ANNALES DE LA FACULTÉ DES SCIENCES DETOULOUSE

GIOVANNI MANCINI

ENZO M ITIDIERI

Positive solutions of some coercive-anticoerciveelliptic systems

Annales de la faculté des sciences de Toulouse 5e série, tome 8, no 3(1986-1987), p. 257-292.

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Positive solutions of some coercive-anticoercive

elliptic systems

GIOVANNI MANCINI(1) AND ENZO MITIDIERI(1)

Annales Faculte des Sciences de Toulouse VoI.VIII, n°3, 1986-1987

Dans ce travail nous traitons des résultats d’existence ou non-existence pour une classe de systemes elliptiques semi-linéaires, avec desnonlinearites critiques, en connexion avec un principe de maximum pour unsysteme elliptique associe.

ABSTRACT.2014 In this paper we discuss existence-nonexistence results for aclass of semilinear elliptic systems with critical nonlinearities in connectionwith a maximum principle for a related elliptic system.

o. Introduction

In dealing with elliptic systems,

where H is a smooth bounded domain and f, 9 : R x R -~ R are givenmappings, we must face [2,7-11,15,17,18,21,24] all the typical problems as-sociated with the scalar case, like nonuniqueness, nonexistence, breaking ofsymmetries, lack of compactness related to critical growth of the nonlinea-rities f and g, as well as some specific features, mainly the lack of a generalmaximum principle.We will investigate in this paper existence and nonexistence results for

a special class of systems of the type (0.1) in connection with a kind of

(1) Dipartimento di Scienze Matematiche, Università degli Studi di Trieste PiazzaleEuropa, 1 - 341©4 Trieste (Italia)

"maximum principle" and critical nonlinearities. Namely we concentrateour attention on the problem

(here a, ~y, S are real constants).Special cases of (0.2)-(0.3) (e.g. h = 0) were considered previously in

[8,13,15-17,21] where those authors treat essentially sublinear nonlinearities(i.e. f sublinear).

Here we are concerned with the problem of existence and non-existenceresults for (0.2)-(0.3) for nonlinearities of the type,

i.e. for systems which are sublinear in v, but superlinear with limiting growthin u. The main motivation of our study is the following nonexistence result :

Let ~~ be the first eigenvalue of --L~ with Dirichlet b.c.

PROPOSITION 0.1. Assume ( f ) and (h)1, and let S2 be a smooth

starshaped domain contained in RN (N > 3).

then (0. ~~-(0. ~3~ has no positive solutions (u, ~0, 0~ provided

In contrast, we can prove the following existence result.

Let A, be j-th eigenvalue of -0 with Dirichlet boundary condition.

THEOREM I.-Let SZ ~ RN be a smooth bounded domain (N > 4).Assume that ( f) (hi) - (1~2 ) hold and

Then there exists a non trivial solution (u, v) to (0.1)-(0.2).If, furthermore

holds, then (0.2)-(0.9) has a solution (u, v) E (Ha n with u > v > 0.

As far as the nonexistence result is concerned, we have to point out thatwhile the inequality A > Ai + is sharp, we don’t know wether theinequality ~y + 2b ~ 0 can be improved. This is because the argumenta la "Pohozaev" we use, requires a kind of maximum principle which ingeneral does not hold without some restrictions on the parameters ~, ~y and~.

Because of this we obtain different bounds on A, in Theorem 1, as far asexistence and, respectively, non existence of positive solutions are concerned.

The paper is organized as follows :

In Section 1. we recall some kind of maximum principle for an integro-differential operator related to (0.2)-(0.3) and we prove radial properties ofpositive solutions in case n is a ball.

In Section 2. we prove some non existence results, first analyzing thecase of a general starshaped domain, secondly concentrating our attentionto the special case S2 = and N = 3.

In Section 3. following Brezis-Kato [5], we prove L°°-regularity andobtain L°°-bounds for solutions of an approximate problem related to (0.2)-(0.3).

In Section 4. we prove some existence results in the case of ~2 C RNwith N > 4.

Finally Section 5. deals with the special case of h = Q, ^~ = 0 and TV = 3.

Notations. Throughout the paper wTe will use standard notations forSobolev spaces HJ and corresponding norms

-, - > will denote the scalar product in while (-, -) will be the scalarproduct in R~ .

1. A maxiinum principle and radial propertiesof solutions

Let us first consider the linear system,

Throughout this Section we will always assume

For completeness, we present here some results contained in [13].

PROPOSITION 1.1. 2014 Assume (Ai) - (A2) and

If ~ 0 ~ f(x), f,h E and (u, v) E Ha X solves (1.1)-(1.2), then u is non negative.

Proof. . Let us denote by G the Green operator

If (u, v~ E H© x H~ solves (1.1~-(1.~), then

Setting B = the system rewrites

So it is enough to prove that the operator

is order preserving. But A > -y + 28 implies L = o- - ~(I - - ~3B)-1where , ,Q"1 are the two positive solutions of c~2x2 + (-y - a)~ + 1 = 0. Sothat = ~~ , a -f- ,(3 = a - ^y and hence a + $2cY-~ _ ,Q + b~,~3-1 == a - .~, al - ~ + ~ ~- This implies oe -y, ,t3 -y, because the functiont -4 t + S2 t-~ is convex, with minimum value at t = a given by 2r, and

> 8. In turns, this implies ~~a~3~~ = -y)-1 1, 1 andhence (I - (I - are order preserving.

Remark.- is a necessary condition for the validity of Prop. i . l,because it is necessary for L, B to be order preserving. In fact (Ai) justmeans that the positive eigenfunction for LB corresponds to=the largesteigenvalue of LB, since the spectrum of LB is given by (03BBj - 03B3 + 03B42(03BBj -03B3)-1 - (03BBj - 03B3)-1, (-039403C6j = 03BBj 03C6j, 03C6j E and if > S, it is nolonger true, in general, that al -y-~-S2(~1-y)-1 a? -y-f-S2(a? -~y)-1 j > 1.

PROPOSITION 1.2. - Assume f, h E C’1 satisfy the following assumption : :

Then, if (03B4, 03B3, 03BB) satisfy (1)-(2)-(3), every Ho -solution (u, v) of (0.1)-(0.2) satisfies

Proof - Let (u, v) be a solution of (0.1)-(0.2).

Using Prop. 1.1 with f (x) := f (u(x)), := 1a(v(x)) we easily see thatu~ 0.

Setting

equation (0.2) gives

and since a E a > 0, the weak maximum principle implies v > 0.

Finally, if fi is a solution of (i = 1.2) ’

we have

which implies in particular

We end this Section with a result concerning symmetry properties ofsolutions, which seems to be new and of interest in itself.

PROPOSITION 1.3. If in addition to the assumptions in Prop, 1.2 ~ve

assume

an SZ = BR(o), then the solutions of (0.1~-(D.,~~ are radially symmetric.

Proof . - After introducing the new variable w := u - v we see that(u, v, w) is a positive solution of the system

Since ~f 2 .

> - 0 ( far i ~ ~ ) we can apply a result by Troy [24], to infer that

(u, v, w) is radial and w’(r) = ~c’ (r) - v’ (r) 0, r E (0, R).

Remark 1.4. As one easily sees the above result can be used to obtainL~-bounds of positive solutions to (0.2)-(0.3) in the same spirit as Cosner~9~. .

2. Some non existence results

In this Section we first give a general result concerning non existence of

positive solutions for (0.1)-(0.2) in a starshaped domain contained in RS.

In case .1V = 3 we will sharpen our result, following BREZis-NIRENBERG[6] in case 11 is a ball. In both cases our technique heavily relies on themaximum principle for (0.1)-(0.2) given in Section 1, and consequently werestrict our attention to parameters (A, 6,,) satisfying (1~1 ) - (A2) - (A3).

Nevertheless, we don’t know whether our limitations an (À, 6, -~) are sharpor not.

POHOZAEV identity for (0.1)-(0.2). (see [13]). Let (u, v) be a smoothsolution of (0..~~-~D.,~,, then

PROPOSITION 2.1. Let S2 c N > 3) be a smooth bounded domain.Assume that (t~~ ) - (~12) - (A3 ) and ( f ), (~1 ) ‘ (h2) hold. Then (D.1~-(D.,~~has no positive solutions provided either

Proof. - The proof of case (i) is straighforward and will be omitted. Todeal with the case (ii), we use (P) together with the assumptions on f ~ ~ )and h(.).

It is easy to see that (Ai) - (A2 ) - (A3 ) imply that the right hand sideof (P) is non positive, while by Prop. 1.2, the left hand side is positive inview of (1.4) and the starshapeness of ~.

Remark 2.2. It is not difficult to check that the above Proposition holdsunder the more general assumptions on the non linearities :

We now show, as in the case of a single equation considered by BREZIS-NIRENBERG [6], we have a more delicate situation in case N = 3 and n is a

ball. For simplicity, we will limit ourselves to the case 1 = 0, h(.) = 0 andn = Bi(0).

PROPOSITION = {x E R3 : ~x~ 1~, and let (u,v) be aC2-solution of

Assume that 0 s C / 16, and let

be the lowest solution of

in such an interval. Then

Remark 2.4. Let 0 03B4 03C02 16 the function L(s) is positive in (203B4, 03C0/2)and becomes negative for s = 1r, hence there is a lowest solution of L(s) = 0in (2B/o, Jr) . If we denote by ~(b~ such a solution, clearly ~(b) = ~ ;thus, "in the limit" we get a non existence interval for values of A, given by(0, ~r2 /4) , which is the one obtained by BREZIS-NIRENBERG in the case of asingle equation.

Nevertheless we don’t know if 03C32(03B4) 4 +

4S2 is a sharp bound for non-

existence, as well as our method, relying on a maximum principle, doesn’tapply (in the case / = 0) to negative and hence it is an open questionto know whether non-existence results hold true in this case.

Proof . Thanks to Prop. (1.3), we are reduced to prove that our system

has no positive radial solutions for a e 2S 03C32(03B4) 4 + 4S2

. Again, ourproof relies on a appropriate :

POHOZAEV identity for (radial) solution of (2.1)-(2.2).Let (u, v) E be a radial solution to (2.1)-(2.2). Let 03C8 E C‘°°(R)

be such that ~(0) = 0 and ~( 1 ) > 0. Then,

Proof o f (PR ) . - The radial solution (u, v) of (2.1)-(2.2) satisfies

Following BREZIS-NIRENBERG [6], we first multiply the first equation(respectively the second equation) by and by (1/2r203C8’ -

((1/2r203C8’ - r03C8)v). After integration by parts we obtain

and

Taking (2.3)-(2.4), after integration by parts we get (PR).Now using Prop. 1.2 we get,

with strict inequality if (i.c, v) is not identically zero.

Dur claim is that if 03BB E 2S 03C32(03B4) 4 4S2 we can choose (.) to etOur claim is that if A ~ (203B4, 03C32(03B4) 4 + 403B42 03C32(03B4) ) we can choose 03C8(.) to get

A ~ 0, which will imply t? ~ 0 . We among the solutions

(~, cp) of the Q.D.E. system :

with such a choice of ~~, 5p~ we have the following :

LEMMA 2.5. Zet 03BB > 203B4, and set 0 = 03BB2 - 403B42, c = 2 . Then

Proof. 2014 Since c(a - c) = fi2, we have

Thus

because 03C6’ 0. Hence using (2.5) ;

Remark 2.6. Let ~~, ~) be a solution to (2.5) ; then

Conversely if cp solves (x), the pair (~, with

solves (2.5).Assuming A > 2S and denoted by the roots of the characteristic

equation of (x), we have the following relations

Since a~ > 4!), we also have p2 4b r~. Now assuming as above A > 2b,let us choose the following solution p) to (2.5)

~ = ~(a, ~), p = b), being given as in (2.6).

Remark 2.7. It is clear that cp chosen as above, satisfies cp’ 0 in (0,1 ),provided ~ ~r, because

u.and 2014 T.

We want also to emphasize that, in view of the existence result; whichwe will give later, and which holds true if

(here obviously Ai = ~r2 ), we have to assume 7- ~’. In fact by (2.6)

and the necessary assumption (in order to get non existence)

yields u 7r.

To complete the Proof of the Proposition 2.3, we need some simpleestimates which we collect in a lemma whose proof can be obtained bydirect calculation.

LEMMA 2.8. Let o(&#x26;~ be the lowest solution of

in Let (~, ~p) be given by (,~.8~. Then

Completiton of the Proof of Proposition ~.,~.

Now the result follows, using Lemma 2.5 and Lemma 2.8, taking intoaccount that by (1.3) and (1.4) we have u > a~2~ v and u’(r) - v’(r) 0, re (0,1).

3. Regularity and L°°-estimates

The main purpose of this section is to get some uniform L°°-estimatesfor weak solutions of families of elliptic systems of the type

where (~, -y, s~ E R3, -y ~1 (the first eigenvalue of -0 with Dirichletboundary conditions) and f ( ~ ), are allowed to have "limiting growth".More precisely we will assume that :

There exist positive constants a, b, aR, bR, such that for every t E R v~Tehave

Our argument follow closely the work of BREZIS-KATO [5] devoted to asingle equation. In what follows we deal with a given system, so that wedrop subscript R.

PROPOSITION (u, v) E Ho be a solution of

where (a, b, e, d) E R~, f, h E C1 satisfy (~.1~ and f (0) = f’(0) = h(0) -h’(0). Then (u, v) E LOO.

For convenience of the reader we state here some facts which are essen-

tially contained in .

LEMMA 3.2. Let a~ E LN~2, (j E N) be such that :

Then there exists E R such that for every f E each equation

has a unique solution u~ E . Furthermore and

y~ 0 ===~ 0.

Proof . Using (3.4) we see that,

which implies

Thus, the bounded selfadjoint linear operators in Hi given by

are (uniformly) positive definite : taking f = 1 /2, we find K, independentof j, such that

Hence, for such K, and any given f E the equations

are uniquely solvable for every j E N and

implies

Finally, if f > 0, multiplying (3.5) by 0) (dropping subscript j) )and using again (3.6), we get 0) = 0.

LEMMA 3.3. Let a E and f ~ LP with p > 2* Let u E Ha be aweak solution of

Then u E LP ~ u E (with T = N/N - 2~, and

~’roo f . Set

Lemma (3.2) applies, to find K E R such that the equations (omittingsubscript j )

have a unique solution = u respectively. Furthermore,

so that we can assume zj H10 ’2’, for some z. Clearly z = ’H, because ajLN/2 aand for every E ~o, and then ~ve can pass to the limit in

Now, let us fix q p. To estimate first notice that taking eventuallythe positive and negative part of f + Ku, it is enough to estimate assuming z; > 0. Dropping subscript j, , let us consider z,~ = min(z, n).Since |zn|q-1 E H10, denoting f = f + Ku, we get from (3.7) :

and hence,

Using (3.6), with u = zn~2 we finally get

Takin g E = 2~q q2 1~ and using Sobolev inequality we obtain,

sending n to infinity and using Fatou lemma we get

with Cq and k independent on j. .

Using lemma 3.2 and taking q = 2* in (3.8) we get

Iterating this procedure, we get, up p r’2*

and then

with c and K independent on j. Passing to the limit (as j 2014~ +00) in (3.9)we set

Combining Lemma (3.3) and elliptic regularity applied to both equations(3.1)-(3.2), we obtain the claim of Proposition 3.1.We are now in position to give L~-estimates for solutions of (0.1)-(0.2)R. .

PROPOSITION 3.4. Let f, h R satisf y ~~.1~ and

Let vR~ be solutions of (0.1~-(0.,~~R such that,(i) {ur} is precompact in L2*, , and supR ~vR~ + Then,

Proo f . According to Proposition 3.1, we know that (u R, vR) E L°° . SetaR(x) := A + where

It is easy to see, using the compactness assuptions, that

So that, by Lemma 3.2, there exists p E R, independent on R, such thatuR(.) is the unique H10-solution of

To get a uniform bound we first prove LP-estimates. To this

extent, let us introduce z~, wR, solutions of (3.12) corresponding to righthand side 6v R)+ and respectively : they exists andare unique because of Lemma 3.2. We claim that, setting T = N/N - 2, wehave

, ,

where Cp is independent on R.Since for p = 2, the right hand side in (3.13) are bounded with respect

to R, we will get

because obviously u R = w R + zR In turn this implies supR |uR|2 2 +00,and thus, by iteration, we will get

To prove (3.13) for, say zR (the same procedure works as well for wR) wefirst need

where Dp does not depend on R. Let us prove (3.15) with vR replaced byvR, solution of

A similar argument, taking instead uR, will give (3.15). If uR >o is the solution of (3.16), multiplying the equation by vR 1, we get(observe that (vR (x)) 0) .

and using Poincare, inequality,

Since I Ai, if p > 2 is sufficiently close to 2, we get

Starting with such a p, and using Sobolev inequality, we have

and hence

which implies By iterating the above procedure, we thus

get ~p > 2, ~ Cp > 0 : Cp|u+R|p (uniformly in R) .We now go back to the proof of (3.13). Multiplying the equation

we get as above

Now, in view of (3.11) we can apply (3.6) to get, with a suitable choiceof E and using Sobolev’s and Holder’s inequalities :

for some positive constants C7p, kE independent on R.

Using (3.15), we finally get

Now, elliptic regularity, Sobolev imbedding and (3.14)-(3.15) applied to(O.l)R give uniform (w.r. to R) ~°°-bound for u,R.

Finally, if sup R M, setting

we get,

and multiplying the above inequality by VR = max~vR - ~, 0~, we obtain

Thus (u~ - = 0, i.e. ~ E Q, VR.

Similarly we get a uniform lower bound for VR. .

4. The variational principle andexistence results

In this section we first consider system (0.1) (0.2) for some classes ofsuperlinear-sublinear nonlinearities, reducing our system to an integrodiffe-rential equation of variational type. The early stage, when we consider f(. )of the type ( f) and h(.) satisfying (hl ) - (h2 ~ and asymptotically linear wealready deal with the main difficulty, i.e. a "mountain pass" situation withthe lack of .P. 5’., which we over come in the spirit of ~6~ .

Nevertheless, while the existence technique is quite the same as in [19](see also [3]), due to the lack of a general maximum principle, we are ableto produce two quite different existence results, as far as we are concernedwith existence and with existence of positive solutions respectively.We believe that this is a peculiar difference between anticoercive-coercive

systems considered by us and elliptic equations.Our first result is the following :

THEOREM C RN (N > 4) be a smooth bounded domain.

Assume

and

Then, there exists a non-trivial solution (u, v) E H10 x Ho to

provided,

If, in addition, we assume

then (1.1~-(1.,~~ has a solution (u, v) satisfying u > v > 0.

In what follows (see Th. 4.9 below), we will improve our result, as far asgrowth conditions on 1~~ ~ ~ are concerned, making use of the a priori estimatesand the results of Section 3.

Finally, we will see that a more delicate situation occours if N = 3 (seeSection 5 and [6] for the scalar case).

In order to prove th. 4.1 we first state a variational principle and weanalyse its geometric features.

Setting,

one can easily see, using assumptions ( h 1 ~ - ( h2 ) and -y that U ( ~ ~ is astrongly monotone operator. In addition using standard tools one can easilyfind that the operator T = U-1 G with

satisfies the following properties ; which we collect in

LEMMA 4.2. -

Now, let us denote

Notice that

and hence

because of - (h2). .

Using the above facts, we easily get the following :

A variational principle

is a C2-functional on with

Henceforth, if = 0, the pair (u, v) with v := is a solution to

(0.1)-(0.2). This easily follows from the relation VJ(v) = U(v), so that

Thus if v = T(8u) and VE(u) = 0, we have

i.e. (u, v) is a solution to (1.1)-(1.2).

Remark 4.3.

(i) If (u, v) is an H10-solution to (0.1)-(0.2) then

(ii) If v = T(u) then for u E Ho it follows that

In the following lemma we describe the geometric properties of "moun-tain-pass" type of the energy E(.). ..

LEMMA 4.4. -

P~-oof . Clearly E(0) = 0, ~E(0) = 0 because U(0) = 0. FurthermoreE"(0) = I - AG + S2G(I - has positive eigenvalues given by~~ 2 (~~ + SZ (~~ - S)-~ - ~) , which are bounded away from zero becauseof (A)

(ii) Since, in view of Lemma 4.2 and the positivity of J

we immediately get as t --~ +00. To make use of the geometricinformations given by Lemma 4.4 we need the following crucial compactnessproperty of E(.) .

Denote by

the best Sobolev constant, we have :

LEMMA 4.5.-Let E .Ho be such that ~ 0 as n -

- f-oo. Then (un) has an Ho -convergent subsequence provided supn E(un) 1

SN/2 .

N °

The proof of such (P.S.) condition, which makes use of arguments from[18] (see also (23~~, will be postponed, as well as the following estimate ofthe "mountain pass level" : Setting

we have :

Proof of Th. 4.1. In view of Lemmas 4.4-4.6; the "mountain-passlemma" (see [1]) applies to get a critical point of E( ~ ) at positive energyC. Our variational principle implies the result.

In order to get existence of positive solution under the additional assump-tion (A1 ) - (A2), we have to slight modify our arguments, first replacing

in (6.1) and h(v) in (0.2) respectively with

The new system

has only L°°-positive solutions by Prop. 1.4. In turns, solutions to (0.1 )’-(0.2)’ can be obtained as critical points of the modified functional

where F is defined in the obvious way. Slight modifications of our arguments,show that Lemma 4.4 (with (ii) only for positive u’s) - 4.5 and 4.6 hold withE replaced by E, and hence the result follows.Now we go back to the proofs of Lemma 4.5 and 4.6. .

Proof of Lemma .~.5. It will require several steps : :

Step ~.2014 supn -f-oo, so that, w.l.o.g. : u, u a.e; andu in La for every a E (1, 2*).

Furthermore there exists (see [19]) positive measures ~ and v andnumbers a~ > 0 such that for every ~ E it results

The lemma will follow noticing that assumption sup E(un) and Steps 2,3 imply aj = 0, (i.e. J~ ’~’ Jn in view of (4.1)-(4.2))which joint with u in L2* imply un - u in 1~ . Finally, setting

we easily see from

that .

Proof of Step 1. By Assumption we have

and hence _ , _ _

implies supn ~un ~ --f-oo.

Proof of Step 2. Passing to the limit in

we see that, taking in the limit cp = u, -

On the other hand, since T is completely continuous and

we get

But > 03A9 |~u|2 + by (4.3), so that using (4.4) and (4.5)we conclude

Proof of Step 3. From (4.2)-(4.3) and (4.5) we obtain

Proof of Lemma 4.6. - It is enough to find a point u such that

supt~0 E(tu) 1 NSN/2. For this purpose, we will use "instanton" like func-tions employed by BREZIS-NIRENBERG [6].

For a given cut-off function 03C6 E 1 in a neighbourhood ofzero, we set

It is well know that (see BREZIS-NIRENBERG [6]) for a suitable choice ofthe normalization constant K, one has the following estimates

(here mean bounded, and bounded away from zero).

In particular

Now, if maxt>o E(tuE), then d/dt 0, i.e. tE~Esatisfies

,

Using (4.11) and Lemma 4.2-(iii) we get

and hence,

Thus, writing = t~u~ v~ = from

and since const.S2 by (h3), we obtain

Using (4.12), inequality and (4.10) we finally obtain

provided (u~ ~2 is sufficiently small, i.e. if E is small.

Remark 4.7. In dealing with the modified functional ~, zve need a modification to get Lemma 4.5. As far as Step 1 is concerned, uTe get insteadof (4.7) :

and then

Assuming, by contradiction, ( ~ -f-oo, we get from (4.14) u n : _

u+n ~ 0 in and consequently, if u := 03C9 - lim un it results u 0..

Furthermore, from

here

we get

strongly in because ~ cos : thus, setting

(~c, u) satisfy

On the other hand, since

by definition of vn := T (bu n ), we also have, dividing and passingto the limit (using (~t3 ~~ : :

But since u 0, by the maximum principle v 0 and hence

Using assumption (t1~1 - (A~2, we see that (4.17~-(4.18) ha only the trivialsolution, = 6 = 0, a contradiction. This proves step 1 for the energyof the modified problem. As for step 2, it is enough to observe that if

un --~ u, v~. := ---~ v = (u, v~ solve the system (o.l~’-(0.2~’and hence they are positive functions in view of assumptions (t~1 ~ - (A2) and Prop. 1.2. But then we still have (4.4), while (4.5) follows observingthat limn 03A9|+n|2* = 03A9|un|2*. In fact,

implies ~~ I ~u ~ ~ 2 --~ 0 because The same remark allows

to work out Step 3, and thus the proof of Lemma 4.5, for the modifiedproblem, is complete.

Remark 4.8. Step 1 in the proof of Lemma 4.5 can be obtainedreplacing (~)3 with the more general assumption

Similarly, assumption could be replaced by

We want to show now how to drop the growth assumption on h(.), makinguse of Theorem 4.1 and of the a priori bounds given by Proposition 3.4.

THEOREM 4.9. Le~t N > 4, and assume and (A2). Then if

are satisfied, the problem

has a non trivial solution (u, v) E I~o (SZ)2 , If in addition

holds, then the above system has a solution (u, v) satisfying

Remark 4.10. (h5 ) ~ (h2 ).Proof of Theorem .~/9. - Given M > 0, let us consider the truncated

nonlinearity :

Notice that h M satisfies (h5), and hence (h2) too (h3). Thus Theorem 4.1applies to get solution (uM, vM) of the modified system :

The proof will be now performed in several steps. Indicating with the

energy corresponding to h M, (u~, v~) can be chosen such that

Clearly step 4 implies the result.

We will chosse (uM, vM) as given by Theorem 4.1, so that they satisfythe inequality 1/N

Proof of Step 1. Here, and in what follows we will use the obviousnotation :

Notice that VM = so that ~~ vMuM = ~~ Now, given as above, we have

and also,

and thus, since u( u)u > 0 (u ~ 0,

Proof of Step 2. It is enough to remark that (4.19~ rewrites :

uniformly with respect to M, because the inequality 03A9 TM(03B4t~u~)03B4t~u~ cti (p E [2N N+2’ 2*]) holds true with c independent on M. As aconsequence, following the proof of Lemma 4.6, we get

which implies

if e is sufficiently small.

Proof of Step 3. - This is a slight modification of the arguments used inthe proof of Lemma 4.5. We already know that, given passing eventuallyto a subsequence, we have Mj := v~ := for some u and

v, because by lemma 4.2-ii, with some c independent onFollowing the proof of Lemma 4.5, we have just to prove that

(where i~( ~ ) an v( ~ ~ are limits, in the sense of measures, respectively of and and v = + ~

Now, (4.8) and (4.9) easily follow from

and

and the fact that v; -j v, u3 --y u in As far (4.24), using Remark 4.3and (h2 ), we see that

Proof of Step ,~. - This is just a consequence of the precompactness ofin HJ and sup~ in view of Proposition 3.4.

5. The case N = 3

In this section we concentrate our attention on the existence of solutionsto the very special system

or equivalently to the problem

obtained from (4.1) by inverting the second equation (i.e. > =

Jn E

We will follow closely the approach of Bahri-Coron (see also BREZIS [4],SCHOEN [22], Mc LEOD [20]) which describes the important role played bythe regular part of the Green function of the linear operator involved in thegiven nonlinear problem, in estimating

For definitness, let G(~, ~~ be the Green function relative to

c2

~i

As usual one can split G~-, -) into a singular and regular part

where y) is, for every x E S~, the only solutions to

PROPOSITION Q c R3 be a smooth bounded domain such that

0 E Q.

If

then

and consequently (4.1~ is solvable.

Proof - Since the proof is the same as in BREZIS [4] we only give theideas involved.

Let E > 0 and consider

a direct calculation shows that

We consider as "test" functions for estimating .S’E the solution to

By introducing

we see that he satisfies

Since E2 + 1 x Z 1 /2

~ 1 |x| in L2 as ~ ~ 0, elliptic theory gives

uniformly in Q, where ho is the solution to

that is ho(x) = 0~. At this point the calculations are exactly the sameas in BREZIS [4]. So we shall omit them.

COROLLARY 5.2. ~et SZ C R3 be a smooth bounded domain such that

Assume that

and

(here g± is the regular. part of the Green function o,f the problem -OG -

(03BB ± 03BB2 - 403B42 ) G = 03B4(. - y)) holds.

Then (5.1 ) has cc positive solution.

Proof . The positivity of the solution is guaranted by (i~ . (see Prop.1.1~. For the existence part, it is sufficient to observe that if (i) holds thenthe solution of (5.7~ is exactly

Rernark 5.3. In the limiting case (from the point of view of the

maximum principle) A = 2b, for the existence of positive solution it is

sufficient to assume that > 0 where in this case g is the regularpart of G, defined by

5.4. - In the special case S~ = B1 (0} and a > 2b, a1 > s, «eknow by Prop. 1.3 that all positive solutions of (5.1) must be radial. In thiscase (ii) of Corollary 5.2 reads :

So we can see that in view of the nonexistence results given by Proposition2.3, (5.10) is "almost sharp" in the sense that there is only a small intervalof .in which we do not know what happen. However, since by usingthe original technique of BREZIS-NIRENBERG [6] we can arrive at the samesolvability condition (5.10), we can conjecture that (5.10) is sharp.

Remark 5.5.2014If 03A9 = B1 (0) and 03BB = 203B4 03BB1 > 03B4 (here 03BB1 = 03C02) then (5.1)has a positive radial solution if ~2014 A 2~r2.

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(Manuscrit reçu Ie 4 novembre 1986)