A critical elliptic problem for polyharmonic operators

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A critical elliptic problem for polyharmonic operators

Yuxin Ge, Juncheng Wei, Feng Zhou

To cite this version:

Yuxin Ge, Juncheng Wei, Feng Zhou. A critical elliptic problem for polyharmonic operators. Journal of Functional Analysis, Elsevier, 2011, 260, pp.2247-2282. �10.1016/j.jfa.2011.01.005�. �hal-00798829�

A critical elliptic problem for polyharmonic operators

Yuxin Ge, Juncheng Wei and Feng Zhou ^∗

Abstract

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturba- tion of the critical nonlinearity when the dimension of domain is bigger than critical one. In particular, for the critical dimension, we prove also the existence of solutions in domains perforated with the small holes. Some instable solutions are obtained in higher level sets by Coron’s topological method, provided that the minimizing solution does not exist.

AMS classification scheme numbers: 35J35, 35J40, 35J60

1 Introduction

This paper is a sequel to [21] on some semilinear critical problems for polyharmonic op- erators. Let K ∈ N _{and Ω ⊂} R^N_{(N ≥ 2K} + 1) be a smooth bounded domain in R^N_. We consider the semilinear polyharmonic problem with homogeneous Dirichlet boundary condition

( (−∆)^Ku=|u|^s−2u+f(x, u) in Ω

u=Du=· · ·=D^K−1u= 0 on ∂Ω (1) wheres:= _N−2K^2N denotes the critical Sobolev exponent for (−∆)^K and f(x, u) is a lower- order perturbation of |u|^s−2u (see the assumption (H2) below). The equation (1) is of variational type: Solutions of (1) correspond to critical points of the energy functional

E(u) = 1

2kuk²_K,2,Ω−1 s

Z

Ω|u|^s− Z

ΩF(x, u), (2)

defined on the Hilbert space

H₀^K(Ω) =ⁿv∈H^K(Ω)| Dⁱv= 0 on∂Ω ∀ 0≤i < K^o which is endowed with the scalar product

(u, v)_Ω =





 Z

Ω((−△)^Mu)((−△)^Mv) if K = 2M is even Z

Ω(∇(−△)^Mu)(∇(−△)^Mv) if K = 2M + 1 is odd

(3)

∗The first author is supported by ANR project ANR-08-BLAN-0335-01. The second author is partially supported by a research Grant from GRF of Hong Kong and a Focused Research Scheme of CUHK. The third author is supported in part by NSFC No. 10671071 and by Shanghai Priority Academic Discipline.

and k · k_K,2,Ω is the corresponding norm,F(x, u) :=

Z _u

0

f(x, t)dt is the primitive off. We assume that

(H1) f(x, u) : Ω×R_→Ris continuous and sup

x∈Ω,|u|≤M

|f(x, u)|<∞ for everyM >0;

(H2) f(x, u) = a(x)u+g(x, u) with a(x) ∈ L^∞(Ω)∩C^∞(Ω), g(x, u) = o(u) as u → 0 uniformly in x andg(x, u) =o(|u|^s−1) as u→ ∞ uniformly inx.

From (H1) to (H2), it follows f(x,0) = 0 and that f is a lower-order perturbation of

|u|^s−2u at infinite in the sense that lim

u→∞

f(x, u)

|u|^s−1 = 0 uniformly in x ∈ Ω. Moreover, we assume that f(x, u) satisfies

(H3) ^∂f_∂u(x, u) is continuous on Ω×R_;

(H4) |^∂f_∂u(x, u)| ≤C(1 +|u|^s−2),∀u∈Runiformly in x∈Ω;

(H5) f₁(x, u) := ^f(x,u)_u is non-decreasing in u > 0 and non-increasing in u < 0 for a.e.

x∈Ω.

For K = 1, f(x, u) = λu and λ∈ (0, λ₁) where λ₁ is the first eigenvalue of −∆ for Dirichlet boundary condition, the problem has a strong background from some variational problems in geometry and physics, like as the Yamabe’s problem with lack of compactness.

This was considered by Brezis and Nirenberg for positive solutions in their pioneer work in [5]. Then it has been studied extensively in the last three decades. We recall briefly some results about the existence and multiplicity of sign-changing solutions to the problem (1) for K = 1 and f(x, u) = λu. For any fixed λ > 0, the first multiplicity result was due to Cerami, Fortunato and Struwe [8]. They obtained the number of the solutions of (1) is bounded below by the number of the eigenvalues of −∆ lying in the open interval (λ, λ+S|Ω|^−2/N), where S is the best constant for the Sobolev embeddingD^1,2(R^N_{) ֒→} L^2∗(R^N) (see the definition below) and|Ω|denotes the Lebesgue measure of Ω. Capozzi, Fortunato and Palmieri in [7] established the existence of a nontrivial solution for λ > 0 which is not an eigenvalue of −∆ when N ≥4 and for any λ >0 whenN ≥5 (see also [44]). In [11], Devillanova and Solimini proved that, if N ≥ 7, then (1) has infinitely many solutions for every λ > 0. They proved also in [12] that, ifN ≥4 and λ∈ (0, λ₁), then there exist at least ^N₂ + 1 pairs of nontrivial solutions. Clapp and Weth [9] has extended this last result to all λ >0 with N ≥4. In the same paper they also obtained some extensions to critical biharmonic problems forN ≥8. When the domain Ω is a ball and N ≥ 4, Fortunato and Jannelli [15] proved there are infinitely many sign-changing solutions which are built using particular symmetry of the domain Ω. Schechter and Zou in [36] showed the same result for any domain Ω when n≥7. In particular, if λ≥λ₁, it has and only has infinitely many sign-changing solutions except zero. Their work is based on the estimates of Morse indices of nodal solutions.

Pucci and Serrin in [34] has studied the problem (1) for K = 2 and λ > 0 when Ω is a ball. They proved that it admits nontrivial radial symmetric solutions for all λ ∈(0, λ1) if and only if N ≥ 8. If N = 5,6,7, then there exists λ∗ ∈ (0, λ1) such that

the problem admits no nontrivial radial symmetric solutions whenever λ∈ (0, λ_∗]. Here λ₁ is understood as the first eigenvalue of ∆² for Dirichlet boundary conditions. This is the counterpart of the well known result of [5] on the nonexistence for radial symmetric solutions for small λ in dimension N = 3 and K = 1 (where λ∗ = λ1/4). They called these dimensions as critical dimensions. They conjectured that for general K ≥ 1, the critical dimensions are 2K+ 1,· · ·,4K −1. The conjecture is not completely solved for all K≥1. Grunau [23] defined later the notion of weakly critical dimensions as the space dimensions for which a necessary condition for the existence of a positive radial solution of (1) in B₁ is λ∈(λ^∗, λ₁) for some λ^∗ >0. He proved that the conjecture is true in the weak sense. Gazzola, Grunau and Squassina [18] proved nonexistence of positive radial symmetric solutions for Navier boundary condition for small λ > 0. They established also some existence results for λ= 0. Their result strongly depends on the geometry of domains. For biharmonic operators, Bartsch, Weth and Willem in [3] and Ebobisse and Ahmedou in [13] have studied the problem (1) on domains with nontrivial topology under Dirichlet boundary condition and Navier boundary condition respectively. For related problems, we infer to [4], [14], [16], [17], [22], [24], [31] and the references therein.

For general caseK ≥1, Ge has studied in [21] the same type of equation (1) for Navier boundary condition when f(x, u) = λu with 0 ≤ λ < λ₁ and λ₁ the first eigenvalues of (−∆)^K. He established the existence of positive solutions in some general domain under the suitable assumptions. In particular instable solutions in higher level set are obtained by Coron’s topological method in domains perforated with the small holes.

The purpose of this paper is to continue the study of the semilinear polyharmonic problem (1) to general K ≥1 with Dirichlet boundary condition for general domain. Let us denote the polyharmonic operator

L:= (−∆)^K−a(x)

and λ₁(Ω) ≤ λ₂(Ω) ≤ · · · ≤ λ_n(Ω) ≤ · · · the eigenvalues of L under the homogeneous Dirichlet boundary condition. It is well known that each eigenvalue λ_k(Ω), k≥1, can be described as the minimax value

λ_k(Ω) = min

V⊂H₀^K(Ω),dimV=k

maxv∈V

Z

Ω

vLv Z

Ω

v² .

It follows that λ_k(Ω) is a non-increasing functional on the domains, that is, if Ω₂ ⊂ Ω₁, then λ_k(Ω₂) ≥ λ_k(Ω₁). Moreover, from the unique continuation principle, we have λ_k(Ω₂)> λ_k(Ω₁) for any k≥1, provided Ω₁ is connected (see [27, 33]). For the perforated domain Ω := Ω₁\Ω₂ with the smooth bounded domains Ω₂ ⊂ Ω₁, with the help of the above description, we have

limλ_k(Ω₁\Ω₂) =λ_k(Ω₁),

where the limit is taken as the diameter of Ω₂ goes to 0. To this aim, it suffices to consider Ω₂ =B(x, ǫ) balls with small radiusǫ >0 in the sequel.

Assume nowλ_n(Ω)≤0 andλ_n+1(Ω)>0 for somen≥1. Lete_i(x) be an eigenfunction associated to λ_k(Ω) with ke_ik_K,2,Ω= 1 for any 1≤i≤n. Define

M:={v∈H₀^K(Ω)\ {0}| dE(v)(w) = 0, ∀w∈Span(v, e1,· · ·, en)}.

We prove in Section 2 that under the hypothesis (H1) to (H5), Mis then a complete C¹ Finsler manifold and it will be a C^1,1 Finsler manifold with additional assumptions (H6) to (H7) (see section 2). This permits to consider the following minimization problem

κ:= inf

v∈ME(v).

We can prove then κ≤ ^K_N(SK(Ω))^2KN for any f satisfying (H1) to (H5), where we denote S_K(Ω) := inf

v∈H₀^K(Ω)\{0}

kvk²_K,2,Ω kvk²_Ls(Ω)

the best constant for the embedding H₀^K(Ω) ֒→ L^s(Ω). Here, as for K = 1, it is well known thatS_K(Ω) is independent of Ω andS_K(Ω) =S_K(R^N_{) := inf}

v∈H^K(R^N)\{0}

kvk²_K,2,RN

kvk²_Ls(R^N)

. Therefore we denote it by S_K in the sequel (see [17, 20]). We prove then that for non critical dimension case N ≥4K andf(x, u) =µu, the infimum above is achieved by some u ∈ M which is a solution of (1). This method can be seen as an alternative approach to the linking method (see [38]). For the critical dimension 2K < N <4K, the existence of solutions to (1) is a delicate issue. To our knowledge, there are few results on it. This fact comes from the minimizing method fails, for example, for K = 1, when Ω⊂R³ _{is a} ball and when f(x, u) =λuwith 0< λ < ^λ₄¹. It is well known that there are no positive solutions. In Section 3, we study the existence of solutions for some perforated domains in such critical dimensions. We analyze the concentration phenomenon when κ equals to

K

N(SK)^2KN . Then following Coron’s strategy of topological argument, we obtain the exis- tence of instable critical points in higher level sets for domains perforated with small holes.

In all this paper,C, C^′ and c denote generic positive constant independent of u, even their value could be changed from one line to another one. We give also some nota- tions here. The space D^K,2(R^N_{) (resp. D}^K,2₍R^N

+)) is the completion of C₀^∞(R^N_{) (resp.}

C₀^∞(R^N

+)) for the normk · k_K,2,RN (resp. k · k_K,2,RN +).

2 Study of the energy functional E on M

We begin this section by studying some properties of the set M. Observe thatv ∈ M is equivalent to say v6= 0 and satisfying

l₀(v) := kvk²_K,2,Ω− kvk^s_Ls(Ω)− Z

Ω

f(x, v)v= 0 l_i(v) := (v, e_i)_Ω−

Z

Ω

|v|^s−2ve_i− Z

Ω

f(x, v)e_i = 0, ∀1≤i≤n.

(4)

Let us denoteV₀ := Span(e₁,· · ·, e_n) then−dimensional vector space spanned bye₁,· · ·, e_n. We prove now the following proposition.

Proposition 1 Suppose (H1) to (H5) are satisfied. Then M is a complete C¹ Finsler manifold. Furthermore, suppose that

(H6) ^∂_∂u^2f2(x, u) is continuous on Ω×R_{and u7→ |u|}^s−2_{u isC}² _on R_; (H7) |_∂u^∂22f(x, u)| ≤C(|u|+ 1)^s−3, ∀u∈R uniformly in x∈Ω.

Then Mis a complete C^1,1 Finsler manifold.

Proof. The proof is divided into several steps.

Step 1. Mis not empty.

By the assumptions (H1)-(H2),E is a continuous functional on H₀^K(Ω). Fixingv6∈V0

and let V := Span(v, e₁,· · ·, e_n). Clearly, for all w∈V, we have E(w)≤ 1

2kwk²_K,2,Ω−1 2

Z

Ωa(x)w²−1 s

Z

Ω|w|^s, (5)

since it follows from (H2) and (H5) that ^g(x,u)_u ≥0 andF(x, u)≥ ¹₂a(x)u²for allu∈R_\{0}

and for a.e. x ∈ Ω. As V is a finite dimensional vector space, all the norms on it are equivalent. In particular, the norms k · k_K,2,Ω and k · k_L^s_(Ω) are equivalent on V. This implies

w∈V,w→∞lim E(w) =−∞. (6)

On the other hand, again from (H2), we infer for any givenε >0, there existsC >0 such that for allu∈Rand for a.e. x∈Ω

g(x, u)≤ε|u|+C|u|^s−1, F(x, u)≤ 1

2(a(x) +ε)u²+C

s|u|^s, (7) so that for all w∈V

E(w)≥ 1

2kwk²_K,2,Ω−1 2 Z

Ω

(a(x) +ε)w²−1 +C s

Z

Ω

|w|^s.

Since v6∈V₀, we can choose v^′ ∈V ∩(V₀)^⊥ such that ¹₂kv^′k²_K,2,Ω−¹₂^R_Ωa(x)(v^′)² >0. By taking a sufficiently small ε >0, we have

1

2kv^′k²_K,2,Ω− Z

Ω

1

2(a(x) +ε)(v^′)² ≥εkv^′k²_K,2,Ω. (8) As a consequence, we obtain

sup

w∈V

E(w)>0. (9)

Together with (6), there exists ˜v ∈ V such that E(˜v) = max_w∈V E(w) since V is a finite dimensional vector space. Clearly, ˜v∈ M.

Step 2. Mis closed.

We define the map

L: H₀^K(Ω) → Rⁿ⁺¹

v 7→ (l₀(v),· · ·, l_n(v)).

In view of the assumptions (H1)-(H2), L is continuous on H₀^K(Ω). Let (v_k) ⊂ M be a sequence in M such that v_k → v in H₀^K(Ω). Then we get L(v) = 0. Now it suffices to show v6= 0. First, we notev_k6∈V0 for all k∈N. Indeed, we have

kv_kk²_K,2,Ω− Z

Ω

a(x)v²_k=kv_kk^s_Ls(Ω)+ Z

Ω

g(x, v_k)v_k. (10) If we have v_k ∈V₀ for some k ≥ 1, the term on the left hand is non-positive. But that one on the right hand is non-negative. Thus, kv_kk^s_Ls(Ω)= 0 and the desired contradiction v_k6= 0 gives the result. Now, we claim there exists some positive number c >0 such that kv_kkK,2,Ω> c. We denote the orthogonal projection ofv_k on V0 by

v^k_k:=

n

X

i=1

(v_k, e_i)_Ωe_i and v_k^⊥ its orthogonal complementary

v_k^⊥:=v_k−v^k_k. Asv_k ∈ M, we obtain

(v_k, v^k_k)_Ω− Z

Ω

a(x)v_kv_k^k= Z

Ω

(|v_k|^s−2+g(x, v_k) v_k )v_kv_k^k. Together with (10), we have

kv_k^⊥k²_K,2,Ω− Z

Ωa(x)(v^⊥_k)²−(kv_k^kk²− Z

Ωa(x)(v_k^k)²)

= Z

Ω

(|v_k|^s−2+g(x, v_k)

v_k )((v_k^⊥)²−(v_k^k)²) which implies

kv^⊥_kk²_K,2,Ω− Z

Ωa(x)(v_k^⊥)² ≤ Z

Ω(|v_k|^s−2+g(x, v_k)

v_k )(v_k^⊥)², (11) since

kv_k^kk²_K,2,Ω− Z

Ω

a(x)(v_k^k)² ≤0.

Gathering (5), (8) and (11), we get 2εkv_k^⊥k²_K,2,Ω ≤ kv^⊥_kk²−

Z

Ω(a(x) +ε)(v^⊥_k)²

≤(1 +C) Z

Ω|v_k|^s−2(v^⊥_k)²

≤(1 +C)kv_kk^s−2_Ls(Ω)kv^⊥_kk²_Ls(Ω)≤C^′(1 +C)kv_kk^s−2_Ls(Ω)kv^⊥_kk²_K,2,Ω

Finally, kv_kk_L^s_(Ω)≥c >0 and the desired claim follows.

Step 3. dL(v) is surjective and its kernel splits for all v∈ M.

By (H3) and (H4),f(x, u)u and f(x, u) areC¹ on Ω×R _and

∂(f(x, u)u)

∂u (x, u)

≤C(1 +|u|^s−1), uniformly in x∈Ω and∀u∈R_{. (12)} Therefore, L isC¹ on H₀^K(Ω) provided the assumptions (H1)-(H4) hold. A direct calcu- lation leads to

dl₀(v)(w) = 2(v, w)_Ω−s Z

Ω

|v|^s−2vw− Z

Ω

(f(x, v) +v∂f(x, v)

∂v )w, dl_i(v)(w) = (w, e_i)_Ω−(s−1)

Z

Ω

|v|^s−2we_i− Z

Ω

∂f(x, v)

∂v we_i, ∀1≤i≤n.

(13)

We claim dL(v)|_V, the restriction on V ofdL(v), is a bijective endomorphism fromV on Rⁿ⁺¹_{. AsV and} Rⁿ⁺¹ have the same dimension, it suffices to prove Ker(dL(v)|_V) ={0}.

Letw∈Ker(dL(v)|_V) and writew=µv+^Pn_i=1µ_ie_iwhereµ, µ_i ∈R_{for each}i. Combining (4) and (13), we get

dl₀(v)(w) =−(s−2) Z

Ω

|v|^s−2vw− Z

Ω

(−f(x, v) +v∂f(x, v)

∂v )w= 0, dli(v)(w) =−

Z

Ω(−f(x, v)

v +∂f(x, v)

∂v )µvei−(s−2) Z

Ω|v|^s−2µvei

+(^Pn_j=1µjej, ei)Ω−(s−1) Z

Ω|v|^s−2ei n

X

j=1

µjej− Z

Ω

∂f(x, v)

∂v ei n

X

j=1

µjej = 0, (14) for all 1≤i≤n. On the other hand, we have

µdl₀(v)(w) +

n

X

i=1

µ_idl_i(v)(w) = 0.

Together with (14), we infer (s−2)

Z

Ω|v|^s−2w²+ Z

Ω|v|^s−2(

n

X

j=1

µ_je_j)²+ Z

Ω(−f(x, v)

v +∂f(x, v)

∂v )w² +

Z

Ω

g(x, v) v (

n

X

j=1

µ_je_j)²−(

n

X

j=1

µ_je_j,

n

X

i=1

µ_ie_i)_Ω+ Z

Ω

a(x)(

n

X

j=1

µ_je_j)² = 0.

We know from (H2) and (H5) that−^f(x,v)_v +^∂f_∂v^(x,v) ≥0, ^g(x,v)_v ≥0 and (

n

X

j=1

µ_je_j,

n

X

i=1

µ_ie_i)_Ω− Z

Ωa(x)(

n

X

j=1

µ_je_j)²≤0.

Finally, we deduce

vw(x) = 0, v

n

X

j=1

µjej(x) = 0 for a.e. x∈Ω (15)

and

(

n

X

j=1

µ_je_j,

n

X

i=1

µ_ie_i)_Ω− Z

Ωa(x)(

n

X

j=1

µ_je_j)²= 0. (16) Thus we have

µv² =vw−v

n

X

j=1

µ_je_j = 0 which yields µ= 0. Moreover, it follows from (16) that

Lw= 0.

By the unique continuation principle, we have either w ≡0 or w(x) 6= 0 for a.e. x ∈ Ω.

Indeed, we state first w is regular. All the derivatives of w vanish a.e. on the set {x ∈ Ω;w(x) = 0} provided this set is not a negligible measurable set. Thus, w vanishes of infinite order at such points. By the strong unique continuation principle [27],wvanishes.

Going back to (15), we have w ≡0 and the desired claim follows. As a consequence, for all v ∈ M, dL(v) is surjective and H₀^K(Ω) = ker(dL(v))⊕V. Mis thus a complete C¹ Finsler manifold (see [26]). Furthermore, Mis a completeC^1,1 Finsler manifold provided (H6) and (H7) are satisfied.

For anyv∈H₀^K(Ω)\V₀, we denote by V⁺:={tv+

n

X

i=1

µ_ie_i| for all t >0, µ_i ∈R_},

the (n+ 1)−dimensional half space spanned byv and {e_i}for all 1≤i≤n. We have the following

Lemma 1 Under the assumptions (H1) to (H5), then there exists an uniquev₀ ∈ Msuch that

M ∩V⁺={v₀}. (17)

Moreover we have

E(v0) = max

w∈V⁺E(w). (18)

Proof. Given v ∈ H₀^K(Ω)\V₀, we define for any t > 0 the n−dimensional affine vector space

V_t:=tv+V₀. We divide the proof into several steps.

Step 1. For any t > 0 there exists an unique v(t) ∈ V_t such that E(v(t)) = max

Vt

E.

Moreover, {v(t), t >0} is a C¹ curve in V⁺.

From (H1) to (H4), it is known that E is C² on V⁺. Thanks to (6), we have

w∈Vlimt,w→∞E(w) =−∞

Thus there exists some v(t) ∈V_t such that E(v(t)) = max_w∈VtE(w). A direct calculation leads to

d²E(v)(w, w) =kwk²_K,2,Ω− Z

Ω

a(x)w²− Z

Ω

((s−1)|v|^s−2+ ∂g(x, v)

∂v )w² By (H5), we infer

g(x, v)

v ≥0 and ∂g(x, v)

∂v ≥ g(x, v) v ≥0.

Hence,d²E(v)<0 onV_t, that is, the functional Eis strictly concave onV_t. This yields the uniqueness. We note {v(t), t >0} ={w ∈V⁺| dE(w)|_V0 = 0}. As the second variation d²E of E is negative define on V₀, it follows from the Implicit Function Theorem that {v(t), t >0}is aC¹ curve in V⁺ which finishes the proof of step 1.

Step 2. For allw∈ M ∩V⁺, the restriction ofE on V⁺ has a strictly local maximum at w.

Recall V := Span(v, e₁,· · ·, e_n). Let v 6= 0 satisfying dE(v)|_V = 0 and w = µv+ Pn

i=1µ_ie_i∈V. As in the proof of Proposition 1, we have by (H2), d²E(v)(w, w) = −(s−2)

Z

Ω

|v|^s−2w²− Z

Ω

|v|^s−2(

n

X

j=1

µ_je_j)²

− Z

Ω

(−f(x, v)

v +∂f(x, v)

∂v )w²− Z

Ω

g(x, v) v (

n

X

j=1

µ_je_j)² +(

n

X

j=1

µ_je_j,

n

X

i=1

µ_ie_i)_Ω− Z

Ωa(x)(

n

X

j=1

µ_je_j)² which implies from (H1) to (H5)

d²E(v)(w, w)<0 providedw6= 0.

Therefore, the desired claim follows.

Step 3. There exists an uniquet₀ >0such thatv(t₀)∈ M. Moreover,dE(v(t))(v(t))>

0 for any 0< t < t₀ and dE(v(t))(v(t))<0 for any t > t₀.

With the same arguments as in the proof of Proposition 1, we have sup

w∈V⁺

E(w)>0. (19)

On the other hand, it follows from (5) that ∀w∈V0

E(w)≤0. (20)

In particular, we obtain sup

w∈V⁺

E(w) = sup

w∈V⁺

E(w) = sup

t>0E(v(t)),

where V⁺ is the closure ofV⁺. Combining (6), (19) and (20) and using the continuity of E on V⁺, there exists somev₀ ∈ M ∩V⁺ such that

E(v₀) = sup

w∈V⁺

E(w).

We know

M ∩V⁺ ⊂ {w∈V⁺| dE(w)|_V0 = 0}={v(t)| t >0} (21) so that there exists t₀ > 0 such that v(t₀) = v₀. Set α(t) := E(v(t)) then α^′(t) = dE(v(t))(v^′(t)) = ^l0(v(t))_t since v^′(t)−v ∈ V₀ and dE(v(t))|_V0 = 0. We claim M ∩V⁺ = {v(t)| α^′(t) = 0}. Obviously, M ∩V⁺ ⊂ {v(t)| α^′(t) = 0}. Conversely, for any v(t) with α^′(t) = 0, by the method of Lagrange multipliers, there exists µ1,· · ·, µn ∈ R _such that dE(v(t))|_V +

n

X

i=1

µ_idl_i(v(t))|_V = 0. Hence, we have on V₀,

n

X

i=1

µ_id²E(v(t))(·, e_i) = 0.

By virtue of the fact d²E(v)|V0 <0 for all v ∈Vt, we infer µ1 = · · ·µn = 0 which proves the claim. Applying (6), we infer

t→+∞lim α(t) =−∞, since

t→+∞lim inf

w∈Vt

kwk_K,2,Ω= +∞.

It follows from Step 2 that there exists only strictly local maximum points forα(t). Hence, t₀ is the only critical point ofα(t). Moreover, α^′(t) >0 for any 0< t < t₀ and α^′(t)<0 for any t > t₀. The lemma is proved.

Now let us consider the minimization problem κ:= inf

v∈ME(v). (22)

We have then

Lemma 2 Under assumptions (H1) to (H5), there holds κ≤ K

N(S_K)^2KN . (23)

Proof. LetB(x₀, R) ⊂Ω for somex₀ ∈Ω andR > 0. We consider for some small ν >0 and for all ǫ∈(0, ν), the function

u_ǫ(x) :=C_N,K ǫ^(N−2K)/2

(ǫ²+|x−x₀|²)^(N−2K)/2,