Multiple solutions of a nonlinear elliptic equation involving Neumann conditions and a critical Sobolev exponent

Multiple Solutions of a Nonlinear Elliptic Equation Involving Neumann Conditions

and a Critical Sobolev Exponent.

J. CHABROWSKI(*) - JIANFU YANG(**)

ABSTRACT- In this paper we prove the existence of two solutions of the nonhomo- geneous Neumann problem (1.1) involving a critical Sobolev exponent. It is assumed that the coefficientQis positive and smooth onVandlD0 is a par- ameter which does not belong to the spectrum of2D. We examine the com- mon effect of the mean curvature of the boundary¯V and the shape of the graph of the coefficient Q on the existence of a second solution.

1. Introduction.

In this paper, we study the existence of multiple solutions of the superlinear problem

. / ´

2Du

¯

¯nu(x)

4lu1Q(x)u₁^2*211f(x) in V 40 on ¯V,

(1.1)

where 2*4_N^2N

22, NF3 is the critical Sobolev exponent, lF0 is a parameter and V%R^N is a bounded domain with a smooth boundary (*) Department of Mathematics, University of Queensland, St. Lucia, Bri- sbane Qld 4072, Australia.

(**) Institute of Physics and Mathematics, Chinese Academy of Science, P.O.

Box 71010, Wuhan 430071, P.R. of China.

¯V. We assume that the coefficient Q is smooth and positive on V and fL^r(V) with rDN. We use the notation u₁4max (u, 0 ).

This problem belongs to a class of problems referred to as the Ambro- setti-Prodi type. More precisely, in the case of the Dirichlet problem

. /´

2Du u

4g(u)1f(x) in V, 40 on ¯V

the limits

g₂4 lim

sK 2Q

g(s)

s and g₁4 lim

sK Q

g(s) s

play an important role. We can basically distinguish three types of pro- blems using the location ofg₂andg₁with respect to the spectrum of the operator2D with the Dirichlet boundary conditions. Denoting by]lk( the sequence of the eigenvalues of2Dwith the Dirichlet boundary con- ditions, the following types of problems have been considered:

(I) 2Q Gg₂El1Eg₁G 1Q,

(II) g₂ andg₁are both finite and the interval (g₂,g₁) contains an eigenvalue. In this case the problem is asymptotically linear,

(III) g₂ lies between two consecutive eigenvalues and g₁41Q. We refer to the paper [12] where the extensive bibliography concer- ning these problems can be found. We point out here that conditions (I) and (III) cover the cases of subcritical, critical and supercritical growth forg. In the case of the Neumann problem the literature is rather scar- ce. In this paper we consider the nonlinear Neumann problem of type (III) with the nonlinearity of one-sided critical growth. We follow some ideas from [12], which considered a similar problem with the Dirichlet boundary conditions. First we consider the case lD0 . The case l40 will be treated separately.

Problem (1.1) may have constant solutions in contrast to the Dirichlet problem. We now discuss a number of conditions guaranteeing that a po- sitive solution of (1.1) is not constant. If for some lD0 and a constant cD0 , the functions Q and f satisfy the equation

lc1Q(x)c^2*211f(x)40 (˜)

for everyxV, thenu4cis a solution of (1.1). IffandQare differentia- ble on some open subset of V then the following condition

(a) ˜f(x) is not parallel to ˜Q(x) for some xV

ensures that a positive solution of (1.1) is not constant. IffandQare not differentiable we can proceed as follows. Integrating the equation (˜) we get

lcNVN 1c^2*21

V

Q(x)dx1

V

f(x)dx40 , (˜˜)

where NVN denotes the Lebesgue measure of V. From (˜) and (˜˜) we derive the equation

c^2*21

u

V

Q(x)dx

v

u

V

f(x)dx

v

We immediately obtain a contradiction if

(b) either Q(x)4const and f(x)cconst , or Q(x)cconst and f(x)4const .

If both functions Q(x) and f(x) are not constant we define a set V04

{

V

Q(x)dx4Q(x)

}

which is nonempty. Then a positive solution cannot be constant if (c) either f(x)4 ¹

NVN

s

V

f(x)dx for all xV2V0, or f(x)c c ¹

NVN

s

V

f(x)dx for some xV₀.

Finally, if (c) does not hold we require (d) the ratio

f(x)NVN 2

s

V

f(x)dx

s

V

Q(x)dx2Q(x)NVN

is either not constant on V2V₀, or it is constant and nonpositive on V2V0.

Therefore one of these conditions will be assumed throughout this work.

We assume that f(x)4t1h(x), where tis a constant andhL^r(V) withrDN. We start by finding a negative solution of (1.1). We denote by l₁40El₂ER the sequence of eigenvalues for2Dwith the Neumann boundary conditions. The first eigenvalue is simple and has constant eigenfunctions.

Let lclk for every k. Then there exists a unique solution u₀ H¹(V)OL^Q(V) of the problem

. /

´

2Du

¯u

¯n

4lu1h(x) in V, 40 on ¯V. The function u_t4 2_l^{t 1u0,} with tDlsup

V Nu₀(x)N is negative and sati- sfies (1.1). We look for a second solution of the formu4v1ut, wherev satisfies

. / ´

2Dv

¯v

¯n

4lv1Q(x)(v1u_t)₁^2*21 in V, 40 on ¯V.

(1.2)

Problem (1.2) will be solved through the min-max based on a topological linking. To this end, we define a variational functional

J(v)41 2

V

(

)

2*

V

Q(x)(v1u_t)₁^2*dx

for vH¹(V). In the next section we examine Palais-Smale sequences for J. In particular, we find the energy level of the functional J below which the Palais-Smale condition holds. In Section 3 we verify that the functional J has the geometry of a topological linking. Conditions gua- ranteeing the existence of critical points ofJ will be given in Sections 4 and 5. The existence results of this section depend on a relation between Qm4max

x¯VQ(x) andQM4max

xV

Q(x). Section 6 is devoted to the casel40 . The existence of a critical point in this case is obtained through the im- plicit function theorem. The distinction of two cases involving the quanti- ties QM and Qm envisaged in Section 4 disappears in the case l40 .

2. The Palais-Smale condition.

We need two quantities:

Q_m4max

x¯VQ(x) and Q_M4max

xV

Q(x).

We set

S_Q4min

g

h

where S denotes the best Sobolev constant, that is,

S4 inf

uD^{1 , 2}(R^N)2 ]0(

s

R^NN˜uN²dx (

s

R^N Nu(x)N^2*dx)^{2 /2* .}

Here D^{1 , 2}(R^N) denotes a Sobolev space obtained as the completion of

C₀^Q(R^N) with respect to the norm VuV_D21 , 2(R^N)

4

R^N

N˜uN²dx.

In what follows, VQV denotes the norm in H¹(V), which is given by VuV²4

V

(

)

In this paper we frequently use the Sobolev inequality:

u

V

NuN^2*dx

v

V

(

)

for all uH¹(V), where C_sD0 is a constant.

PROPOSITION 2.1. Let l_kElEl_k11. If

J(u_n)KcES_Q and J8(u_n)K0 in H²¹(V) then ]u_n( is relatively compact in H¹(V).

PROOF. We commence by showing that ]um( is bounded in H¹(V).

We write

un4un2

1un1, un2E² and un1E¹, where

E²4span of all eigenfunctions corresponding to l1,R,lk,

and E¹4(E²)^». If fH¹(V), then

V

˜u_n˜fdx2l

V

u_nfdx4

V

Q(x)(u_n1u_t)₁^2*21fdx1enVfV (2.1)

with enK0 . Taking f4u_n¹, we get

V

N˜u_n¹N²2l

V

(u_n¹)²4

V

Q(x)(u_n1u_t)₁^2*21u_n¹dx1e_nVu_n¹V.

Let dD0 be such that l1dElk11. Then (2.2)

g

h

V

N˜u_n¹N²dx1d

V

(u_n¹)²dxG

G

V

Q(x)(u_n1u_t)₁^2*21u_n¹dx1e_nVu_n¹V. We now use (2.1) withf4un2and letd1D0 be such thatl2d1Dlk. Then

(2.3)

g

h

V

N˜u_n²N²dx1d₁

V

(u_n²)²dxG

G 2

V

Q(x)(u_n1ut)₁^2*21un2dx1enVun2V.

On the other hand for nFn0, we can write c1e_nVu_nV11FJ(u_n)2 1

2aJ8(u_n),u_nb 4 1

2

V

Q(x)(u_n1ut)₁^2*21undx2 1 2*

V

Q(x)(u_n1ut)₁^2*dx

4 1 N

V

Q(x)(u_n1u_t)₁^2*dx2 1 2

V

Q(x)(u_n1u_t)₁^2*21u_tdx

F 1 N

V

Q(x)(un1ut)₁^2*dx.

Applying the Young inequality, we deduce from (2.2) and the above

estimate that for hD0 we have (2.4)

g

h

V

N˜u_n¹N²dx1d

V

(u_n¹)²dxG

G

V

Q(x)(u_n1u_t)₁^2*21u_n¹dx1enVu_n¹VG

Gh

u

V

Q(x)Nu_n¹N^2*dx

v

u

V

Q(x)(u_n1u_t)₁^2*dx

v

GC_sQ_M^{2 /2*}hVu_n¹V²1C₁

u

V

(u_n1u_t)₁^2*dx

v

GCsQM2 /2*hVun1V²1C1VunV^{(N12 ) /N}1C2Vun1V1C3

for some constants C1D0 , C2D0 and C3D0 . In a similar way, we obtain

(2.5)

g

21

h

V

N˜u_n²N²dx1d1

V

(u_n²)²dxG

GC_sQ_M^{2 /2*}hVu_n²V²1C₄

(

)

for some constant C₄D0 . Estimates (2.4) and (2.5) imply that ]u_n( is bounded inH¹(V). We may therefore assume that unuinH¹(V). By the concentration-compactness principle there exist sequences of points ]x_j(%R^N, sequences of numbers ]n_j( and ]m_j( such that

Nu_nN^2*˜NuN^2*1

!

j

n_jd_xj

and

N˜unN^2˜N˜uN²1

!

j

mjdx_j

in the sense of measures, where Sn2 /2*j

Gmj if xjV

and

Sn_j^{2 /2*}

2^{2 /N} Gmj if xj¯V.

Fixx_j. Let]fd(be a family of smooth and positive functions concen- trating at xj as dK0 . Then using the Brézis-Lieb Lemma, we ob- tain

V

N˜unN²f_ddx1

V

˜unun˜f_ddx1l

V

un2f_ddx

4

V

Q(x)(u_n1u_t)₁^2*21u_nfddx1o( 1 )

4

V

Q(x)(u_n1u_t)₁^2*fddx2

V

Q(x)(u_n1u_t)₁^2*21u_tfddx1o( 1 )

G

V

Q(x)Nu_n1u_tN^2*fddx2

V

Q(x)(u_n1u_t)₁^2*21u_tfddx1o(1)

4

V

Q(x)Nu_nN^2*fddx2

V

Q(x)NuN^2*fddx1

V

Q(x)Nu1u_tN^2*fddx

2

V

Q(x)(u_n1u_t)₁^2*21u_tfddx1o( 1 ).

LettingnK Qand thendK0 we deduce that in both casesxj¯V and xjV,

mjGQ(x_j)nj. If mjD0 for some x_j, then

m_jF S^N/2

Q(x_j)^{(N22 ) /2} if x_jV and m_jF S^N/2

2Q(x_j)^{(N22 ) /2} if x_j¯V.

We now write J(u_n)2 1

2*aJ8(u_n),u_nb 4 1

N

V

(

)

V

Q(x)(u_n1u_t)₁^2*dx

1 1 2*

V

Q(x)(u_n1u_t)₁^2*21u_n1o( 1 )

4 1 N

V

(

)

V

Q(x)(u_n1ut)₁^2*21utdx1o( 1 )

F 1 N

V

(

)

Since u is a solution of (1.1) we also have

V

(

)

V

Q(x)(u1u_t)₁^2*21u dx4

4

V

Q(x)(u1u_t)₁^2*21u₁dxF0 .

We aim to show thatm_j40 for everyj. If not, the concentration-com- pactness principle implies that

cF 1 N

V

(

)

!

j

mjF 1 N

!

j

mj.

If mjD0 for some j with xj¯V, then cF 1

2N

S^N/2

Q(xj)^{(N22 ) /2}F 1 2N

S^N/2 Qm(N22 ) /2.

This is obviously impossible. Similarly if m_jD0 for some j with x_jV. Thus

V

Q(x)(u_n1u_t)₁^2*dxK

V

Q(x)(u1u_t)₁^2*dx

and also

V

N˜unN²dxK

V

N˜uN²dx and the result follows. r

3. Topological linking.

We assume that l(lk,lk11). Let

E²4span]e₁,R,e_l(,

wheree₁,R,e_l are eigenfunctions corresponding tol1,R,lk. We set E¹4(E²)^». Let

S_r4¯B_rOE¹ and D4[ 0 ,Re]5(B_rOE²), eE¹,

whereB_rdenotes the ball of radiusrwith centre at 0 . To apply a topolo- gical linking we need to verify that

JN^SrFaD0 , rER, JN^¯DEa and max

uD J(u)ES_Q.

LEMMA 3.1. There exist r0D0 and a: ( 0 , r0]K( 0 ,Q) such that J(u)Fa(r) for every vS_r.

PROOF. We choose hD0 so that lkEl1hElk11. Then

V

N˜uN²dxFlk11

V

u²dx

for every uE¹. Since u_tE0 on V, we have J(u)F

V

g

h

V

u²dx2C_s^22*/2Q_M

u

V

(

)

v

Fb

V

(

)

u

V

(

)

v

where

b4min

g

,h

h

Letting r4VuV we obtain the following estimate J(u)Fbr²2Cs22* /2

QMr^2*. To complete the proof we set

a(r)4br²2C_s^{22* /2}Q_Mr^2*, with r0D0 such that r20

2Cs22* /2

QMr02*

D0 . r From now on, we use the instanton

U_e(x)4 c_Ne^{(N22 ) /2}

(

)

in the definition of the set D, where cND0 is a constant and we set e_e4P¹U_e. It is well-known that U4U1 satisfies the equation

2DU4U^2*21 in R^N and moreover

R^N

N˜UN²dx4

R^N

U^2*dx4S^N/2.

With the choice ofe4e_ewe verify the remaining conditions of the to- pological linking. Without loss of generality we may assume in Lemma 3.2 below that 0V.

LEMMA 3.2. There exist r₀D0 , R₀D0 and e0D0 such that for rFr₀, RFR₀ and 0EeGe0 we have

J(u)Ea for every u¯D. PROOF. We set

¯D4G1NG2NG3,

where

G14BrOE²,

G24 ]vH¹(V); v4w1se_e, wE², VwV4r, 0GsGR(, G34 ]vH¹(V); v4w1Re_e, wE²OB_r(. For vE² we have

V

N˜vN²dxGlk

V

v²dx

and

J(v)G1

2

g

h

V

N˜vN²dx2 1 2*

V

Q(x)(v1ut)^2*dxG0 .

We now consider G2. Let vG2 and define d²4 sup

0EeG1

V

N˜e_eN²dx.

Let r²4V˜wV²and choose h₁D0 so that l_kEl2h₁. Then for v4w1 1se_e we have

J(v)G 1 2

V

N˜vN²dx2l 2

V

v²dx

G 1

2

V

N˜wN²dx2 l 2

V

w²dx1s²

2

V

N˜e_eN²dx

G1

2

g

h

V

N˜wN²dx2h1

2

V

w²dx1 s²

2

V

N˜e_eN²dx.

Let h24max

g

2

h

2

V

N˜e_eN²dx.

We sets₀4^k_d^2a . ThenJ(v)Ga for 0GsGs₀. We now consider the

case sDs₀. Put

K4sup

m ^V

V

Q

; s₀GsGR, VwV4r, wE²

n

We now estimate P²U_e. Let

P²U_e4

!

j41 l

ajej, aj4

V

U_eej(x)dx.

Since the first eigenfunction corresponding to l40 is constant, we see that P²U_eg0 . Hence

VP²U_eV₂²4

!

j41 l

aj2

4

!

j41 l

u

V

U_eejdx

v

G

!

j41 l

Ve_jV

Q

2 VU_eV₁²GCe^N22.

Therefore

P¹U_e( 0 )4U_e( 0 )2P²U_e( 0 )

FCe^{2(N22 ) /2}2VP²U_eV

QFCe^{2(N22 ) /2}. By the continuity of P¹U_e there exists a d4d(K) such that

B_d( 0 )%]xV; P¹U_e(x)DK(.

We also need the following inequality: if v%V and u1vD0 on v, then

N

N

where C4C(p). We apply this estimate with v4Ve, where V_e4 ]xV; P¹U_e(x)DK(.

Letting Q

*4min

xVQ(x) we get

V

Q(x)

g

h

Ve

g

h

FQ

*

u

e

Ne_eN^2*dx1

Ve

N

N

2C

V_e

g

_N

N

N

N

h

h

FQ

*

u

V

Ne_eN^2*dx1

V_e

N

N

v

2C1

(

1Ve_eV_L1(V_e)

)

Since

VP¹U_eV_L2*21 2*21

GCe

N22

2 and VP¹U_eV_L1GCe

N22 2

we deduce from the previous estimate that J(v)Gh2r²1 s²

2 S^N/22 s^2*

2*Q

*S^N/21Cs^2*e^{(N22 ) /2} 4h2r²1F_e(s).

It is easy to check that F_e(s)G 1

2

g

h

⁽

⁾

Increasing r, if necessary, we get

J(v)E0 for vG2. If vG3, then

J(v)41

2

g

h

V

N˜wN²dx1 R²

2

V

N˜e_eN²dx

2R^2*

2*

V

g

h

Let KD0 be such that Vw1u_tV_LQGK. Then there exists an e0D0 (small enough) such that P¹e_e( 0 )D2K for every 0EeGe0. Then for

every 0EeGe0 we can find R₀4R₀(e) and h4h(e)D0 such that

N m

n _N

for RFR₀. Hence J(v)G0 for vG₃ for eGe₀ and RFR₀. r

4. Case Q_ME2^{2 /(N22 )}Q_m.

Let H(y) denote the mean curvature of the boundary ¯V at y¯V. Throughout this section we assume that:

(A) the coefficient Q satisfies the following conditions:

Q_ME2^{2 /(N22 )}Q_m,

andNQ(x)2Q(y)N 4o(Nx2yN) for somey¯VwithQ(y)4Q_m,H(y)D D0 and x close to y.

Obviously in this case we have S_Q4 ^S

N/2

2NQm(N22 ) /2.

PROPOSITION 4.1. Let NF5 and suppose that (A) holds. Then max

vD J(v)E S^N/2 2NQ_m^{(N22 ) /2}. (4.1)

PROOF. Without loss of generality we may assume that y40 . Let vD. Then v4w1se_e and

J(w1se_e)4 1

2

V

(

)

2

V

(

)

2 1 2*

V

Q(x)(w1se_e1u_t)₁^2*dx.

For 0EsGs₀, with s₀ sufficiently small, we have J(w1se_e)G s²

2

V

N˜e_eN²dxG S^N/2 2NQ_m^{(N22 ) /2}. If sFs₀, then repeating the estimates from Lemma 3.2 we get

J(w1se_e)G s²

2

V

(

)

2*

V

Q(x)e_e^2*dx1Cs^2*e^{(N22 ) /2}

for some constant CD0 . Hence

J(w1se_e)G 1 N

(

s

V

(N˜e_eN²2le_e²)dx)^N/2 (

s

V

Q(x)Ne_eN^2*dx)^{(N22 ) /2 1O(e}

(N22 ) /2) .

Since

V

NP¹U_eN^2*dx4

V

U_e^2*dx1O(e^N22) and

V

N˜P¹U_eN²dx4

V

N˜U_eN²dx1O(e^N22) , we obtain

J(w1se_e)G 1 N

(

s

V

(N˜U_eN²2lU_e²)dx)^N/2 (

s

V

Q(x)U_e^2*dx)^{(N22 ) /2 1O(e}

(N22 ) /2) .

We need the following asymptotic formulas (see [17])

V

N˜U_eN²dx4 K1

2 2I(e)1o(e) ,

V

U_e^2*dx4K2

2 2P(e)1o(e) , where K₁4

s

R^N

N˜UN²dx, K₂4

s

R^N

U^2*dx, S4^K1/K2 (N22 ) /N

, I(e)4O(e) and P(e)4O(e). Moreover, we have (see (3.17) in [17])

lim

eK0

I(e)

P(e) DN22 N

K2

K1

. (4.2)

By assumption (A) we have

V

Q(x)U_e^2*dx4 Q_mK₂

2 1o(e) .

Thus

J(w1se_e)G 1 N

g

h

g

h

(4.3)

According to (4.2) we can find an e0D0 and a rD0 such that I(e)D N22

N K₁ K2

P(e)1r (4.4)

for 0EeGe0. It then folllows from (4.3) and (4.4) that J(w1se_e)G

gg

h

g

h

h

3

gg

h

g

h

h

E S^N/2

2NQ_m^{(N22 ) /2}2Cr

for some constant CD0 and the result follows. r

We are now in a position to formulate the following result

THEOREM 4.2. Suppose that the assumptions of Proposition 4.1 hold. Then problem (4.1) has at least two solutions.

5. Case QMF2^{2 /(N22 )}Qm.

If Q_MF2^{( 2 /N)22}Q_m, then S_Q4 ^S

N/2

NQ_M^{(N22 ) /2}. We assume that

NQ(x)2Q(y)N 4o(Nx2yN²) (5.1)

for some yV with Q(y)4Q_M and x close to y. Assuming that y40 , we have

V

N˜U_eN²dx4K11O(e^N22) ,

V

Q(x)U_e^2*dx4K₂Q_M1o(e²)

and

V

U_e²dxFc₁e²

for some constantc₁D0 independent ofe. As in the proof of Proposition 4.1 we have

J(w1se_e)G (

s

V

(N˜U_eN²2lU_e²)dx)^N/2 N(

s

V

Q(x)U_e^2*dx)^{(N22 ) /2 1O(e}

(N22 ) /2)

G (K₁1O(e^N22)2c₁e²)^N/2 N(K2QM1o(e²))^{(N22 ) /2 1O(e}

(N22 ) /2).

If NF7 , taking eD0 sufficiently small, we can check that maxvD J(v)S^N/2/Q_M^{(N22 ) /2}.

THEOREM 5.1. Let NF7 . Suppose that Q_MF2^{2 /(N22 )}Q_m and that (5.1) holds. Then problem (1.1) has two solutions.

6. Existence of solutions in the case l40.

In this case problem (1.1) takes the form

. / ´

2Du

¯

¯nu(x)

4Q(x)u₁^2*211f(x) in V 40 on ¯V,

(6.1)

Obviously a necessary condition for the existence of a solution of pro- blem (6.1) is the condition

V

f(x)dxE0 . (6.2)

Since the eigenfunctions corresponding tol40 are constant, we decom- pose H¹(V) as H¹(V)4span]1(5E¹, where

E¹4

{

V

v dx40

}

Thus for every function uH¹(V) we have u4t1v, where tR and

s

V

v dx40 . The variational functional J for (6.1) is given by

J(u)4 1 2

V

N˜vN²dx2 1 2*

V

Q(x)(t1v)₁^2*dx2

V

f(x)(t1v)dx.

It is easy to show that the function tKJ(t1v) is bounded above. Let vE¹ and set

g(t)4J(t1v).

It is clear that for every vE¹ there exists t(v)D0 such that g(t(v) )4max

tR g(t) ,

that is, J(t1v)GJ(t(v)1v) for every tR. Thus by the implicit fun- ction theorem we can define a continuously differentiable mapping

vE¹ ¨ t(v)R,

such that J(t1v)GJ(t(v)1v) for every tct(v). Since 04g8(t(v) )4 2

V

Q(x)(t(v)1v)^2*21dx2

V

f dx, we see that

V

Q(x)(t(v)1v)₁^2*21dx1

V

f(x)dx40 (6.3)

for every vE¹. In particular, if v40 , then

V

Q(x)(t( 0 ))₁^2*21dx4 2

V

f(x)dx. This combined with (6.2) yields t( 0 )D0 and we have

t( 0 )^2*21

V

Q(x)dx4 2

V

f(x)dx.

(6.4)

We now claim that the functional

F(v)4J(v1t(v)) attains its minimum on some ball B_r( 0 ). We set

A4 2

V

f(x)dx and B4

V

Q(x)dx.

By easy computations using (6.4), we verify that F( 0 )4N12

2N

A^{2N/(N12 )} B(N22 ) /(N12 ) . We now estimate F(v) from below:

F(v)FJ(v)4 1 2

V

N˜vN²dx2 1 2*

V

Qv₁^2*dx2

V

fv dx

F1 2

V

N˜vN²dx2 1 2*

V

Qv₁^2*dx2VfV₂VvV₂.

We now observe that

V

N˜vN²dxFl₂

V

v²dx

for everyvE¹. Since

s

V

v dx40 , we can use the Sobolev inequality to obtain

F(v)F 1 2

V

N˜vN²dx2 QM

2* S^{2N/(N22 )}

u

V

N˜vN²dx

v

2VfV₂l221

u

V

N˜vN²dx

v

Letting r4V˜vV₂ we derive from the above estimate F(v)Fr²

2 2Q_M

2* S^2N/N2r^2*2VfV₂l₂^{21 /2}r

4rz

g

2* S^2N/N22r^2*212VfV₂l221 /2

h

Since j(r) achieves its maximum at

r04

g

h

we see that

F(v)Fr0

g

g

2(N12 )

h

h

(6.6)

4r₀

g

h

We now assume that VfV₂G l1 /22

N12

g

h

(6.7) and

(6.8) 2

V

f(x)dxG

G

g

h

^u

V

Q(x)dx

v

As an immediate consequence of (6.5), (6.6), (6.7) and (6.8) we can sta- te the following lemma:

LEMMA 6.1. Suppose that(6.7)and(6.8) hold. Then F(v)DF( 0 )for all vE¹ such that VvV4r0, and moreover

F( 0 )E S^N/2 NQ_M^{(N22 ) /2}.

We can now formulate the existence result in the case l40 . THEOREM 6.2. Suppose that(6.7)and (6.8)hold. Then problem(6.1) has a solution.

PROOF. It follows from Lemma 6.1 that m4 inf

vBr0( 0 )F(v)E S^N/2 NQ_M^{(N22 ) /2}. (6.9)

Let ]v_n( be a minimizing sequence for (6.9). Since ]v_n( is bounded in H¹(V), we may assume that v_nv₀in H¹(V) and v_nv₀ inL^q(V) for every 2GqE2* . By the low semicontinuity of norm with respect to