SINGULAR ELLIPTIC EQUATION INVOLVING THE GJMS OPERATOR ON COMPACT RIEMANNIAN MANIFOLD

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SINGULAR ELLIPTIC EQUATION INVOLVING THE GJMS OPERATOR ON COMPACT RIEMANNIAN

MANIFOLD

Mohammed Benalili, Ali Zouaoui

To cite this version:

Mohammed Benalili, Ali Zouaoui. SINGULAR ELLIPTIC EQUATION INVOLVING THE GJMS OPERATOR ON COMPACT RIEMANNIAN MANIFOLD. 2016. �hal-01312595�

OPERATOR ON COMPACT RIEMANNIAN MANIFOLD

MOHAMMED BENALILI AND ALI ZOUAOUI

Abstract. In this paper we consider a singular elliptic equation involv- ing the GJMS (Graham-Jenne-Mason-Sparling) operator of orderk on n-dimensional compact Riemannian manifold with 2k < n. Mutiplicity and nonexistence results are established.

Let (M, g) be an n-dimensional Riemannian manifold. The k-th GJMS operator (Graham-Jenne-Mason-Sparling, see ([3])Pgis a differential opera- tor defined for any integerkif the dimension n is odd, and 2k≤notherwise.

In the following, we will consider the case 2k≤n. Pg is of the form P_g= ∆^k+lot

where ∆ = −div_g(∇) is the Laplacian-Beltrami operator and lot denotes the lower terms. One of the fundamental property of Pg is its behavior with respect to conformal change of metrics: for ϕ ∈C^∞(M) , ϕ >0 and g=ϕ^n−2k4 g a conformal metric to g,

ϕ^n+2kn−2kP

egu=P_g(ϕu).

P_g is self-adjoint with respect to the L²-scalar product. To P_g is associ- ated a conformal invariant scalar function denoted Qg and is called the Q-curvature. For k = 1, the GJMS operator is ( up to a constant ) the conformal Laplacian and the corresponding Q-curvature function is simply the scalar curvature. Fork= 2, the GJMS operator is the Paneitz operator introduced in ([8]). For 2k < n, the Q-curvature is Qg = _n−2k² Pg(1). Many works was devoted theQ-curvature equation in the last two decades (see [1], [2], [5], [12]). Many authors investigated the interactions of conformal meth- ods with mathematical physic which led them to study the Einstein-scalar fields Lichnerowiz equations (see [4], [6], [9], [10], [11],). These methods have been extended to scalar fields Einstein-Licherowicz type equation in- volving the Paneitz operator, (see [7]). In this work we analyze an Einstein- Lichnerowicz scalar field equation containing thek-th order GJMS operator on a Riemannian n-dimensional manifold with 2k < n. Our work is or- ganized as follows: in a first step we show the existence of a solution to equation (1.1) obtained by means of the mountain-pass theorem. In the

1991Mathematics Subject Classification. 58J99-83C05.

Key words and phrases. GJMS operator,Critical Sobolev Growth.

1

second step we prove, by the Ekeland lemma, the existence of a second so- lution to equation (1.1). In the last section we give a result of nonexistence of solution.

1. Existence of a first solution

Let (M, g) be an n-dimensional compact Riemannian manifold and k ∈ N^∗, with 2k < n. Consider the following equation

(1.1)

(

Pg(u) =B(x)u^2]−1+ ^A(x)

u^2]+1 + ^C(x)_up

u >0

where 2^]= _n−2k²ⁿ ,Pg is thek-th GJMS operator andp >1. In all the sequel of this paper we assume that the operator P_g is coercive which allows us ( see Proposition 2, [2]) to endow H_k²(M) with the following appropriated equivalent norm

kuk= v u u t Z

M

u.Pg(u)dvg.

So we deduce from the coercivity of P_g and the continuity of the inclusion H_k²(M)⊂L^2](M), the existence of a constant S >0 such that

(1.2) kuk₂]≤Skuk

where 2^]= _n−2k²ⁿ .

We assume also that the GJMS operator has a positive Green function.

Proposition 1. Suppose that the metric g is Einstein with positive scalar curvature of dimension n >2k, then the GJMS P_g admits a Green positive function.

Proof. Onn-dimensional Einstein manifold, the GJMS operator of order k (see [3]) is given by

Pg =

k

Y

l=1

(∆−c_lSc) wherecl= (n+2l−2)(n−2l)

4n(n−1) ,Scis the scalar curvature. If the scalar curvature is positive it is well known that the operator ∆−c_lSchas a positive Green function. Denote by Ll = ∆−clSc, l = 1, ..., k; by definition of the Green function of L_l we know that for allu∈C^∞(M),

(L_lu) (x) = Z

M

G_l+1((x, y) (L_l+1L_lu) (y)dvg(y). So

u(x) = Z

M

G_l(x, z) (L_lu) (z)dvg(z) + 1 V ol(M)

Z

M

u(x)dvg(x)

= Z

M

Z

M

G_l(x, z)G_l+1((z, y)dvg(z)

(L_l+1L_lu) (y)dvg(y)+ 1 V ol(M)

Z

M

u(x)dvg(x)

and letting

Gl,l+1(x, y) =Gl∗Gl+1(x, y) = Z

M

Gl(x, z)Gl+1((z, y)dvg(z).

And by induction, we get u(x) =

Z

M

G₁∗...∗G_k(x, y)P_g(y)dv_g(y).

ThusP_g admits a positive Green function.

In this section we establish the following theorem.

Theorem 1. Let(M, g)be a compact Riemannian manifold with dimension n > 2k and A > 0, B > 0, C > 0 are smooth functions on M. Suppose moreover that the operatorP_g is coercive and have a positive Green function.

If there exists a constant C(n, p, k)>0 depending only onn, p, ksuch that (1.3) kϕk^2]

2^] Z

M

A(x)

ϕ^2\ dv_g ≤C(n, p, k)

Smaxx∈MB(x) ²⁺²

] 2−2]

and

(1.4) kϕk^p−1 p−1

Z

M

C(x)

ϕ^p−1dvg ≤C(n, p, k)

Smaxx∈MB(x) ^p+1

2−2]

for some smooth function ϕ >0, then equation (1.1) admits a smooth solu- tion.

To prove the existence of solutions to (1.1), we consider the following -approximating equations

(1.5) Pg(u) =B(x) (u)^2]−1+ A(x)u (ε+u²)^2[+1

+ C(x)u (ε+u²)^p+12

where 2^[ = ²₂^],p > 1.Which gives us a sequence (u_ε)_ε of solutions to (1.5).

The solution of equation (1.1) is then obtained as the limiting of (uε)_ε. Following Hebey-Pacard-Pollak ([4]), to get rid of negative exponents, we consider the energy functional associated to (1.5) defined by, for anyε >0

I_ε(u) =I¹(u) +I_ε⁽²⁾(u) where I⁽¹⁾ :H_k²(M)→R is given by

I⁽¹⁾(u) = 1 2

Z

M

uP_g(u)dv_g− 1 2^]

Z

M

B(x) u⁺2^]

dv_g

and I_ε⁽²⁾:H_k²(M)−→Ris I_ε⁽²⁾(u) = 1

2^] Z

M

A(x)

ε+ (u⁺)²

2^[dv_g+ 1 p−1

Z

M

C(x)

ε+ (u⁺)^{2 p−1}

2

dv_g. It is easy to check the following inequality

(1.6) Φ (kuk)≤I⁽¹⁾(u)≤Ψ (kuk) with

Φ (t) = 1 2t²−

Smax

M B(x) t^2]

2^] and

Ψ (t) = 1 2t²+

Smax

M B(x) t^2]

2^].

The Φ (t) is increasing on [0, t0] and decreasing on ]t0,+∞[, where

(1.7) t0 =

Smaxx∈MB(x) ^2k−n_4k

and

(1.8) Φ (t0) = 1

2 − 1

2^] Smax

x∈MB(x) ^2k−n

2k

= k nt0. Lemma 1. Let θ >0 such that

a 2

²

2\

< θ² < 1

2 (n−k) =a^22\ where

a= 1

(2 (n−k))²

\ 2

and put

t1 =θt0. Then we have the following inequality (1.9) Ψ (t1)≤θ²2^]+ 2

2^]−2Φ (t0)< 1

2kΦ (t0). Proof. In fact

Ψ (t₁) = 1 2t²₁+

Smaxx∈MB(x) t²₁^]

2^]

=θ² 1

2t²₀+Smax

x∈MB(x)θ^2]−2t²₀^] 2^]

! . Since

(1.10) t²₀^]

Smaxx∈MB(x)

=t²₀

we get

Ψ (t₁) =θ² 1

2t²₀+θ^2]−2 2^] t²₀

!

=θ²t²₀

"

1

2+ θ^2]−2 2^]

#

≤θ²t²₀ 1

2 +n−2k 2n

≤θ²t²₀2n−2k 2n

≤(n−k)θ²t²₀ n. And since

2^]+ 2

2^]−2 = n−k

k and Φ (t0) =kt²₀ n we infer that

Ψ (t1)≤θ²2^]+ 2

2^]−2Φ (t0)< 1

2kΦ (t0) .

Now we check the Mountain-Pass lemma conditions for the functionalIε. Lemma 2. The functionalI satisfies the following conditions

i) There exit ρ >0 and t₀ >0 such that I(u)≥ρ for everyu∈H_k²(M) with kuk=t₀

ii)There exits isv∈H_k²(M) such that

kvk=t₂> t₀ and I(v)<0.

Proof. Letϕ∈C^∞(M),ϕ >0 on M and without loss of generality we can assume kϕk= 1. Put

(1.11) C(n, p, k) = (2k−1)θ^2]+p

4n ≤C1(n, k) = (2k−1)θ^2]

4n .

The inequality (1.3) becomes

(1.12) 1

2^] Z

M

A(x)

(t₁ϕ)^2]dvg ≤ 2k−1 4k Φ (t0) . Indeed, we have

(1.13) Φ(t_o) = k

nt²_o. and by (1.11), we get

1 2^\

Z

M

A(x)

(t₁ϕ)^2]dvg ≤ C1(n, k) t²_o^\θ^2\

S.max

M B(x) ²⁺²

] 2−2]

= 2k−1 4k Φ(t_o).

Analogously and by putting

C₂(n, p, k) = (2k−1)θ^p−1 4n we obtain

(1.14) 1

p−1 Z

M

C(x)

(t₁ϕ)^p−1dv_g ≤ 2k−1 4k Φ(t_o).

Let us prove know (i). By relations (1.6), (1.9), (1.12) and (1.14), we infer that

I(t₁ϕ)≤Ψ (kt₁ϕk)+1 2^]

Z

M

A(x)

ε+ (t1ϕ)²

2^[dv_g+ 1 p−1

Z

M

C(x)

ε+ (t₁ϕ)^2p−1₂ dv_g

(1.15)

≤Ψ (t₁)+1 2^]

Z

M

A(x)

ε+ (t1ϕ)²

2^[dv_g+ 1 p−1

Z

M

C(x)

ε+ (t1ϕ)^2p−1₂

dv_g ≤Φ (t₀). Again from (1.6), we deduce that

I(t₀ϕ)≥Φ (t₀) + 1 2^]

Z

M

A(x)

ε+ (t0ϕ)²

2^[dv_g+ 1 p−1

Z

M

C(x)

ε+ (toϕ)^{2 p−1}

2

dv_g

and since A andC are assumed with positive values, we obtain (1.16) I(t0ϕ)≥Φ (t0) .

Finally from (1.15) and (1.16), we get

I(t₁ϕ)<Φ (t₀)≤I(t₀ϕ) Now to have (i) just take

u=t₀ϕand ρ= Φ (t₀) . it remains to prove (ii); to do so we remark that

t→+∞lim I(tϕ) = lim

t→+∞





 1

2ktϕk²_Pg − 1 2^]

Z

M





B(x) (tϕ)^2]dvg− A(x)

ε+ (tϕ)²2^[





dvg







− lim

t→+∞

1 p−1

Z

M

C(x)

ε+ (tϕ)^2p−1₂ dv_g

= lim

t→+∞t^2] 1

2t^2]−2 − 1 2^]

Z

M

B(x)ϕ^2]dv(g)

. Since by assumptionB >0, we have

t→+∞lim I(tϕ) =−∞

consequently there ist2 such that

t₂ > t₀ and I(t₂ϕ)<0

Hence, puttingu2=t2ϕ,we get the assumption (ii) of Lemma 2.

Lemma 1 allows us to apply the Mountain-Pass Lemma to the functional I. Let

C= inf

γ∈Γmax

u∈γI(u)

where Γ denotes the set of paths in H_k²(M) joining the functions u₁ =t₁ϕ and u2 =t2ϕ.

So C_ε is a critical value of I_ε and moreover C>Φ (t0)

and by puttingγ(t) =tϕ, for t∈[t₁, t₂], we see thatC_ε is uniformly when εgoes to 0, so we have

(1.17) 0<Φ (t0)< C≤C forεsufficiently small and C >0 not depending onε.

Consequently there exists a sequence (um)_m of functions in H_k²(M) such that

(1.18) I(um) →

m→+∞C and DI(uk) →

m→+∞0

By Lemma 2 the sequence (um)_m∈N ofH_k²(M) is a Palais-Smale sequence (P-S) for the functional I.

Theorem 2. The Palais-Smale sequence (um)_m∈N is bounded in H_k²(M) and converges weakly to nontrivial smooth solutionu_εof equation (1.5).

Proof. By (1.18) we get for anyϕ∈H_k²(M) DI(um)ϕ=o(1) i.e. for any ϕ∈H_k²(M) one has

(1.19)

Z

M

ϕPgumdvg = Z

M

B(x) u⁺_m2^]−1

ϕdvg

+ Z

M

A(x)u⁺_mϕ

ε+ u⁺m22^[+1dvg+ Z

M

C(x)u⁺_mϕ

ε+ u⁺_m2^p₂+1dvg+o(1) in particular, forϕ=u_m we have

Z

M

u_mP_gu_mdv_g− Z

M

B(x) u⁺_m2^]

dv_g = Z

M

A(x) (u⁺_m)²

ε+ u⁺m

22^[+1dv_g

+ Z

M

C(x) (u⁺_m)²

ε+ (um)^{2 p}

2+1dv_g+o(1) .

or

−1 2

Z

M

u_mP_gu_mdv_g+1 2

Z

M

B(x) u⁺_m2

dv_g

(1.20) +1 2

Z

M

A(x) (u⁺_m)²

ε+ u⁺_m22^[+1dvg+ Z

M

C(x) (u⁺_m)²

ε+ u⁺m

2^p₂+1dvg =o(1) . On the other hand it comes from (1.18) that

1 2

Z

M

u_mP_gu_mdv_g− 1 2^]

Z

M

B(x) u⁺_m2^]

dv_g (1.21)

+1 2^]

Z

M

A(x) (u⁺_m)²

ε+ u⁺m

22^[+1dv_g+1 p

Z

M

Z

M

C(x) (u⁺_m)²

ε+ u⁺m

2^p

2+1dv_g =C+o(1) . So by adding (1.20) and (1.21) we get

(1.22) 2 n

Z

M

B(x) u⁺_m2^]

dvg+1 2

Z

M

A(x) (u⁺_m)²

ε+ u⁺m

22^[+1dvg+1 2^]

Z

M

A(x) (u⁺_m)²

ε+ u⁺m

22^[+1dvg

1 2

Z

M

C(x) (u⁺_m)²

ε+ u⁺m

2^p₂+1dv_g+ 1 2^]

Z

M

C(x) (u⁺_m)²

ε+ u⁺m

2^p₂+1dv_g =C+o(1) . For sufficiently largem we deduce that

2 n

Z

M

B(x) u⁺_m2^]

dvg ≤2C+o(1)

or 1

2^] Z

M

B(x) u⁺_m2^]

dvgdv(g)≤ n

2^]C+o(1) and plugging this last inequality with in (1.21) we obtain

1 2

Z

M

umPgumdv(g)≤C+ n

2^]C+o(1)

≤nC+n(n−4)

2n C+o(1)≤2 (n−2)C+o(1) . Hence for m large enough

Z

M

umPgumdv(g)≤4nC+o(1)≤4nC+ 1 i.e.

(1.23) ku_mk²≤4nC+ 1

Thus we prove the sequence (u_m)_mis bounded inH_k²(M) so we can extract a subsequence, still denoted (u_m)_m which verifies:

1. u_m →u_ε weakly inH_k²(M).

2. um →uε strongly in L^p(M),∀p < _n−2k²ⁿ 3. u_m →u_ε a.e. in M.

4. (um)^2]−1→u²_ε^]−1 weakly inL

2] 2]−1 (M).

Furthermore, putting g(x) = _ε¹q, where ε > 0 and q > 0, we get by Lebesgue’s dominated convergence theorem that

∀k∈N:

u⁺_m2

+ε −q

< ε^−q and ε^−q ∈L^p(M) ∀p≥1 thus

(u⁺_m)²+ε−q

→ u^◦2

+ε−q

strongly in L^p(M) ∀p≥1 and with (ii) we infer that ^u+m

(^u+m)^2+εq → ^u+

ε+(^u+ε)^2q strongly inL²(M). So if we let m go to +∞ in (1.19) we obtain that u is a weak solution to the equation (1.24) Pguε=B(x) u⁺_ε2^]−1

+ A(x)u⁺_ε

ε+ u⁺ε

22^[+1 + C(x)u⁺_ε

ε+ u⁺ε2^p+1₂ where 2^b= ²₂^\ and p >1.

Our solution u is not identically zero; indeed by (1.21) we have

−1 2^]

Z

M

B(x) u⁺_m2^]

dv(g)≤C_ε+o(1)≤2C_ε+o(1) or

−2 n

Z

M

B(x) u⁺_m2^]

dv(g)≤ 4

n.2^]C_ε+o(1) . In addition by (1.22), we get

1 2^]

Z

M

A(x)

ε+ (u^◦_m)²

2^[dv(g)≤C_ε− 2 n

Z

M

B(x) u⁺_m2^]

dv(g)

≤Cε+ 4

n2^]Cε+o(1)≤ 2^]−1

Cε+o(1) . Letting k→+∞ and taking into account of (1.17) we infer that

(1.25) 1

2^] Z

M

A(x)

ε+ u⁺_ε22^[dv(g)≤

2^]−1 C

whereC is the upper bound ofCε. Now if for a sequence ε_j →0

j→+∞

( with_j >0,∀j ∈N); u_εj goes to 0, then it follows

(1.26) 1

2^](2^]−1)ε²_j^[ Z

M

A(x)dv(g)≤C.

So if j →+∞,it leads to a contradiction since by assumptionA >0.

Finally, forεsufficiently small,uε is a solution not identically zero of the equation (1.5).

By writing the equation (1.24) in the form

∆^k_gu+

k−1

X

l=1

A_l(x)

∇^lu,∇^lu +









A₀(x)− A(x)

ε+ u⁺22^b+1 − C(x)

ε+ u⁺

2^p+1

2







 u⁺_ε

=B u⁺_ε2^\−1

where A_l is a smooth tensor field of type T⁰_2l and symmetric in the sense thatAl(X, Y) =Al(Y, X) for tensorsX,Y of type T^l₀ on M (see [12]).

Noticing that

A0(x)− A(x)

ε+ u⁺

22^b+1 − C(x)

ε+ u⁺2^p+1₂

∈L^∞(M)

we deduce by standard regularity theory that u∈C^2k(M).

Now we are in position to prove Theorem 1.

Proof. From what precedes u is a C^2k(M) nontrivial solution to equation (1.5), moreover u is a weak limit of the sequence (u_k)_kwhich allows us by the lower semicontinuity of the norm to write

kuk ≤ lim

k→+∞infku_kk.

And by the inequalities (1.17), (1.23) we deduce that the sequence (uε)ε of the ε-approximating solutions is bounded in H_k²(M) for sufficiently small >0 i.e.

(1.27) kuk²≤4nC+ 1

thus we can extract a subsequence still labelled (uk)_k satisfying:

i)uk−→u weakly inH_k²(M)

ii) u_k−→u strongly in L^p(M) for p <2^] iii)u_k−→u a.e. in M.

vi)u²_k^]−1−→u^2]−1weakly in L

2] 2]−1.

Furthermore the sequence (u_k)_kis bounded below: indeed as the functions u_k are continuous, denote by x_k their respective maximums on M and put xo = limxk ( a subsequence of (xk)_k still labelled (xk)k ). Since by assumption the operator P_g admits a positive Green function, then we can write

u_k(x_k) = Z

M

G(x_k, y)









B(y) u⁺_k (y)2^]−1

+ A(y)u⁺_k (y)

ε+ u⁺_k (y)22^[+1 + C(y)u⁺_k(y)

ε+ u⁺_k (y)2^p+1

2







 dv_g

and by the Fatou’s lemma, we get lim inf

k u_k(x_k)≥ Z

M

lim inf

k G(x_k, y)









B(y) u⁺_k (y)2^]−1

+ A(y)u⁺_k (y)

ε+ u⁺_k (y)22^[+1 + C(y)u⁺_k (y)

ε+ u⁺_k (y)2^p+1₂







 dvg

= Z

M

lim inf

k G(x_k, y)









B(y) u⁺(y)2^]−1

+ A(y)u⁺(y)

ε+ (u⁺(y))²

2^[+1 + C(y)u⁺(y)

ε+ (u⁺(y))^{2 p+1}₂







 dv_g.

And since the functions A, B, C are positive, then lim inf_ku_k(x_k) = 0 implies that u⁺ = 0. This contradicts relation (1.26). Thus, there exists δ > 0, such that u_k ≥ δ. We can once again use Lebesgue’s dominated convergence theorem to get

1

ε_k+ (u_k)²q → 1

(u)^2q strongly in L^p(M) ,∀p≥1,∀q ≥1.

Since for klarge enough u_k>0 there is ˜ε >0 such that 1

ε_k+u²_kq ≤ 1

˜

ε^q withq >0.

Thus by the Lebesgue’s dominated convergence theorem, we infer that 1

εk+ (uk)²

q −→ 1

u^2q strongly inL^p(M) , ∀p≥1,∀q >0.

Finally, with ii), it follows that u_k

ε_k+ (u_k)²

2^[+1 −→ 1

u^2]+1 strongly in L²(M) .

with u >0. Letting ε_k → 0 in (1.24) as k→ +∞,we get that u is a weak solution of equation (1.1) and since the regularity of u is the one of the functionγ(x) =|x|^n−2k4k and since u >0, thenu∈C^∞(M).

2. Existence of a second solution

According to the previous section our functional admits a local maximum Cε , this means the following inequalities

Iε(t1ϕ)<Φ(t0)< Iε(t0ϕ)≤Cε

where t₀, t₁ are real numbers satisfying 0< t₁ < t₀ and ϕ∈C^∞(M) with ϕ >0 andkϕk= 1.

On the other hand and as Iε(tϕ) tends to −∞ astgoes to +∞, there is t₂ >> t₀ such thatI_ε(t₂ϕ)<0. Now if we lett and εtend both to 0⁺, the functional laIε tends to +∞. Indeed,

t→0lim⁺

ε→0lim⁺Iε(t.ϕ)

= lim

t→0⁺

h

I⁽¹⁾(t.ϕ) +I₀⁽²⁾(t.ϕ) i

= lim

t→0⁺

Z

M

(t.ϕ)P_g(t.ϕ)− 2

2^]B(x)(t.ϕ)^2]

dv(g)

+ lim

t→0⁺



 1 2^]

Z

M

A(x)

(t.ϕ)^2]dv(g) + 1 p−1

Z

M

C(x)

(t.ϕ)^p−1dv(g).





= +∞.

and it follows that for ε small enough, there is near 0 a real number 0 <

t⁰ << t₁ such that I_ε(t⁰ϕ) > Φ(t₀ > I_ε(t₁ϕ). What let us see, from this effect, that our function has a local lower bound. We will give the necessary conditions for this lower bound to exist, then we show by Ekeland’s lemma that this lower bound is reached.

We will need the following version of the Ekeland’s lemma

Lemma 3. Let V be a Banach space,J be aC¹lower bounded function on a closed subsetF ofV andc= inf_F J. Letu_ε ∈F such thatc≤J(u_ε)≤c+ε.

Then there is uε ∈F such that







c≤J(uε)≤c+ε ku_ε−u_εk_V ≤2√

ε

∀u∈F, u6=uε, J(u)−J(uε) +√

εku−uεk_V >0.

If moreover, u_ε is in the interior of F, then kDJ(u_ε)k_V0 ≤√

ε.

We can consider the sequence ( u_ε )_ε in the interior of F. Indeed if u_ε is on the border of F then by the continuity of J there is u_ε belonging to interior of F such that |J(uε)−J(uε)|< ε. Which gives, for ε sufficiently, c−ε < J(u_ε)< c+ 2ε and J(u)−J(u_ε) +√

εku−u_εk_V =J(u)−J(u_ε) + J(uε)−J(uε) +√

εku−uε+uε−uεk_V

≥ J(u)−J(u_ε)−ε+√

εku−u_εk_V −√

εku_ε−u_εk_V > J(u)−J(u_ε) +

√εku−uεk_V −2ε >0. So we can speak about the differential DJ(uε).

Before stating the main result of this section we will establish some pre- liminary lemmas.

Lemma 4. Let θ >0 such that (2.1)

a 2

²

2]

< θ² < a

2 2\

where

a= 1

(2 (n−k))²

\ 2

and put

t3 = a

2 ¹

2]

t0

then we have the following inequality

(2.2) Φ(t3)> a

2Φ(t0).

Proof. Since (t1 being defined as in Lemma 1),

(2.3) t3 =

a 2

¹

2]

t0 < θt0 =t1. and

(2.4)

a 2

²

2]

> a 2. by (1.7), we get

Φ(t₃) = 1 2t²₃−

S.max

M B(x) t²₃^]

2^]

= 1 2

a 2

¹

2]

.t₀ 2

− 1 2^]t²₀a

2

= 2n k

1 2

a 2

²

2]

−a 2

n−2k 2n

k 2nt²₀. Knowing by (1.13) that

k

2nt²₀= 1 2Φ(t0) we deduce

Φ(t3) = n

k a

2 ²

2]

−n a 2k +a

1 2Φ(t0)

= n

k a

2 ²

2]

− a 2

+a

1 2Φ(t₀)

> a 2Φ(t₀).

Where we used the inequality (2.4) in the last line.

Lemma 5. Given a Riemannian compact manifold (M, g ) of dimension n >2k, k∈N^? and 3< p <2^]+ 1.

So there is a constantλ^∗>0such that: ∀ε∈]0, λ^?[the following inequalities take place

(2.5) Z

M

A(x) (ε+ (t3ϕ)²)²

] 2

dvg ≥ 2 a.



 Z

M

pA(x)dvg





2

1 t₀a₁

2^]

and

(2.6) Z

M

C(x) (ε+ (t3ϕ)²)

p−1 2

dvg≥ 2

a ^p−1

2]



 Z

M

pC(x)dvg





2

1 t0a2

p−1

where a₁, a₂ are positive constants andt₃, a are chosen as in Lemma 4.

Proof. Letϕ∈C^∞(M), ϕ >0 in M with kϕk= 1. Put

(2.7) β₁=

a 2V (M)

²

2] t₀ a1

√2 2

and

(2.8) β₂ =

1 V (M)

_p−1² a

2 ²

2]

t₀a₂

√2 2

whereV(M) denotes the volume ofM. Let λ^? = min (β₁, β₂) . By H¨older’s inequality, we get



 Z

M

pA(x)dvg





2

=



 Z

M

pA(x) [ε+ (t₃ϕ)²]²

] 4

ε+ (t3ϕ)²²

] 4 dvg





2

≤



 Z

M

A(x) (ε+ (t₃ϕ)²)²

] 2

dv_g



.



 Z

M

ε+ (t₃ϕ)²²

] 2 dv_g





(2.9) ≤I

kε+ (t3ϕ)²k₂] 2

²

] 2

where

I = Z

M

A(x) (ε+ (t3ϕ)²)²

] 2

dv_g.

Independently, the Minkowski’s inequality can be written kε+ (t₃ϕ)²k₂]

2

≤ kεk₂] 2

+t²₃kϕ²k₂] 2

consequently (2.10)

kε+ (t3ϕ)²k₂] 2

²

] 2

≤

kεk₂] 2

+t²₃kϕ²k₂] 2

²

] 2

. Notice that

kεk₂] 2

=ε.[Vg(M)]

2 2]

and

kϕ²k₂] 2

=kϕk²₂]

By continuity of inclusion H_k²(M) ⊂L^2](M), we deduce the existence of a positive constant S1 such that

kϕk₂]≤S1kϕk_H2 k

and as the operator Pg is coercive, the norms k.k_H2

k and k.k are equivalent (this results from Proposition2, reference [12]) and the exists an another constantC >0 such that

kϕk_H2

k ≤Ckϕk_Pg ∀ϕ∈H_k²(M) and therefore (2.10) becomes

kε+ (t3ϕ)²k₂] 2

²

] 2

≤

ε(V(M))

2

2] +t²₃(S1C)^{2 2}

] 2

where we have used the fact thatkϕk_Pg = 1.

Taking account of

t3 = a

2 ¹

2]

t0

and letting a₁ =√

2S₁C, (2.9) becomes



 Z

M

pA(x)dvg





2

≤I ε[Vg(M)]

2 2] +

a_ε 2

²

2]

t²₀ a₁

√ 2

2!²

] 2

.

And since 0< ε < λ^? ≤β1,we get



 Z

M

pA(x)dv_g





2

≤I 2a_ε 2

²

2] t₀α₁

√2 2!²

] 2

.

Finally we deduce I ≥ 2

a_ε



 Z

M

pA(x)dv_g





2

1 t₀.α₁

2^]

.

Let us now prove the second inequality; again with the H¨older’s inequality, we have



 Z

M

pC(x)dvg





2

=



 Z

M

pC(x) [ε+ (t₃ϕ)²]^p−14

. ε+ (t3ϕ)^2p−1₄ dvg





2

(2.11)

≤



 Z

M

C(x) (ε+ (t3ϕ)²)

p−1 2

dv_g



.



 Z

M

ε+ (t₃ϕ)^2p−1₂ dv_g





≤J.

h

kε+ (t3ϕ)²kp−1 2

i^p−1 (2.12) 2

where

J = Z

M

C(x) (ε+ (t₃ϕ)²)^p−12

dv_g. Again with Minkowski’s inequality, we have

kε+ (t3ϕ)²kp−1 2

≤ kεkp−1

2 +t²₃kϕ²kp−1 2

(2.13)

≤ε.[V_g(M)]^p−12 +t²₃kϕk²_p−1 (2.14)

and since the embeddingH_k²(M)⊂L^p−1(M) is continuous, there is a posi- tive constant S₂ such that

kϕk_p−1≤S2kϕk_H2 k. The norms k.k_H2

k,k.kbeing equivalent, there is a positive constantC1 such that

kϕk_H2

k ≤C₁kϕk, ∀ϕ∈H_k²(M).

consequently (2.11) becomes



 Z

M

pC(x)dv_g





2

≤J

εV_g(M)^p−12 +t₃²S₂²C^2p−1₂ .

Replacing t3 by his expression and posing a2 =√

2S2C1 it comes that



 Z

M

pC(x)dvg





2

≤εJ.V(M)^p−12 + a

2 ²

2]

t0

a₂

√ 2

2

now since 0< ε < λ^? whereλ^?≤β2 , then it follows that



 Z

M

pC(x)dv_g





2

≤J 2a 2

²

2] t₀ a₂

√2

2!^p−1₂

i.e.:

Z

M

C(x) (ε+ (t3ϕ)²)

p−1 2

dv_g≥



 Z

M

pC(x)dv_g





2

2 a

^p−1

2] 1 t0a2

p−1

. Now we are able to prove the existence of a second solution to equation (1.1)

Theorem 3. Let (M, g) be a compact Riemannian manifold of dimension n >2k, (k∈N^?). Suppose the operator P_g is coercive, has a Green positive

function and there is a constant C(n, p, k)>0which depends only on n, p , k such that:

kϕk^2] 2^]

Z

M

A(x)

ϕ^2\ dv_g ≤C(n, p, k)

Smaxx∈MB(x) ²⁺²

] 2−2]

and

kϕk^p−1 p−1

Z

M

C(x)

ϕ^p−1dv_g ≤C(n, p, k)

Smaxx∈MB(x) ^p+1

2−2]

for some smooth function ϕ > 0 on M. If moreover for every ε∈ ]0, λ^?[ , where λ^? is a positive constant, the following two conditions occur

(2.15) 2

a



 Z

M

pA(x)dv_g





2

1 t0a1

2^]

>2^]kt²₀

4n(2−a) and

(2.16)

2 a

^p−1

2]



 Z

M

pC(x)dv_g





2

1 t0a2

p−1

>(p−1)kt²₀

4n(2−a) where a₁, a₂ are positive constants, 2^] = _n−2k²ⁿ , 3 < p < 2^]+ 1. Then the equation (1.1) admits a second smooth solution.

Proof. The proof will be done in three steps

1^th-step. The functionalI_ε has a local lower bound.

This consists to find a strictly positive real number λ^? such ∀ε∈ ]0, λ^?[ one has the following inequality

I_ε(t₃ϕ)>Φ(t₀) ∀ϕ∈C^∞(M), kϕk= 1, with t₃ < t₁. Indeed, according to Lemma (4) inequality (2.2), one has

I_ε(t₃ϕ) =I⁽¹⁾(t₃ϕ) +I⁽²⁾(t₃ϕ) (2.17)

> a

2Φ(t₀) + 1 2^]

Z

M

A(x) (ε+ (t3ϕ)²)^2b

dv_g

+ 1

p−1 Z

M

C(x)

(ε+ (t₃ϕ)²)^p−1dvg. and as by assumption

2 a.



 Z

M

pA(x)dvg





2

1 t₀a₁

2^]

>2^]kt²₀

4n(2−a) and

λ^? = min (β₁, β₂)

where β1, β2 are given by (2.7) and (2.8) and knowing that Φ(t₀) = k

nt²₀ it follows by Lemma5 that,∀ε∈]0, λ^?[

(2.18) 1

2^] Z

M

A(x) (ε+ (t₃ϕ)²)²

] 2

dv_g >

1 2 −a

4

Φ(t₀).

Similarly, one has 2

a ^p−1

2]



 Z

M

pC(x)dvg





2

1 t0.a2

p−1

>(p−1)kt²₀

4n(2−a) which implies with Lemma (5) that ∀ε∈]0, λ^?[

(2.19) 1

(p−1) Z

M

C(x) (ε+ (t₃ϕ)²)^p−12

dvg>

1 2 −a

4

.Φ(t0).

Finally, by combination of (2.17), (2.18) and (2.19) we get

I_ε(t₃ϕ)> a

2Φ(t₀) + 2 1

2 −a 4

Φ(t₀) (2.20)

>Φ(t₀).

Hence our result.

2^th− step. The infimum of the functionalI_ε is reached.

Denote by B(0, t1) =

u∈H_k²(M) :kuk ≤t1 the closed ball centred at the origin 0 of radius t₁ in H_k²(M). In this section we will show that cε = inf_B(0,t1)Iε ( cε < Φ(t0) ) is reached. By Ekeland’s Lemma, there exists a sequence (um)_m∈N in B(0, t1) such that Iε(u) → cε =Inf_B(0,t1)Iε

and DI_ε(u_m) → 0 strongly in the dual space of H_k²(M). That is to say (um) is a Palais-Smale sequence, so by the same arguments as in Theorem2 and Theorem1, we get that equation (1.1) has a smooth solution v. Since the ε-approximating solutions are obtained as weak limit of sequences of functions from B(0, t₁), it follows by the weak lower semi-continuity of the norm that these ε-approximating solutions are in B(0, t₁). As in turn v is obtained as a limit of a sequence ofε-approximating solutions,v∈B(0, t₁).

3^th-step. The two solutions are distinct. Indeed, we have on the one other hand the energy of the solution v given by c = lim_ε→0+c_ε ≤ Φ(t₀) and secondly the energy of the solution u obtained in the first section is C = lim_ε→0+C_ε where C = inf

γ∈Γmax

u∈γI(u) and Γ denotes the set of paths joining the functionsu₁=t₁ϕand u₂ =t₂ϕ. Now since any path joiningu₁ andu₂ intersects the sphere centred at the origin and of radiust₀ inH_k²(M)

then C = inf

γ∈Γmax

u∈γI(u)≥(I_ε(t₀ϕ)) =I⁽¹⁾(t₀ϕ) +I_ε⁽²⁾(t₀ϕ)≥Φ (t₀) +I_ε⁽²⁾(t₀ϕ)

≥Φ (t₀) + 1 2^]

Z

M

A(x)

ε+ (t₀ϕ)²2^[dv_g+ 1 p−1

Z

M

C(x)

ε+ (t0ϕ)^{2 p−1}₂

dv_g

Thus the Lebesgue’s dominated theorem, we get C= lim

ε→0⁺Cε≥Φ (t0) + 1 2^]

Z

M

A(x)

(t₀ϕ)^2\dvg+ 1 p−1

Z

M

C(x) (t₀ϕ)^p−1dvg

>Φ (t0).

So the solutionsu and v are of different energies and therefore are distinct.

3. Nonexistence of solution

In this section we will be placed in a closed ballB(0, R) ofH_k²(M) centered at the origin 0 and of radius R > 0, we prove that under some condition (inequality 3.1) that the equation (1.1) admits no solution.

Theorem 4. Given (M, g) a compact Riemannian manifold of dimension n > 2k, (k ∈ N^?) and A, B, C are positive smooth functions on M and 2< p <2^]+ 1. Assume that

(3.1) C(n, p, k)





 R

M

√B.Cdv_g

R

M

Bdvg







2. ^2]

p−1+2]

Z

M

Bdv_g >(SR)² where S, R are positive constants and

C(n, p, k) = 2^]+p−1 p−1

p−1 2^]

²

] 2]+p−1

.

Then the equation (1.1) has no smooth positive solutionuwith energykuk_H2 k(M) ≤ R.

Proof. Suppose that there exists a smooth positive solutionu∈H_k²(M) such that kuk_H2

k(M) ≤R. By multiplying both sides of equation (1.1) by u end integrating overM, we get

Z

M

uP_g(u)dv_g= Z

M

B(x)u^2]+A(x)

u^2] +C(x) u^p−1

dv_g.

And since kuk_pg =rR

M

uPg(u)dvg is a norm equivalent to kuk_H2

k(M), there exists a constant S >0 such that

kuk ≤Skuk_H2 k(M).