Non existence of bounded-energy solutions for some semilinear elliptic equations with a large parameter

Non Existence of Bounded-Energy Solutions for Some Semilinear Elliptic Equations

with a Large Parameter.

DANIELE CASTORINA(*) - GIANNI MANCINI(**)

ABSTRACT- Following previous work of Druet-Hebey-Vaugon, we prove that the energy of positive solutions of the Dirichlet problem2Du1lu4u

N12 N22 inV, u40 in¯Vtends to infinity aslK 1Q. We also prove, extending and simpli- fying recent results, that bounded energy solutions to a mixed B.V.P. have at least one blow-up point on the Neumann component as lK 1Q.

1. Introduction.

LetVbe a smooth bounded domain inR^N,NF3,lF0. We consider the problem

(D)_l

. / ´

2Du1lu4u^p uD0

u40

xV xV x¯V

(*) Indirizzo dell’A.: Università di Roma La Sapienza, Dipartimento di Ma- tematica, Piazzale Aldo Moro 2, Roma 00100 Italy.

E-mail: castorinHmat.uniroma1.it

(**) Indirizzo dell’A.: Dipartimento di Matematica, Università di Roma Tre, Via S. Leonardo Murialdo 1, Roma, Italy.

E-mail: manciniHmatrm3.mat.uniroma3.it

The authors’ research is supported by M.U.R.S.T. under the national project Variational methods and nonlinear differential equations.

where p4^N_N¹²

22. In case V is starshaped (D)_l has no solutions: an ob- struction to existence is given by the well known Pohozaev identity.

However, (D)_lmight have solutions for any l: this is the case ifV is an annulus.

In [6], Druet-Hebey-Vaughon, investigating the role of Pohozaev type identities in a Riemannian context, discovered that such identities still provide some kind of obstruction to existence: if u_j solve (D)_lj on some compact (conformally flat) Riemannian manifoldM and ljK 1Q, then

s

M

u_j^p11K 1Q (see [7] for extensions to fourth order elliptic PDE’s and [10] for more questions). In other words, there are no positive sol- utions with energy below some given bound, if l is too large.

This result does not carry over to manifolds with boundary under general boundary conditions (e.g. homogeneous Neumann boundary conditions, see [1], [2] for this non trivial fact.)

The main purpose of this note is to show that the result quoted above is indeed true for the homogeneous Dirichlet B.V.P. (D_l); see Theorem 1 below.

Our approach is as in [6]: to show that a sequence of solutions cannot blow-up at a finite number of points (as it should be assuming a bound on the energy). The obstruction found in [6] is given by localL²2estimates.

In turn, these estimates are based on inequalities obtained localizing the standard Pohozaev identity on balls centered at blow-up points (see (13) below). Now, differently from [6], where there is no boundary, we have to take into account possible blow-up at boundary points. Since Pohozaev type inequalities on balls centered at boundary points do not hold, in general, the main issue here is to get local L² estimates at boundary points.

Notice that a more or less straightforward application of arguments from [6] would only lead to the statement: bounded energy solutions have to blow up at least at one boundary point, which is the (quite inter- esting in itself) correct statement for the Neumann problem (see Th. 2 below) but which is not the result we are looking for the Dirichlet problem.

Since it does not seem easy to rule out blow up at boundary points, we stick to the approach in [6], but we have to deal with the new difficulty coming from possible blow up at boundary points.

To handle this difficulty, we will establish Pohozaev-type inequalities suitably localized at interior points(see Lemma 5 below), which, used

in a clever way, will allow to obtainestimates up to boundary points(see Lemma 6, which also provides a simple adaptation and a self contained exposition of the main arguments in [6]).

This is the main technical contribution in this paper, and we believe that such estimates might be of interest by themselves. Our first result is THEOREM 1. Let NF3. Let ujbe solutions of (D)_lj,withljK 1Q. Then

s

V

u_j^p11K 1Q.

NOTE. Such a result is quite obvious ifpis subcritical, i.e.pE^N_N¹²

22. In this case, it holds true, in contrast with the critical case, also for the homogeneous Neumann B.V.P. (see [11] for an explicite lower bound on the energy of ground state solutions).

A second question we address in this note is concerned with the mixed boundary value problem

(M)_l

. ` / `

´

2Du1lu4u

N12 N22

uD0 u40

¯u

¯n 40

xV xV xG0

xG₁

Here, ¯V4G0NG1 with Gi disjoint components.

As for the Neumann problem, (M)_l possesses low energy solutions for anylpositive, at least if the mean curvature ofG1is somewhere posi- tive, and hence non existence of bounded energy solutions forl large is false, in general. Indeed, several existence results for (bouded energy) solutions blowing up at (one or several) boundary points are known (see [12] for an extensive bibliography). Nothing is known, to our best knowl- edge, about existence of solutions blowing up both at interior and at boundary points. On the other hand, the extreme case of purely interior blow-up has been widely investigated:

Are there solutions which blow up only at interior points?

(Q)

Let us review the known results, all of them actually concerning the ho- mogeneous Neumann problem (i.e. G04¯).

A first, negative, answer has been given in [4] for NF5:

(M)_l has no solutions of the form

u_l4w_l1

!

j41 k

U_m^j_l,yl^j, w_lK0 in H¹(V)

withm^j_lK 1Q,y_l^jKy^jVaslgoes to infinity andyⁱcy^j(icj. Here, U(x)4^{[N(N22 ) ]}

N22 4

( 11 NxN²)

N22 2

and U_m,y4m

N22

2 U(m(x2y) ).

Now, bounded energy solutions u_l (with l going to infinity) are known to be of the form given above, apart from the property yⁱcy^j (icj, which has to be regarded as a9no multiple concentration at a sin- gle point9assumption. Under the even more restrictive assumptionk4 41, a corresponding non existence result has been proven in [8] in any di- mension NF3.

At our best knowledge, the only result fully answering (Q), and due to Rey [12], is limited to the dimension N43. According to Rey, 9 the main difficulty is to eliminate the possibility of multiple interior peaks9, and he accomplishes this task through a very careful expansion of sol- utions blowing up at interior points: this is the basic tool to obtain a negative answer to (Q) in dimension N43. It is henceforth remarkable that, as a byproduct of ourL²estimates, we can bypass this difficulty and easily prove

THEOREM 2. Let NF3. Let u_jbe solutions of(M)_lj, withljK 1Q. If sup

j

s

V

ujp11

E Q, then uj has at least one concentration point which lies on the Neumann component G1.

Actually, we expect that solutions for the mixed problem, with a uni- form bound on the energy, should not even exist, forllarge, if the mean curvature of the Neumann component is strictly negative. This is fairly obvious in the case of one peak solutions, which, by the above, should blow-up at one boundary point. However, in such a point the mean curva- ture should be non negative (see [3], or [13]).

Acknowledgment. We wish to thank E. Hebey for bringing to our at- tention the results in [6].

2. L² global concentration.

To be self contained, we review in this section some essentially well known facts. Let u_n be solutions of (D)_ln (but also homogeneous Neu- mann, or mixed, boundary conditions might be allowed, with minor changes, here). Multiplying the equation byu_n, integrating by parts and using Sobolev inequality, we see that

V

N˜u_nN²FS

N

2,

V

u_n^p11FS

N

(1) 2

where S denotes the best Sobolev constant. We start recalling concen- tration properties of u_n, assuming u_n0 in H₀¹.

LEMMA 3. Let unbe solutions of(D)_ln. Assume un0in H0¹. Then there is a finite set C%V such that u_nK0 in H_loc¹ (V0C) and in C_loc⁰ (V0C).

PROOF. Let C:4 ]xV: lim sup

nK Q

s

B_r(x)OVN˜unN²D0 , (rD0(. Because of (1) and compactness of V, C cannot be empty. We claim that

(xC, (rD0 , lim sup

nK Q

B_r(x)OV

unp11

FS

N 2 . (2)

To prove the claim, let WC₀^Q(B_2r(x) ), Wf1 on B_r(x), 0GWG1.

Notice that 2

s

V

u_nW²Du_n4

s

VN˜u_nN²W²1ⁱ(1)4

s

VN˜u_nWN²1ⁱ(1) because u_n0.

From the equation, and using Holder and Sobolev inequalities, we get

_N˜unWN²_{1i( 1 )G}

_un^N242_(unW)²_G ¹

S

u

B_2r(x)

u_n

2N

N22

v

^N2

^N˜unWN2

(3)

For xC,

s

_N˜u_nWN² is bounded away from zero along some subse- quence, and then (2) follows by (3). Also, ifx₁,R,x_kC, choosingB_r(x_j) disjoint balls, and eventually passing to a subsequence, we get by (2)

kS

N

2G

!

j

Br(xj)OV

unp11

Gsup

n

V

unp11

E 1Q

ThusCis finite. Also, from the very definition ofC, it follows that˜u_nK K0 in L_loc² (V0C), and, by Sobolev inequality, u_nK0 in L_loc^p11(V0C) as well.

Finally,Cloc⁰ (V0C) convergence will follow by standard elliptic theory once one has proved thatu_nK0 inL_loc^q (V0C)(q. In turn, this fact readily follows iterating the (Moser type) scheme

(4) qF2 ,

B_2r(x)

u^p11G

g

^Sq

h

^N2¨

u

B_r(x)

u^sq

v

^1sG_Sr⁸²

B_2r(x)

u^q, s:4p11 2

To prove (4), we can procede as for (3), choosing now as test function W²u^q21, V˜WV_QG²_r. We now obtain

(5)

_˜u˜(u^q21_W²_)G

_u^p21_(Wu^q2₎²_G ¹

S

u

B_2r(x)

u^p11

v

^N2

^N˜Wuq2N2

On the other hand

(6)

_˜u˜(u^q21_W²₎₄

_{(q21 )W}²_u^q22_N˜uN²_12u^q21_W˜u˜W

(7) 2

q

_N˜(Wu

q

2)N²4

^q

2W²u^q22N˜uN²12u^q21W˜u˜W12

qN˜WN²u^q

Substracting (6) from (7) and then using (5), we obtain

(8) 2

q

_N˜(Wu^q2_)N²_G

_˜u˜(u^q21_W²₎₁ ²

q

_N˜WN²_u^q_G

G 1 S

u

B2r(x)

u^p11

v

^N2

^N˜(Wuq2)N21qr⁸²

B2r(x)

u^q

Hence, using the assumption

s

B2r(x)

u^p11G

g

^Sq

h

^N2 and Sobolev inequality,

we get (4). Now, iterating (4) withxV0C_d, 2rEd,u4u_nand using el- liptic estimates ([9], page 194), we obtain

sup

V0Cd

u_nG cN

d

N

2

u

V0Cd/2

u_n²

v

^12K0

(9)

Points inCare called9geometric9concentration points and Cis the concentration set. A crucial observation is that L² norm concentrates around C (see [6]):

LEMMA 4. Let u_n be solutions of (D)_ln. Assume u_n0 , and let C:4 ]x₁,R,x_m( be its concentration set. Let C_d»4_j

0

₄₁^{m Bd(xj), Bd(xj)}

disjoint closed balls. Then, for n large,

V0Cd

u_n²G 16 d²ln

V

u_n²

In particular, if l_nK 1Q, then

s

V0Cd

u_n²

s

C_d

u_n² K0 . (10)

PROOF. Let WC^Q(R^N), 0GWG1 , Wf0 in Cd 2

, Wf1 in R^N0C_d, V˜WV

QE⁴_d. Multiplying the equation by u_nW² and integrating, we get

_N˜unN²_W²₁₂

_{unW˜un˜W1ln}

_un²_W²_{4i( 1 )}

_un²_W²

(11)

because u_n^p21(x)K0 uniformly in V0C_d/2 by Lemma. After setting g_n²»4s_N˜u_nN²W²

s

V0Cd/2

un2 we get from (11) the desired inequality

ln

s

V0Cd

u_n²

s

V0Cd/2

u_n² Gⁱ( 1 )12V˜WV

Qg_n2g²_nGⁱ( 1 )1V˜WV

Q

2 G 16

d²

3. Pohozaev identity and «reverse» L² concentration.

In this Section, after briefly recalling the Pohozaev identity, we first derive suitably localized Pohozaev inequalities which will allow to get uniform L²-local estimates up to boundary points. These up-to-the- boundary estimates are the main novelty with respect to [6]: combined with the L² global concentration reviwed in Section 2, they will readily imply Theorems 1 and 2.

Given any vC²(V), x0R^N, an elementary computation (see [14]) gives

ax2x₀,˜vbDv2 N22

2 N˜vN²4div

g

^{ax2x0,˜vb˜v2N˜v}2^N2(x2x₀)

h

If in additionvC₀¹(V), so that ˜v(x)4_¯n^¯vn(x) for any x¯V, where n(x) is the exterior unit normal at x¯V, an integration by parts yields

(12)

V

ax2x₀,˜vbDv1 N22

2 vDv4 1 2

¯V

ax2x₀,n(x)bN˜vN²ds

If furthermore 2Dv4g(x,v), g(x,t)fb(x)NtN^p21t2la(x)t, through another integration by parts we obtain

1

N

V

o

^{x2x0,vp11 p˜b}₁^12l2v2˜a

p

^2Nl

V

av²41

2

¯V

ax2x₀,nbN˜vN²ds

A straightforward and well known consequence of this identity is thatv has to be identically zero ifVis starshaped with respect to some x₀and ax2x₀,˜bbG0Gax2x₀,˜ab. However, this conclusion is false, in gen- eral. The idea, following [6], would be to localize (12) to obtain, for every givenx₀V and some d4dx₀D0 and for allWC₀^Q(B_4d(x₀) ), inequali- ties of the form

B4d(x₀)OV

ax2x₀,˜(Wv)bD(Wv)1 N22

2 WvD(Wv)F0 , (13)

However, while this can be done at interior points (e.g. with 4d4 4d(x₀,¯V)), (13) is in general false,(dsmall, ifx₀¯V. So, we have to lo- calize (12) at interior points but in a carefull way, to cover, in some sense,

also boundary points. Our basic observation is that (13) holds true forx₀ as long asd(x₀,¯V)Fdifdis sufficiently small, and this will be enough to get control up to the boundary. The first statement is the content of the following simple but crucial lemma.

LEMMA 5. There is d4d(¯V) such that, if 0EdGd, vC²(V)O OC¹(V),vf0 on ¯V and x0V with d(x₀,¯V)Fd, then(3.13) holds true.

PROOF. Letdbe such that, for everyz¯V, anyx¯VOB_8d(z) can be uniquely written in the form

(i) x4z1h1g^z(h)n(z), ah,n(z)b40, with Ng^z(h)N Gc(¯V)NhN², for some smoothg^z, withg^z( 0 )40,˜g^z( 0 )40 ,NhN G8dand some con- stant c only depending on ¯V. We will also require

(ii) Nn(z8)2n(z9)N G¹

8 for any z8,z9¯V with Nz8 2z9N G8d (iii) dE_128c¹ , c4c(¯V).

Let 0EdGd. We are going to apply (12) with V replaced by B_4d(x₀)OVandvbyWv,WC₀^Q(B_4d(x₀) ). Ifd(x₀,¯V)F4d, then equal- ity holds in (13), so, let us assume 0EdGd(x₀,¯V)G4d. We can write x₀4z2tn(z) for some z¯V and dGtG4d. For xB_4d(x₀) we have Nx2zN G8d and hence, (i) holds:x4z1h1g^z(h)n(z),NhN G8d. Now, using (ii)-(iii), we see that

ax2x₀,n(x)b4ax2x₀,n(x)2n(z)b1

1az1h1g^z(h)n(z)2(z2tn(z) ), n(z)bF d

2 264cd²F0 . Hence the r.h.s. in (12) (withV replaced byVOB_4d(x₀) andvby Wv) is nonnegative and the Lemma is proved. r

In the Lemma below we will show how Pohozaev inequalities lead to

«reverse L²-concentration» of solutions at any blow up point. We will adapt arguments from [6], where, however, it is made a crucial use of the validity of (13) at any point, which is not the case here. Still, a clever use of Lemma 5, i.e. of (13) limited to points which are d-away from the boundary, will unable to get the desired estimates up to boundary points.

LEMMA 6. There is a constant c4c_N, only depending on N, such that if u_l is a solution of (D)_l and 0EdGd(¯V)G1 , then

l

B_d(x)

u_l²G c d³

B4d(x)0B_d(x)

(u_l²1u_l^p11) , (xV (14)

PROOF. We are going to apply (13) withx04xifd(x,¯V)Fd, while, if d(x,¯V)Ed, x4z2tn(z) for some z¯V and 0GtEd, we will choosex₀4z2dn(z). Let, without loss of generality,x₀40. By Lemma 5 we have

B4dOV

ax,˜(Wu_l)bD(Wu_l)1N22

2 Wu_lD(Wu_l)F0 i.e. (dropping subscript B_4dOV)

(15)

k

^{ax,˜(Wul)b1N2}2 ²Wu_l

l

^{WDul12Wax,˜ulba˜W,˜ulb1R(l)F0 ,}

where R(l)»4R1(l)1R2(l), R1,R2 given by

R₁(l)»4

2u_lax,˜Wba˜W,˜u_lb1ax,˜(Wu_l)bu_lDW

R2(l)»4 N22

2

_{Wul[ulDW12a˜ul,˜Wb]}

In what follows we properly adapt and simplify arguments from [6].

Taking W radially symmetric and radially decreasing, we have

˜W4

o

^˜W, N^xxN

p

N^xxN

, with a˜W(x),xbG0. In particular, a˜W(x),

˜u_l(x)bax,˜u_l(x)b4

o

^˜W(x), Nx^xN²

p

^{ax,˜ul(x)b2G}0 and hence (15) yields

R(l)1

_{Wax,˜(Wul)bDul1} ^N22

2

_W²_ulDulF0

(16)

Now, let us write g(t)»4lt2 NtN^p21t, G(t)»4^l

2u²2^u

p11

p11. We first

rewrite, integrating by parts, the second term in (16) as follows:

(17)

_{Wax,˜(Wul)bDul4}

_{Wax,˜Wbulg(ul)1j}

!

41

N

_W²_xjg(ul)^¯ul

¯x_j 4

4

_Wax,˜(W)b[u_lg(u_l)22G(u_l) ]2N

_W²G(u_l)4

4 2 2

N

Wu_l^p112N

_W²G(u_l) because 2G(u)2ug(u)4 2_N² u^p11. Since NG(u_l)2^N2₂ ²u_lg(u_l)4 2 2lu_l², (16) gives

R(l)2 2

N

_ax,˜WbWu_l^p11F 2l

_ul²

(18)

Let us now transform, integrating by parts,R(l) as an integral against u_l²dx.

R₁(l)4

!

j41

N

y

^{ax,˜Wb ¯W}_¯x

j

1 1

2x_jWDW

z

^¯_¯x^2ul2

j2 1ax,˜Wbu_l²DW4 42

u_l²

!

j41

N

y

^ax,˜Wb¯_¯x^2W

j2 1 ¯

¯xj

ax,˜Wb¯W

¯xj

11

2WDW11 2x_j ¯

¯xj

(WDW)

z

¹

1u_l²ax,˜WbDW42

_ul²

k

^{a˜W,˜(ax,˜Wb)b1N}2 WDW11

2ax,˜(WDW)b

l

^.

R₂(l)4N22

2

k

^{ul2WDW2 1}2

_ul²_DW²

l

^{4 2N2}2 ²

_ul²_N˜WN²

and thus

R(l)42

_ul²

k

^{a˜W,˜ax,˜Wbb1N}2 WDW11

2ax,˜(WDW)b1N22

2 N˜WN²

l

^.

Now, assuming Wf1 on B_2d, Wf0 outside B_3d, we obtain R(l)G c

d³

B3d0B_2d

u_l²

for some c4c(N), and hence, by (18), l

B_2d

u_l²G c d³

B3d0B2d

u_l²1u_l^p11

Since B_d(x)%B2d and B3d0B2d%B4d(x)0B_d(x), the Lemma is proved.

PROOF OF THEOREM 1. Lemma 4 and 6 provide the tools for the proof, which, at this stage, goes like in [6]. We briefly sketch the argument.

We have to prove that ifu_nare solutions of (D)_lnwith sup

n

s

u_n^p11E1Q, then sup

n lnE 1Q. This is clear ifu_nhas a non zero weak limit, so we can assume u_n0.

According to Lemma 3, there are x₁,R,x_kV such that u_nK0 in Cloc0 (R^N0C) with Cf]x1,R,xk(. Let 0EdEmin]d(¯V), ¹

8d(xi,xj), icj(, so that (3.14) in Lemma 6 holds for all x_jC:

ln

B_d(x_j)

u_n²G 2c_N

d³

B4d(xj)0Bd(xj)

u_n²(x_jC

Since the balls B_4d(x_j) are taken disjoint, we get ln

C_d

u_n²G2c d³

V0C_d

u_n² (19)

which, jointly with (10), implies l_n remains bounded.

PROOF OF THEOREM 2. The proof is by contradiction: we assume that there is a sequence u_l of bounded energy solutions with lK 1Q, and no blow-up points onG₁. Hence, for this sequence, (14) holds true at any blow-up point. In addition, Lemma 4 holds true for the problem (M)_l. In fact, arguments in the proof of Lemma 4 are not affected by the presence of Neumann boundary conditions, and the concentration be- haviour assumed therein follows by a simple adjustment in the proof of Lemma 3: in (2) the termS

N

2 becomes ^S

N 2

2 as it follows by replacing in (3), the Sobolev inequality with the Cherrier inequality (see [5]) for any

dD0 there exists C(d)D0 such that for any uH¹(V)

u

₂^S_N^{2 2d}

v u

V

NuN^p11

v

^p121G

V

N˜uN²1C(d)

V

u²

and hence Lemma 3 holds for (M)_l as well, thanks to this inequality, to the fact that

s

V

un2

K0 (so that C(d)NunN²²4ⁱ( 1 )) and since, because of the null boundary conditions, there are no boundary contributions in the estimates. We use Cherrier inequality also in the Moser-type scheme to obtain Cloc0 (V0C) convergence.

Since (10) and (14) are satisfied, the same argument as in the proof of Theorem 1 applies, giving a contradiction.

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Manoscritto pervenuto in redazione il 20 dicembre 2002.