_ = Sl = 0.

Theorem 1 generalizes the result of Amann and Mancini [4] in the case where F satisfies condition (N).

Let J;(x) := lim infl+ fa, f(x, <) and f*(x) := lim supr+* a f(x, r). Moreover, let

w 0 := & - f(-u, r>, and let h, and 6, be defined as above.

To prove the existence theorem for (2.1) by considering the limiting functions f, , we need the following estimates which complement Lemma 5.

LEMMA 7. Suppose that llujjl -+ co as j-+ co. Then (9 ifujllujlI+#EN~- {O), then IIWjll<Y(lujl/“2, and

09 if~,i/ll~jll~wE~~+~-(OJr

then Il~jll~YIl~,~ll”2~

where Uj = vj + wj + yj with ui E N, , wi E Nk+ I and Yj E (N,ON,+,YnW).

ProoJ (i) If Uj/ll Ujll--+ Q E Nk - {O), and suppose that

suP ]I

wjll/ll ujll I” = C0*

j

By (3.12) we can assume (by passing to an appropriate subsequence if necessary) that

‘Yj/IIW.jII-IJEN,+,-(OJ

and

_{LYj/ll wjll + O9}

and that the convergences are almost everywhere in R.

Divided (3.14) by ]] wj]], by Lemma 4, for a.a. x E Q,

lim f(X, uj(x))/ll wjll = fiz dwj(x)/I( wj(l = d?(X). &cc

On the other hand, by (3.3) and (3.4) for a.a. x E Q,

lim inff(x, uj(x)) @(X)/II wj(l Z 0. j+m

Hence

4(x) $6) 2 0 for a.a. x E Q, a contradiction j, 41~7 = 0 if (E) holds.

Condition (ii) can be proved by a similar argument.

THEOREM 2. Let hypotheses (H), (H-l), (H-2), and (E) be satisJied. Zf .I”0 tf+s+ -.w)= co and 1, (6, y’+ - h^- ym) = 03 for all $ E N, - (0) and I,V E Nk+, - (O), then (2.1) is solvable.

ProoJ Suppose that I( ujll + co as j + co. We may assume that for a.a. xE0 andjEN

I uj(x)lll

ujll

I G f2Cx)

with an appropriate fi E L,(G). If u~/I]u~I] + 4 E Nk - (O}, then

“P

UX”j>, uj/ll ujll) < O3

.i

by Lemmas 4, 5, and 7. On the other hand, for a.a. x E 0

uj(x)f(x, uj(x>>lll

ujll > -c(x> I uj(xMluill - d(x)lll ujll

> -c(xIf*(x> - d(x)lll ujll.

By applying Fatou’s lemma,

lim inf (f(“j)T uj/ll ujll> > il, li,tn~fUj(X).OX, uj(x))/ll ujll

_j+cc

a contradiction.

If ~j/llujll+ vENk+l - (O}, then

suP

Cauj

-fC”jh u.j/ll u.jll) < 03.

i

A similar argument leads to a contradiction

1 R (h, ly+ - h^- y-) = co.

The theorem is proved.

We remark that Theorem 2 generalizes Theorem 1.

4. THE RESONANCE CASE

We are now in a position to study the resonance problem in the case I‘, CJ+#’ -L-I< co or I, (h, W+ - h^- w-) < co. We shall decompose the function f into two parts, which are easier to handle. The approach relies on a device of De Figueiredo [5] and simplified by Amann and Mancini [4 ].

We note that Eq. (3.2) can be written as

Lu = nf(z.4) + q&U, (4.1)

or

Mu = qdf(u) - du) - EU, _(4.2)

where L=A-&I and IM=A--J.~+,I, and

for all u E D(A), (see [4, (A.l)]).

If uj are solutions of (4.1) and /I ujll -+ co as j-+ co. Then, by Lemma 6, either uj/llujll -+ Q E Nk - (0} or uj/lluj[l + v E Nk+, - (O}. Equation (4.1) is used in the former case and (4.2) in the latter. Let G(x, <) =f(x, <) and a’ = 1 if (4.1) is used, and G(x, <) = f(x, l) - St and a’ = -1 if (4.2) is used. We also remark that (3.3) and (3.4) imply that

(H-1’) W-(x, 4 -JO <f,(x) ItI,

(H-2’)

I.%, 0 - 64 < 6 ItI +.fi(x).

For every fixed r > 0, we define functions

f,(.> 4 = (3.7 0,

if <>l and a’G(., <) < r, = a’r, if l>l and a’G(., <) > r, = G(-, t-1, if (G-1 and a’G(.,<)>-r, = -a’r, if << -1 and a’G(., l) < -r,

and G,: QxR-+R by

GA.7

0 = G(., 4 - c&C-,

0,

for ItI > 1,

= t[G(-, 1) - $,(a,

111,

for O<Y< 1,

= t[G(-, -1) - 8,(-, -111, for -1 <r<O, and let g,=G-G,.

Then G,. and g, are Carathtdory functions, and

Moreover,

sup II g,(., u(.))ll < Yr < 03,

UEII (4.5 )

where constant yr depends only on r, (see [4, (A.17)]).

We shall prove the following estimate on (cr’g,(u), u), which is bounded above by assuming [4, (H + (ii)) or (HP(ii)) 1.

LEMMA 8. For every fixed r > 0,

(a’gr(u),

u>

G CA G(u)ll

+ 1)

for all solution u of (4.1) (and (4.2)), where C, depends only on r. Proof Since

(art&>, u> = @‘G(u), u> - @‘G,(u), u>,

we shall give an upper bound on (a’G(u), u) and a lower bound on (a’Gr(u), u).

It is easy to verify that

I G,(x, 01 G 6 ItI + X 6)

for all r E R and a.a. x E Q. By (4.4),

(a’G,@h ~1) = 1, I G,(u)1

I u I

> (l/S> II C@>ll’ - Y IIG,@>ll

for all u E H. Since

II W412 = II G(u) - g,(u)l12

> II Wdl’ - 2 II G@)ll II g,@)ll + II d412

2 II G(u)ll’ - Qr II WII

and

II Gr(u)ll G II G(u)ll + II g,@Il

G II W)/l + yr

by (4.5). Hence

(a’G,(u>, u) 2 (l/S> II W)l12 - Ci(ll G(u>ll + I>?

We next give an upper bound on (a'G(u), u). If a’ = 1, by (4. l), @‘G(u), ~1 = (f(u), u)

= (L% 24) - w II u II2 + (1 - rlKf(uh u)

< (l/S) PII

+ (1 - 9) II 41 Ilf(u)ll

< (l/S) IlfWll’ + rw(~)ll + 1)

by (4.3), (3.9), and Lemma 4. Similarly, if a’ = -1, by (4.2), (a/G(u), u) = -(f(u) - 6u, u)

= -(Mu, u) - E 11 .u II2 - (1 - q)(f(u) - au, U)

< (l/4 lW412 + (1 -v> Ilull IIf

- dull

< u/4 Ilf(u> - dull2 + Yw-(~) - dull + 1).

Hence, in both cases, we have

Therefore,

(a”W), u> G (l/4 II GWI’ + HII W>ll + 1).

@‘g,(u), ~1 Q CAlI W)ll + 11,

where constant C, depends only on r.

THEOREM 3. Let hypotheses (H), (H-l), (H-2), (E), and

(H-3) ~n(J;#+-.?e#-)>Oand~,(h+t/+ -6-w-)>0

foralZ~EN,-{O}andWEN,+,- (O}, be satis$ed. Then (2.1) is solvable. Proof. Suppose that )I uill + co as j -+ co. It is easy to see that

II G(uj>ll G dll ujll”* + 1).

In fact, if uj/ll ujll + 4 E Nk - {O),

Ilf(“j)ll G Y(II ujl11’2

+ l),

and, if uj/ll~jll+ w E Nk+, - (O},

Ilft”j) - 6ujll G Ilf(uj) - 6wjll + 6 II vjll + 6 II Yjll

G Y(lI”jl11’2

+ l)

by Lemmas 5 and 7. By Lemma 8,

Moreover, by the construction of g, and by assuming j uj(x)j/jj ujjJ < fi(x) for a.a. x E Q, all j E N, and an appropriate f, E L,(Q), a’~,~g,.(~~~)/ll u,~!I is bounded below by an integrable function, see [4] for details. Therefore, by applying Fatou’s lemma,

I

lim inf a’g,(u,J u,~/I[ ujll < 0. 12 j+m

By letting r+ co and using B. Levi’s theorem, we have

and

i

R

(h+w+ -LJ-)<O

if a/=-l, which contradict (H-3). The theorem is proved.

We remark that the results obtained in this paper can be applied to semilinear elliptic boundary value problems as in [ 2, 4, 5 ] and semilinear wave equations as in [ 2, 41.

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