Biharmonic Equations Under Dirichlet Boundary Conditions with Supercritical Growth

Theorem 1.1 ([11])

Let N≥7 and 1≤k≤N-6. Then there exist smooth bounded domains Ω⊂ℝN, homotopic to 𝕊k, such that for any continuous function f verifying t⁢f⁢(t)≥p⁢F⁢(t) in ℝ, with

p > 2 ⁢ ( N - k - 1 ) N - k - 5   𝑎𝑛𝑑   F ⁢ ( t ) = ∫ o t f ⁢ ( s ) ⁢ 𝑑 s ,

the only solution of (D0) is u≡0.

Assume that Lτ⁢u=f⁢(u,x) in Ω and u=∇⁡u=0 on ∂⁡Ω. For any smooth vector field h=(hi) from Ω¯ into ℝN and a∈ℝ,

Here n denotes the unit external normal vector on ∂⁡Ω and Dh is the differential map of h.

A bounded smooth domain Ω is said to be Dirichlet h-starlike if ℋ*⁢(Ω)≠∅.

Let τ>0 (resp. τ=0) and Ω be a Dirichlet h-starlike domain in ℝN satisfying that M*⁢(Ω) given by (2.8) (resp. (2.6)) is positive. Assume that f=f⁢(u) verifies F⁢(s)≤a⁢s⁢f⁢(s) in ℝ with 0<a<M*⁢(Ω). Then there exists no nontrivial solution to problem ((Dτ)).

Consider a star-shaped domain Ω with respect to the origin up to a translation. Let h⁢(x)=xN. Clearly h verifies (2.2), μ⁢(h)=12-1N and μ¯⁢(h)=1N. Therefore M*⁢(Ω)>0 for N≥5 and we get some corresponding nonexistence results for ((Dτ)). In particular, M*⁢(Ω)≥N-42⁢N for (D0), hence if f⁢(u)=|u|p-1⁢u, no nonzero solution could exist to (D0) on a star-shaped domain Ω⊂ℝN with N≥5, if p>N+4N-4, which is just the classical result in [25, 21].

Let N≥3. We consider domains Ω in ℝN homotopic to spheres 𝕊k. Precisely, let η>0, 1≤k<N and consider the domain

Ω = 𝕋 k ⁢ ( η ) := { x ∈ ℝ N : ( ∥ P k ⁢ ( x ) ∥ - η - 1 ) 2 + ∥ Q k ⁢ ( x ) ∥ 2 ≤ η 2 } ,

where, for x=(xi)∈ℝN,

P k ⁢ ( x ) = ( x 1 , … , x k + 1 , 0 , … , 0 ) ∈ ℝ N , Q k ⁢ ( x ) = x - P k ⁢ ( x ) .

Set Bk={x∈ℝN:Pk⁢(x)=0} and define the vector field h as follows:

h ⁢ ( x ) = 1 N - k ⁢ [ φ ⁢ ( r k ) ⁢ P k ⁢ ( x ) + Q k ⁢ ( x ) ]   for any ⁢ x ∈ ℝ N ∖ B k ,

with rk⁢(x)=∥Pk⁢(x)∥ where ∥⋅∥ denotes the Euclidean norm and φ:(0,+∞)→ℝ is the function

φ ⁢ ( r ) = 1 k + 1 ⁢ ( 1 - 1 r k + 1 ) .

We have Ω=𝕋k⁢(η)⊂ℝℕ∖Bk. Clearly, Ω is homotopic to 𝕊k, since

𝕋 k ⁢ ( η ) = { x ∈ ℝ N : dist ⁡ ( x , S k ⁢ ( η ) ) ≤ η } ,

where Sk⁢(η)={x∈ℝN:∥x∥=η+1,Qk⁢(x)=0}. Similarly to [11], we see that h verifies (2.2). Let us prove that h satisfies also (2.7). Let y∈ℝN and x∈Ω. The definition of h and a direct calculus give

1 2 ⁢ | y | 2 - y ⋅ D ⁢ h ⁢ y = ( 1 2 - φ ⁢ ( r k ) N - k ) ⁢ ∑ i = 1 k + 1 y i 2 + ( 1 2 - 1 N - k ) ⁢ ∑ i = k + 2 N y i 2 - 1 ( N - k ) ⁢ r k k + 3 ⁢ ∑ i , j = 1 k + 1 x i ⁢ x j ⁢ y i ⁢ y j .

Since

∑ i , j = 1 k + 1 x i ⁢ x j ⁢ y i ⁢ y j ≤ r k 2 ⁢ ∑ i = 1 k + 1 y i 2 ,

we obtain

1 2 ⁢ | y | 2 - y ⋅ D ⁢ h ⁢ y ≥ ( 1 2 - φ ⁢ ( r k ) + r k - k - 1 N - k ) ⁢ ∑ i = 1 k + 1 y i 2 + ( 1 2 - 1 N - k ) ⁢ ∑ i = k + 2 N y i 2 .

As 1≤rk≤2⁢η+1 for any x∈Ω=𝕋k⁢(η), we have

1 2 - φ ⁢ ( r k ) + r k - k - 1 N - k ≥ 1 2 - 1 N - k > 0 ,

for all 1≤k<N-2. Then, we get (2.7) with ch=12-1N-k. We conclude that for k<N-2, the domain Ω is a Dirichlet h-starlike domain, non star-shaped and μ⁢(h)≥N-k-22⁢(N-k).

In order to estimate M*⁢(Ω), we now focus on inequality (2.4). For any v∈H02⁢(Ω),

D ⁢ h ⋅ ∇ 2 ⁡ v = ∑ i , j = 1 N ∂ ⁡ h i ∂ ⁡ x j ⁢ ∂ 2 ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ x j = 1 N - k ⁢ { Δ ⁢ v - k k + 1 ⁢ ∑ i = 1 k + 1 ∂ 2 ⁡ v ∂ ⁡ x i 2 } + ∑ i , j = 1 k + 1 a i ⁢ j ⁢ ∂ 2 ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ x j

with

a i ⁢ j = 1 N - k ⁢ ( x i ⁢ x j r k k + 3 - δ i ⁢ j ( k + 1 ) ⁢ r k k + 1 )   for all ⁢ 1 ≤ i , j ≤ k + 1 .

Since rk=∥Pk⁢(x)∥≥1 in Ω, there holds

∑ i , j = 1 k + 1 a i ⁢ j 2 = k ( N - k ) 2 ⁢ ( k + 1 ) ⁢ r k 2 ⁢ k + 2 ≤ k ( N - k ) 2 ⁢ ( k + 1 )   in ⁢ Ω .

By the Cauchy–Schwarz inequality, we have

∫ Ω ( ∑ i , j = 1 k + 1 a i ⁢ j ⁢ ∂ 2 ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ x j ) 2 ⁢ 𝑑 x ≤ k ( N - k ) 2 ⁢ ( k + 1 ) ⁢ ∫ Ω ∑ i , j = 1 k + 1 ( ∂ 2 ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ x j ) 2 ⁢ d ⁢ x := k ( N - k ) 2 ⁢ ( k + 1 ) ⁢ A ⁢ ( v ) .

Therefore

On the other hand, using ∇⁡v=0 on ∂⁡Ω and integrating by parts,

∫ Ω ∂ 2 ⁡ v ∂ ⁡ x i 2 ⁢ ∂ 2 ⁡ v ∂ ⁡ x j 2 ⁢ 𝑑 x = ∫ Ω ( ∂ 2 ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ x j ) 2 ⁢ 𝑑 x ≥ 0   for all ⁢ 1 ≤ i , j ≤ N .

Hence

Combining (3.1) and (3.2), we get

∫ Ω Δ ⁢ v ⁢ ( D ⁢ h ⋅ ∇ 2 ⁡ v ) ⁢ 𝑑 x ≤ 1 N - k ⁢ [ ∫ Ω ( Δ ⁢ v ) 2 ⁢ 𝑑 x - k k + 1 ⁢ B ⁢ ( v ) ] + 1 N - k ⁢ k k + 1 ⁢ ∥ Δ ⁢ v ∥ 2 ⁢ B ⁢ ( v ) .

As, for any C>0 and D≥0,

max ℝ + ⁡ ( - C ⁢ x + D ⁢ x ) = D 2 4 ⁢ C ,

we obtain

∫ Ω Δ ⁢ v ⁢ ( D ⁢ h ⋅ ∇ 2 ⁡ v ) ⁢ 𝑑 x ≤ 5 4 ⁢ ( N - k ) ⁢ ∫ Ω ( Δ ⁢ v ) 2 ⁢ 𝑑 x   for all ⁢ v ∈ H 0 2 ⁢ ( Ω ) .

So (2.4) holds with Ch=54⁢(N-k), hence μ¯⁢(h)≤54⁢(N-k) for any k<N.

Finally, for k<N-2,

M * ⁢ ( Ω ) ≥ min ⁡ ( 1 - 4 ⁢ μ ¯ ⁢ ( h ) 2 , μ ⁢ ( h ) ) ≥ min ⁡ ( 1 2 - 5 2 ⁢ ( N - k ) , N - k - 2 2 ⁢ ( N - k ) ) = N - k - 5 2 ⁢ ( N - k ) .

Let τ>0. There exist smooth bounded domains in ℝN (N≥7), homotopic to 𝕊k with 1≤k≤N-6 which are Dirichlet h-starlike, satisfying M*⁢(Ω)>0. Assume that f⁢(u) verifies the growth condition F⁢(s)≤a⁢s⁢f⁢(s) in ℝ with 0<a<M*⁢(Ω). Then there exists no nontrivial solution to problem ((Dτ)).

More precisely, if f⁢(t)=|t|p-1⁢t and k≤N-6, then we get a nonexistence result on 𝕋k⁢(η) for p>N-k+5N-k-5.

Let b>0 and φ∈C2,α⁢(-b,b) such that φ⁢(t)>0 in (-b,b), φ⁢(±b)=0 and φ<0 outside (-b,b). For x=(x1,x¯)∈ℝ×ℝN-1, we set Φ⁢(x):=|x¯|2-φ⁢(x1). Consider the domain

Ω = { x = ( x 1 , x ¯ ) ∈ ℝ × ℝ N - 1 : - b < x 1 < b , Φ ⁢ ( x ) < 0 } .

Moreover, suppose φ⁢(t)=eξ⁢(t) in (-b,b) with limt→±∞⁡t⁢ξ′⁢(t)=-∞ and assume that Ω is a regular domain. As ∇⁡Φ⁢(x)=(-φ′⁢(x1),2⁢x¯)≠0 on ∂⁡Ω, the exterior normal to ∂⁡Ω is given by

n ⁢ ( x ) = ∇ ⁡ Φ | ∇ ⁡ Φ | = 1 | ∇ ⁡ Φ | ⁢ ( - φ ′ ⁢ ( x 1 ) , 2 ⁢ x ¯ ) .

For δ≥0, we consider the vector field

h δ ⁢ ( x ) = ( δ ⁢ x 1 , 1 - δ N - 1 ⁢ x ¯ ) .

We have div⁡hδ≡1 and Δ⁢hδ=0 in Ω for any δ. The condition hδ⋅n≥0 holds on ∂⁡Ω if and only if

2 ⁢ 1 - δ N - 1 ⁢ φ ⁢ ( x 1 ) - δ ⁢ x 1 ⁢ φ ′ ⁢ ( x 1 ) ≥ 0   for all ⁢ x ∈ ∂ ⁡ Ω ,

which is equivalent to

δ ≤ 1 φ ¯ , where ⁢ φ ¯ := max - b < t < b ⁡ ( 1 + N - 1 2 ⁢ t ⁢ φ ′ ⁢ ( t ) φ ⁢ ( t ) ) .

Note that the maximum exists and there holds always 1φ¯≤1. Hence for 0≤δ≤1φ¯, we have hδ∈ℋ⁢(Ω).

Now, for any x∈Ω and y=(y1,…,yN)∈ℝN, we have

1 2 ⁢ | y | 2 - y ⋅ D ⁢ h δ ⁢ ( x ) ⁢ y = ( 1 2 - 1 - δ N - 1 ) ⁢ | y | 2 + 1 - N ⁢ δ N - 1 ⁢ y 1 2 .

So, for N≥3, 0<δ≤δ*:=min⁡(1φ¯,1N), we have

1 2 ⁢ | y | 2 - y ⋅ D ⁢ h δ ⁢ ( x ) ⁢ y ≥ c h δ ⁢ | y | 2   with ⁢ c h := N - 3 + 2 ⁢ δ 2 ⁢ ( N - 1 ) > 0 .

Hence the domain Ω is Dirichlet h-starlike, i.e., ℋ*⁢(Ω)≠∅ and μ⁢(hδ)≥N-3+2⁢δ2⁢(N-1).

If v∈H02⁢(Ω), then

As δ≤1N, it holds δ-1-δN-1≤0. And since ∇⁡v=0 on ∂⁡Ω, we have

∫ Ω v x 1 ⁢ x 1 ⁢ Δ ⁢ v ⁢ 𝑑 x = ∑ i = 1 N ∫ Ω v x 1 ⁢ x i 2 ⁢ 𝑑 x ≥ 0 .

Therefore, we are led to

∫ Ω Δ ⁢ v ⁢ ( ∇ ⁡ h δ ⋅ ∇ 2 ⁡ v ) ⁢ 𝑑 x ≤ 1 - δ N - 1 ⁢ ∫ Ω ( Δ ⁢ v ) 2 ⁢ 𝑑 x ,

so μ¯⁢(hδ)≤1-δN-1. Thus,

M * ⁢ ( Ω ) ≥ sup 0 < δ ≤ δ * ⁡ min ⁡ ( N - 5 + 4 ⁢ δ 2 ⁢ ( N - 1 ) , N - 3 + 2 ⁢ δ 2 ⁢ ( N - 1 ) ) = N - 5 + 4 ⁢ δ * 2 ⁢ ( N - 1 ) .

For N≥5, we have M*⁢(Ω)>0. As φ¯ is an arbitrary function satisfying our assumptions, we can obtain domains with rich geometry as dumbbell shaped or multi-dumbbell shaped domains.