A note on the bifurcation of solutions for an elliptic sublinear problem

A Note on the Bifurcation of Solutions for an Elliptic Sublinear Problem.

ALESSIO PORRETTA(*)

1. Introduction.

We consider the following semilinear problem in a bounded domain V%R^N:

. /

´

Lu1u^u4lu uF0 , ug0 , u40

in V, in V, on ¯V, (1.1)

where 0EuE1 andLuis a linear differential operator defined asLu4 42div (A(x)˜u),A(x) being a symmetric, bounded and coercive matrix, that is such that

(1.2) aNjN²GA(x)jQjGbNjN²

for every jR^N, a.e. xV, with a,bD0 . A necessary condition to find nonnegative nontrivial solutions of (1.1) is that lDl1, the first eigenvalue of the operator L on V with Dirichlet (*) Indirizzo dell’A.: Dipartimento di Matematica, Università di Roma «Tor Vergata», Via della Ricerca Scientifica, 00133, Roma, Italia.

boundary conditions. Notice that (1.1) is the Euler equation of the functional

(1.3) J_l(u)41 2

V

A(x)˜uQ˜u dx2l 2

V

(u¹)²dx1 1 u11

V

NuN^u11dx

uH₀¹(V) , which, since 0EuE1 , is not bounded from below if lDl1.

As it was remarked in [8] (see also [6]), the existence of a solution of (1.1) for everylDl₁ can be obtained applying the abstract fixed point results by Krasnoselskii ([9]). This approach relies on the definition of the inverse operator ofuOLu1 NuN^{u u}

NuN

and makes use of its compact- ness properties and the computation of its derivative at zero and at infin- ity. Similarly, using the results of Rabinowitz in [11], in [6] it is proved the existence of a connected branch of solutions (l,u_l) bifurcating from infinity atl4l1. The same kind of arguments is used in a more general context in [3] to deal also with sublinear eigenvalue problems, following the method developed in [1] (see also [10]).

As a consequence of the previous results, the bifurcation diagram for solutions of (1.1) reads as follows: there is bifurcation from infinity if and only ifl4l₁while no finite value oflcan be a bifurcation point from the zero solution. The aim of this note is to prove that the bifurcation from the trivial solution actually occurs at l4 1Q.

More precisely, we prove that there exists a sequence u_l, lDl1, of Mountain Pass type solutions of (1.1) such thatVu_lV_H1

0(V)tends to zero as lgoes to infinity. In fact, we do not know whether any sequence of sol- utions of (1.1) will converge to zero as l tends to infinity, nevertheless the sublinear behavior of the absorption term both near the origin and at infinity yields a classical Mountain Pass type structure to the functional J_l which allows to construct such a sequence, obtaining the bifurcation from u40 at l4 1Q. This is the result that we prove.

THEOREM 1.1. Let 0EuE1 , and let (1.2) be satisfied. Then for everylDl1problem(1.1)admits a solution u_lH0¹(V)which is a criti- cal point of Mountain Pass type of the functional J_l defined in (1.3).

Moreover this family of solutions u_l satisfies

lim

lK 1Q

Vu_lV_H1 0(V)40 . (1.4)

The bifurcation from the trivial solution at infinity has already been proved for the semilinear problem

. /

´

Lu4lu^p uF0 , uc0 , u40

in V, in V, on ¯V, (1.5)

withpD1 . For a certain range of values ofpthis was obtained in [5] via a priori estimates while a general result (for every subcriticalpD1) was proved in [4] using the variational structure of the equation and esti- mates on the Mountain Pass level of the related functional. Our problem (1.1) is much different with respect to (1.5) since the nonlinearitylu2u^u is asymptotically linear at infinity and the Mountain Pass character is due to a sublinear positive perturbation. This implies that the approach- es of these previous papers do not work, in particular the standard regu- larity (bootstrap) argument is not such helpful and the method used in [4] would only give here an estimate on theL^u11(V) norm ofu. For this reason we use here a refined construction of the Mountain Pass solutions u_l, which are obtained through a recurrence argument, allowing, at fixed l, to estimateJ_lon the paths [ 0 ,u_l], wherelElandu_lis the Mountain Pass solution previously found.

Let us stress that, in dimension N41 , a full study of the equation (1.1) is carried out in the recent work [7]. In that paper the case of thep- laplace operator is also considered and the set of solutions of the prob- lem is fully characterized, giving multiplicity results as well. While our results can be proved without relevant changes for thep-laplace opera- tor as well, we cannot give here a full picture of the behavior of the sol- utions in dependence of l.

2. Proof of Theorem 1.1.

We denote byl₁the first eigenvalue of the operator Lwith Dirichlet boundary conditions on ¯V, and by W1 the positive eigenfunction such that LW₁4l1W1, VW1V_L2(V)41 .

PROOF OFTHEOREM 1.1. The proof of Theorem 1.1 is a consequence of the following two Propositions.

PROPOSITION 2.1. LetlDl1. Then problem(1.1) admits a solution u_lwhich is a critical point of Mountain Pass type of the functional J_l

defined in (1.3). Moreover u_l satisfies:

(2.1) J_l(u_l)G

g

2

h ^g

^s

^h

g

^s

V

w²2

s

V

A(x)˜wQ˜w dx

h

,

for every w:l

s

V

w²dx2

s

V

A(x)˜wQ˜w dxD0 .

PROOF. We are going to use the classical theorem of Ambrosetti and Rabinowitz ([2]) with a little change in the choice of the paths.

We start by proving thatu40 is a local minimum for the functional J_l. Precisely, we claim that

)rD0 : J_l(u)D0 (uB_r0]0( 4 ]VuV_H1

0(V)Gr,uc0(, (2.2)

which will easily imply that

)s : J_l(u)Fs (u:VuV_H1

0(V)4r. (2.3)

In order to prove (2.2) we argue by contradiction. If (2.2) were not true, it would exist a sequence u_n in H₀¹(V) such that:

Vu_nV_H1

0(V)K0 , u_nc0 , J_l(u_n)G0 (nN,

that is 1 2

V

A(x)˜u_nQ˜u_ndx1 1 u11

V

Nu_nN^u11dxG l 2

V

Nu_nN²dx. (2.4)

Setting w_n4_Vu^un

nV_H1 0 (V)

, we have:

V

Nw_nN^u11dxG l(u11 ) 2

Vu_nV

H₀¹(V)

12u

V

Nw_nN². (2.5)

SinceVwnV_H1

0(V)41 , we have that, up to subsequences,wnconverges to a function wH₀¹(V) almost everywhere in V, hence we deduce by Fa- tou’s lemma and (2.5) thatw40 , so thatw_nstrongly converges to zero in L²(V) as well by Rellich theorem. Dividing the equality in (2.4) by Vu_nV_H1

0(V)

2 , using the ellipticity of A(x) we conclude thatw_ntends to zero inH0¹(V), contradicting the fact that VwnV_H1

0(V)41 . Thus (2.2) is proved.

Now, assume that (2.3) is violated. Then we deduce again the existence of

a sequence u_n such that Vu_nV_H1

0(V)4r and lim sup

nK Q

J_l(u_n)40 . Up to a subsequence we have that there exists a functionu in B_r4]VuV_H1

0(V)Gr( such thatun weakly converges to u in H0¹(V). By lower semicontinuity we get thatJ_l(u)G0 , so that (2.2) implies thatu40 . As a consequence, u_n tends to zero strongly in L²(V), hence from

a 2

V

N˜unN²dxG1 2

V

A(x)˜unQ˜undxGJ_l(un)1l 2

V

NunN², we conclude, since lim sup

nK Q

J_l(u_n)40 , thatu_nstrongly converges to zero inH0¹(V), which is not possible sinceVunV_H¹

0(V)4rfor everyninN. Thus (2.3) is proved.

The proof that the functionalJ_lsatisfies the Palais-Smale condition is standard and the details are left to the reader. Let us just recall that the main tool is, as usual, the proof that a Palais-Smale sequenceu_nis bound- ed, in order to apply the compactness properties of the equation. To ob- tain the boundedness of the sequenceun, it is enough to argue by contra- diction, assuming that Vu_nV_H1

0(V) goes to infinity and reasoning on the normalized sequence ^un

Vu_nV_H1 0 (V)

, which can be proved to strongly converge to zero (actually, it will converge to a nonnegative eigenfunction corre- sponding to the eigenvector l, since lDl1 it must converge to zero).

We define now the class of paths to which apply the usual deforma- tion lemma. First, let

C_l»4

{

V

w²2

V

A(x)˜wQ˜w dxD0

}

which is clearly a nonempty subset of H₀¹(V) since W1C_l for every lDl1. Note that for anywC_lwe have that there exists only one value t_wD0 such that J_l(t_ww)40 .

Then we define the set of paths:

X_l»4 ]gC( [ 0 , 1 ],H₀¹(V) ) :)wC_l: g( 0 )40 , g( 1 )4t_ww(. We can then prove that

c»4 inf

gX_l sup

[ 0 , 1 ]

J_l(g(t) )

is a critical value forJ_l. Indeed, let r and s be given from (2.3). Since J_l(t_ww)40 it follows from (2.2) thatVt_wwV_H1

0(V)Dr, hence there exists a value t such that Vg(t)V_H1

0(V)4r and thus cFs. Now, if the set Kc4 4 ]vH0¹(V) :J_l(v)4c,J_l8(v)40(were empty, it would exist a contin- uous deformation h: [ 0 , 1 ]3H₀¹(V)KH₀¹(V) and a value eD0 such that h(t,v)4v for any vH0¹(V) such that NJ_l(v)2cN D^s

2 and J_l(h( 1 , v) )Gc2efor everyvsuch thatJ_l(v)Gc1e. Ifgeis a path such that sup

[ 0 , 1 ]

J_l(g_e(t) )Gc1e, we have that the pathh( 1 , ge(t) ) belongs to X_l and sup

[ 0 , 1 ]

J_l(h( 1 ,g_e(t) ) )Gc2e, contradicting the definition of c. Therefore c is a critical value and there exists a critical point u_l in Kc. Clearly, u_l is a weak solution of (1.1). Moreover, note that the path g(t)4tt_ww belongs to X_l for any wC_l, hence we get:

J_l(u_l)4cGsup

[ 0 , 1 ]

J_l(tt_ww) . A straightforward calculation implies (2.1). r

PROPOSITION 2.2. Let u_l be the solution of (1.1) found in Proposi- tion 2.1. Then we have:

lim

lK 1Q

Vu_lV_H1 0(V)40 . (2.6)

PROOF. We denote, in what follows, by C all possibly different con- stants which do not depend on l. First of all, since J_l8(u_l)40 , we have

g

h

V

u_l^u11dx4J_l(u_l)2 1

2aJ_l8(u_l),u_lb4J_l(u_l) , so that using (2.1) we obtain the estimate:

(2.7)

V

u_l^u11dxG

g ^s

V NwN^u11dx

h

g

^s

V

w²2

s

V

A(x)˜wQ˜w dx

h

V

u_l^u11dxG C l

u11 12u

, (lDl1. (2.8)

Moreover, for every l8 El we have that

V

A(x)˜u_l8Q˜u_l8dx2l8

V

u_l8²dx4 2

V

u_l8^u11dx,

hence

V

A(x)˜u_l8Q˜u_l8dx2l

V

u_l²₈dx4(l8 2l)

V

u_l²₈dx2

V

u_l^u₈¹¹dxE0 .

Thus u_l8C_l for every l8 El, so that (2.7) implies:

V

u_l^u11dxG

g ^s

VNu_l8N^u11dx

h

g

^s

V

ul8

22

s

V

A(x)˜ul8Q˜ul8dx

h

,

which yields, using the equation solved by u_l8,

V

u_l^u11dxG

g ^s

V Nu_l8N^u11dx

h

g

s

V

u_l8²1

s

V Nu_l8N^u11dx

h

(2.9)

Our main task now is proving that the sequence ]u_l( is bounded in H0¹(V). To this purpose, we argue by contradiction. Assume then that Vu_lV_H1

0(V)is not bounded. Then we can construct a subsequencelh, such that:

(2.10) lh2lh21F1 , Vu_lhV_H1

0(V)FVu_lh21V_H1

0(V), lim

hKQ

Vu_lhV_H1

0(V)41Q.

From (2.9) written foru_lh andu_lh21 we obtain (note that lh2lh21F1):

V

u_l^u_h¹¹dxG

g ^s

VNu_lh21N^u11dx

h

g ^s

V

u_l²_h211

s

VNu_lh21N^u11dx

h

.

In order to simplify our notations, we denote henceforth u_lh by u_l and

u_lh21 by u_l21, so that we rewrite the previous inequality as

V

u_l^u11dxG

g ^s

VNu_l21N^u11dx

h

g ^s

V

u_l22 1

1

s

VNu_l21N^u11dx

h

. (2.11)

Let us set v_l4_Vu ^ul

lV_H1 0 (V)

. Then we have:

Vv_lV_H1

0(V)41 , Lv_l1 v_l^u

Vu_lV_H1 0(V) 12u 4lv_l. (2.12)

Moreover, using (2.11) and since Vu_lV_H¹

0(V)FVu_l21V_H¹

0(V) we obtain:

V

v_l^u11dxG 1 Vu_lV

H₀¹(V) u11

g ^s

VNu_l21N^u11dx

h

g ^s

V

u_l21² 1

s

V Nu_l21N^u11dx

h

G 1 Vu_l21V_H1

0(V) u11

g ^s

V Nu_l21N^u11dx

h

g ^s

V

u_l²₂₁1

s

V Nu_l21N^u11dx

h

G

s

VNv_l21N^u11dx

u

_s

^s

VNu_l21N^u11dx

v

which yields:

V

v_l^u11dxG

s

VNv_l21N^u11dx 11

u _s

^s

V Nu_l21N^u11dx

v

(2.13)

We claim now that there exists a constant eD0 such that:

lim inf

lK Q

s

V

u_l²dx

s

VNu_lN^u11dxFeD0 . (2.14)

Indeed, assume that (2.14) is not true: then we get that, for a subse- quence (not relabeled)

lK Qlim

s

V

u_l²dx

s

V Nu_lN^u11dx 40 , so that, using (2.8),

V

v_l²dx4 1 Vu_lV_H1

0(V)

2

V

u_l²dxG C Vu_lV_H1

0(V)

2 l^u1

1 12u

. But, using (2.10) (recall that here u_l4u_lh), this implies that

llimK Q

l

V

v_l²dx40 , which yields, from the equation solved by v_l (2.12):

llimK Q

V

N˜v_lN²dx40 .

This can not hold because of (2.12), hence we deduce that (2.14) must be true. Now, (2.13) and (2.14) together imply that there existslDl1and a constant dD0 such that:

V

v_l^u11dxG 1 11d

V

v_l2u111dx (lDl. Therefore, we also have that there exists a constant CD0:

V

v_l^u11dxGC

g

h

(2.15)

Now, (2.12) implies, on account of standard regularity results (see [12]), that, for a value pD^N₂ :

Vv_lV_LQ(V)GCVlv_lV_Lp(V), which yields, thanks to (2.15) (take also pDu11):

1GCl

V

Vv_lV_LQ(V)

V

V

Vv_lV_LQ(V)

V

p

GCl

g

h

L^Q(V) (u11)/p .

Then we have

Vv_lV_LQ(V)GCl

p

u11

g

h

so that Vv_lV_LQ(V) converges to zero as l tends to infinity. Then l

V

v_l²dxGCl

V

v_l^u11dxGCl

g

h

Since (2.12) implies (recall (1.2)) a

V

N˜v_lN²dxG

V

A(x)˜v_lQ˜v_ldxGl

V

v_l²dx,

we conclude thatv_l strongly converges to zero inH0¹(V), which contra- dicts the fact thatVv_lV_H1

0(V)41 . The contradiction proves that no subse- quence u_lh satisfying (2.10) can exist, and then that Vu_lV_H1

0(V) is bounded.

It is now easy to repeat the same arguments to prove that u_l con- verges to zero in H0¹(V). Indeed, if this were not true it would exist a value eD0 and a subsequence u_lh such that:

l_h2l_h21F1 , Vu_lhV_H1

0(V)Fe (hD0 . (2.16)

In this case (2.9) still implies that

V

u_l^u1_h ¹dxG

g ^s

VNu_lh21N^u11dx

h

g ^s

V

u_lh

21

2 1

s

V Nu_lh

21N^u11dx

h

,

and then

V

u_l^u_h¹¹dxG

s

V Nu_lh21N^u11dx 11

u _s

^s

VNu_lh

21N^u11dx

v

(2.17)

This is the same inequality as (2.13) previously obtained onv_l. Thus, the same arguments can now be applied; indeed, we have the alternative

that either there exists a constant e such that

lim inf

hK Q

s

V

u_l²_hdx

s

V Nu_lhN^u11dx FeD0 (hD0 ,

or there exists a further subsequence (not relabeled) such that

lim

hK Q

s

V

u_l²_h

s

V Nu_lhN^u11dx40 .

In both cases, the same arguments used before for v_lallow to conclude that a subsequence of u_lh strongly converges to zero in H₀¹(V), which contradicts (2.16). This concludes the proof thatu_lstrongly converges to zero in H0¹(V) r

REMARK 2.3. Note that, at least if ^N22

N12EuE1 , we can also argue from the proof of Theorem 1.1 that

llimK Q

Vu_lV_LQ(V)40 . In fact, recall that we have the estimate:

V

u_l^u11dxG C l

u11 12u

(lDl₁.

If ^N22

N12Eu, then ^u11

12uD^N

2, hence we get from the equation solved byu_l, Vu_lV_LQ(V)GClVu_lV

L u11 12u(V). Hence, using that ^u11

12uDu11 ,

1GCl

V

Vu_lV_LQ(V)

V

L u11

12u(V)GCl

V

Vu_lV_LQ(V)

V

L^u11(V) 12u

GC 1 Vu_lV_L12uQ(V) . We first deduce from this inequality thatVu_lV_LQ(V) is bounded, then we obtain, for a value p such that max

m

n

lim

lK Ql^p

V

u_l^pdxGCl^p

V

u_l^u11dxGC 1 l

u11 12u2p

,

so thatlu_l strongly converges to zero in L^p(V) with pD^N₂ . Classical regularity results (see [12]) then imply that u_l converges to zero in L^Q(V). In particular, this proves the convergence of u_l in L^Q(V) for any value of u at least in dimension NG2 .

Aknowledgments. This work originated from a discussion I had with Lucio Boccardo in Trieste at the International School held at the ICTP in April 1996, so I am grateful to the organizer A. Ambrosetti for the pos- sibility to attend that meeting. I also thank J. Hernandez who addressed to me references [8] and [7], and B. Pellacci and E. Montefusco for stim- ulating my interest about this problem with friendly discussions.

R E F E R E N C E S

[1] A. AMBROSETTI- P. HESS,Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73 (1980), pp. 411-422.

[2] A. AMBROSETTI- P. H. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Functional Analysis 14(1973), pp. 349-381.

[3] D. ARCOYA- J. CARMONA- B. PELLACCI, Bifurcation for some quasilinear operators, Proc. Royal Soc. of Edinburgh, Sect. A, 131, no. 4 (2001), pp. 733-765.

[4] D. ARCOYA- J. L. GAMEZ- L. ORSINA- I. PERAL,Local existence results for sub-super-critical elliptic problems, Comm. Appl. Anal.,5, no. 4 (2001), pp.

557-569.

[5] H. BREZIS - R. E. L. TURNER, On a class of superlinear elliptic problems, Comm. Partial Diff. Eq., 2(1977), pp. 601-614.

[6] J-P. DIAS- J. HERNANDEZ,Bifurcation a l’infini et alternative de Fredholm pour certains problemes unilateraux, J. Math. Pures et appl., 55 (1976), pp. 189-206.

[7] J. I. DIAZ- J. HERNANDEZ,Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris, Serie I, 329 (1999), pp. 587-592.

[8] J. HERNANDEZ, PHD Thesis, Universidad Autonoma de Madrid (1977).

[9] M. A. KRASNOSELSKII,Positive Solutions of Operator Equations, Noordhoff, Groningen (1964).

[10] E. MONTEFUSCO, Sublinear elliptic eigenvalue problems in Rⁿ, Atti Sem.

Mat. Fis. Univ. Modena, 47, no. 2 (1999), pp. 317-326.

[11] P. H. RABINOWITZ, On bifurcation from infinity, J. Diff. Eq., 14 (1973), pp. 462-475.

[12] G. STAMPACCHIA,Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15, no. 1 (1965), pp. 189-258.