On Liouville theorem and the regularity of weak solutions to some nonlinear elliptic systems of higher order

On Liouville theorem and the regularity of weak solutions to some nonlinear elliptic systems of higher order

L. Balanda, E. Viszus

Abstract. The aim of this paper is to show that Liouville type property is a sufficient and necessary condition for the regularity of weak solutions of nonlinear elliptic systems of the higher order.

Keywords: regularity of weak solutions, nonlinear elliptic systems Classification: 35J60, 35D10

Introduction.

In this paper, we shall deal with nonlinear elliptic systems. More precisely, we shall consider the following problem.

Let Ω be a bounded domain with Lipschitz boundary inRⁿ,n≥2. Let us denote σ(n, k) = ^n+k−1_k

and̺(n, k) = ^n+k_k

,n, k∈N. We shall study the weak solutions u∈[H^k,∞(Ω)]^N to the system

(0.1)

X

|α|≤k

(−1)^|α|D^α(aⁱ_α(x, γ(u))) = X

|α|≤k

(−1)^|α|D^αf_αⁱ, i= 1, . . . , N; x∈Ω, k≥1,

γ(u) ={D^αuⁱ:i= 1, . . . , N; |α| ≤k}.

By a weak solution of (0.1) we mean a functionu∈[H^k(Ω)]^N (H^k(Ω)≡H^k,2(Ω)

— Sobolev space,u= (u¹, . . . , u^N) — see [4]) such that

(0.2)

N

X

i=1

X

|α|≤k

Z

Ω

aⁱ_α(x, γ(u))D^αϕⁱdx=

N

X

i=1

X

|α|≤k

f_αⁱD^αϕⁱdx, ϕ∈[D(Ω)]^N.

The authors are indebted to Professor J. Neˇcas for the very valuable advice concerning the paper

We shall suppose that:

aⁱ_α ∈C¹(Ω×R^{N ̺(n,k)}), i= 1, . . . , N, |α| ≤k, (0.3)

N

X

i,j=1

X

|α|=|β|=k

∂aⁱ_α

∂η^j_β (x, η)ξ_αⁱ ξ_β^j >0, (0.4)

(x, η)∈(Ω×R^{N ̺(n,k)}), ξ∈R^{N σ(n,k)}, ξ6= 0.

f_αⁱ ∈H^1,pα(Ω), pα= p

k− |α|+ 1, where p > n, (0.5)

p≥2(k+ 1), i= 1, . . . , N, |α| ≤k.

Let us denote forM >0, G >0:

[M] ={u∈[H^k,∞(Ω)]^N :u solves (0.1) and kuk_[Hk,∞

(Ω)]^N ≤M}, [G] ={f_αⁱ ∈H^1,pα(Ω) :

N

X

i=1

X

|α|≤k

kf_αⁱk_H¹,pα(Ω)≤G}.

We shall use the notations

γ₁(u) ={D^αuⁱ:i= 1, . . . , N;|α| ≤k−1}, γ₂(u) ={D^αuⁱ:i= 1, . . . , N;|α|=k},

P_m^N ={(P₁, . . . , P_N) :Pi — polynomial with deg(Pi)≤m, i= 1, . . . , N}, m≥0,

B(x⁰, R) ={x∈Rⁿ:|x−x⁰|< R}.

Definition 0.6. The condition (L) of the Liouville type is satisfied for the sys- tem (0.1) if for ∀x⁰ ∈ Ω, ξ ∈ R^{N ̺(n,k−1)}, the only weak solutions in Rⁿ to the system

(0.7) X

|α|=k

(−1)^|α|D^α(aⁱ_α(x⁰, ξ, γ₂(v))) = 0, i= 1, . . . , N

with bounded derivatives ofk-th order are polynomials of at mostk-th degree (i.e.

v∈P_k^N).

Definition 0.8. We say that the system (0.1) has a property of regularity (R) if for∀x⁰ ∈Ω, ξ ∈R^{N ̺(n,k−1)}, M >0 there exist η >0, c >0 and µ∈(0,1) such that every weak solutionu(inRⁿ) of the system (0.7) with|D^αuⁱ| ≤M, |α|=k, i= 1, . . . , N, belongs to the space [C^k,µ(B(0, η))]^N and

kuk_[Ck,µ(B(0,η))]^N ≤c.

It will be proved in this paper that the property (L) implies the interior regularity, i.e. if u is a weak solution to (0.1) then u ∈ [C^k,µ(Ω^′)]^N, where Ω^′ ⊂⊂ Ω, µ ∈ (0,1−ⁿ_p).

It will be also shown that (R)⇒(L).

These results generalize the results of [2], [3]. In [2], [3], the analogous assertions are proved for nonlinear elliptic systems of the second order. The history of the regularity problem and the Liouville’s property is described in [2], [3], [5].

1. Some lemmas.

By standard arguments (see [6]), we could prove

Lemma 1.1. Letu∈[H^k,∞(Ω)]^N be a weak solution to the system(0.1)and let (0.3),(0.4),(0.5)be satisfied. Thenu∈[H_loc^k+1(Ω)]^N.

In our next considerations, we shall use the result from [1]. This result concerns the solutionsu∈[H^k(Ω)]^N ∩[H^k−1,∞(Ω)]^N of quasilinear elliptic systems of the type

N

X

j=1

X

|α|≤k

|β|=k

(−1)^|α|D^α(Aⁱ_α^j_β(x, γ₁(v))D^βv^j) = X

|α|≤k

(−1)^|α|D^αg_αⁱ, (1.2)

x∈Ω, i= 1, . . . , N, with the following assumptions

Aⁱ_α^j_β∈C(Ω×R^{N ̺(n,k−1)}), (1.3)

gⁱ_α∈L^pα(Ω), pα= p

k− |α|+ 1, p > n, p≥2(k+ 1), (1.4)

N

X

i,j=1

X

|α|=|β|=k

Aⁱ_α^j_β(x, ξ)ζ_αⁱ ζ_β^j >0, (1.5)

for allx∈Ω,ξ∈R^{N ̺(n,k−1)}, ζ∈R^{N σ(n,k)},ζ6= 0.

If we denote

[M^′] ={u∈[H^k(Ω)]^N∩[H^k−1,∞(Ω)]^N :u is a solution to (1.2) and u_[Hk−1,∞(Ω)]^N ≤M^′}, M^′>0, [G^′] ={g_αⁱ ∈L^pα(Ω) :

N

X

i=1

X

|α|≤k

kgⁱ_αk_Lpα(Ω)≤G^′}, G^′>0,

A^′ = sup

|ξ|≤M^′ x∈Ω

{ X

i,j,α,β

|A^α_i^β_j(x, ξ)|},

U(x⁰, R) =R⁻ⁿ Z

B(x⁰,R) N

X

i=1

X

|α|=k−1

|D^αuⁱ(x)−(D^αuⁱ)_x⁰_,R|² dx,

where u ∈ [H^k−1(Ω)]^N, (D^αuⁱ)_x⁰_,R means the integral mean value D^αuⁱ in B(x⁰, R), we shall state the following

Lemma 1.6. Suppose that u∈ [M^′] and the right-hand sides of the system 1.2 belong to [G^′]. Let (1.3),(1.4),(1.5) be satisfied. Let Ω^′ be a domain such that Ω^′⊂⊂Ω. Let

(1.7) lim

R→0⁺ inf U(x, R) = 0

uniformly with respect tox∈Ω^′, u∈[M^′]. Thenu∈[C^k−1,µ(Ω^′)]^N, µ∈(0,1−ⁿ_p) and the a-priori estimate

kuk_[Ck−1,µ(Ω^′)]^N ≤c(M^′, G^′, A^′,Ω^′), c >0, holds uniformly with respect to the class[M^′]∪[G^′].

This lemma (in a slightly generalized form) is proved in [1]. In [1], the problems analogous to those in this paper are solved for quasilinear elliptic systems of higher order. In both papers, the methods of proofs are based on the same idea. The crucial point is to show that the Liouville property implies the assumption (1.7).

But the methods are technically different.

2. Main results.

Theorem 2.1. Let u∈ [M] and the right-hand sides of the system (0.1) belong to[G]. LetΩ^′ be a domain such thatΩ^′⊂⊂Ω. Suppose that(0.3),(0.4),(0.5)and the condition (L)be satisfied. Then there exists a constant c =c(Ω^′, M, G) such that

kuk_[Ck,µ(Ω^′)]^N ≤c, µ∈(0,1−n p).

Proof: For all x⁰ ∈ Ω^′ and R > 0, we define the transformation T_x⁰_R : y = T_x⁰_R(x) =^x−x_R⁰. Foru∈[M] (u= (u¹, . . . , u^N)), we define onO_x⁰_R=T_x⁰_R(Ω) :

uⁱ_x0R(y) = uⁱ(x⁰+R y)

R^k − X

|γ|<k

D^γuⁱ(x⁰) R^k−|γ|γ! y^γ, (2.2)

i= 1, . . . , N.

From (2.2), it follows that fori= 1, . . . , N

(2.3)



















D^αuⁱ_x0R(0) = 0, |α| ≤k−1,

D^αuⁱ(x⁰Ry) =R^k−|α|D^αuⁱ_x0R(y) + P

|γ|<k α≤γ

R^|γ−α|Bγ,αD^γuⁱ(x⁰) γ! y^γ−α,

|α| ≤k, D^αuⁱ(x⁰+Ry) =D^αuⁱ_x0R(y) a.e. in O_x0R, |α|=k.

Bγ,α — constants which are related to the derivatives of “y^γ”. Let us choose a numbera >0. Then there existsR₀>0 such that for∀x⁰∈Ω^′ and 0< R≤R₀ B(0,2a) ⊂ O_x⁰_R. From (2.3) it follows that D^αuⁱ_x0R,|α| ≤ k, i = 1, . . . , N, are bounded uniformly with respect to x⁰ ∈Ω^′ and 0< R≤R₀. Clearly there exists a constantt >0 such that for allx⁰∈Ω^′ and 0< R≤R₀

(2.4) ku_x⁰_Rk_[Hk(B(0,2a))]^N ≤t.

PuttingR^kϕⁱ(^x−x_R⁰)∈ D(Ω) in (0.2) as a test function and using the transformation x=x⁰+Ry we have

(2.5)

N

X

i=1

X

|α|≤k

Z

O_x0R

R^k−|α|aⁱ_α(x⁰+Ry, γ₁(u(x⁰+Ry)), γ2(u(x⁰+Ry)))D^αϕⁱ(y)dy=

=

N

X

i=1

X

|α|≤k

Z

O_x0R

R^k−|α|f_αⁱ(x⁰+Ry)D^αϕⁱ(y)dy.

From Lemma 1.1 it follows thatu_x⁰_R∈[H_loc^k+1(O_x⁰_R)]^N and forv_x⁰_R=^∂u_∂y^x0R

r , the following equation in variations holds:

(2.6)

N

X

i=1

X

|α|≤k

Z

O_x0R

R^k+1−|α|∂aⁱ_α

∂x_r ·D^αϕⁱ(y)dy+

+

N

X

i,j=1

X

|α|≤k

|β|≤k

Z

O_x0R

R^k−|α|∂aⁱ_α

∂η^j_β ·

R^k−|β|D^βv_x^j0R(y)+

+ X

|γ|<k β+λr≤γ

R^|γ−β|B_γ,β+λrD^γu(x⁰)

γ! y^{γ−β−λr}

D^αϕⁱ(y)dy=

=

N

X

i=1

X

|α|≤k

Z

O_x0R

R^k+1−|α|∂f_αⁱ

∂x_r(x⁰+Ry)D^αϕⁱ(y)dy.

(We omitted the arguments in ^∂a_∂x^iα

r,^∂aiα

∂η^j_β, λr= (0, . . . ,0,1,0, . . . ,0)∈Rⁿ.) From (0.5) it follows that

(2.7) R^k+1−|α|k∂f_αⁱ

∂x_r (x⁰+Ry)k_L²_(B(0,2a)) ≤c₁(a)R

p−n pα ·G.

Now putting in (2.6) ϕⁱ = vⁱ_x0Rχ^2k, i = 1, . . . , N, where χ ∈ D(B(0,2a)),0 ≤ χ ≤ 1, χ = 1 on B(0, a) and using the standard argumentation ((0.4), uniform

boundedness of ^∂a_∂x^iα

r,^∂aiα

∂η^j_β, H¨older inequality and (2.7)), we obtain the estimates

J :=

N

X

i=1

X

|α|=k

Z

B(0,2a)

(D^αv_xⁱ0R·χ^k)²dy≤

≤c₂

N

X

i,j=1

X

|α|=|β|=k

Z

B(0,2a)

∂aⁱ_α

∂η^j_β(D^βv^j_x0R·χ^k) (D^αv_xⁱ0R·χ^k)dy andJ ≤c₃J¹² +c₄, wherec₃, c₄>0 are some constants.

The last inequality implies that there exists a constantc₅>0 such that

(2.8)

N

X

i=1

X

|α|=k

Z

B(0,a)(D^αv_xⁱ0R(y))²dy≤J≤c₅

for all x⁰ ∈ Ω^′, u ∈ [M], R ∈ (0, R0]. (2.4) and (2.8) imply that there exists a constantc₆ such that

(2.9) ku_x⁰_Rk_[Hk+1(B(0,a))]^N ≤c₆ for allx⁰∈Ω^′, u∈[M] andR∈(0, R₀].

Now we shall prove that

(2.10) lim inf

R→0⁺ N

X

i=1

X

|α|=k

R⁻ⁿ Z

B(x⁰,R)

|D^αuⁱ(x)−(D^αuⁱ)_x⁰_,R|²dx

= 0

holds uniformly with respect tox⁰ ∈Ω^′ andu∈[M]. Let us suppose the contrary.

Then there exists{x^s}^∞_s=1 ⊂Ω^′, x^s →x∈Ω^′,{Rs}^∞_s=1 ⊂R⁺, R_s →0,{us}^∞_s=1 ⊂ [M] andε >0 such that

(2.11)

N

X

i=1

X

|α|=k

R⁻ⁿ_s Z

B(x^s,Rs)|D^αuⁱ_s(x)−(D^αuⁱ_s)x^s,Rs|²dx≥ε.

Putting a = m, m ∈ N (N = {1,2, . . .}) and using (2.9) and the diagonalization process, we obtain a functionP∈[H^k+1(B(0, m))]^N such that for allm∈N:

u_sx^s_Rs →P in [H^k+1(B(0, m))]^N weakly, (2.12)

u_sx^s_Rs →P in [H^k(B(0, m))]^N, (2.13)

D^αuⁱ_sxsRs →D^αPⁱ a.e. in B(0, m), (2.14)

|α| ≤k, i= 1, . . . , N.

(2.3), (2.14) imply that there exists a constantτ >0 such that (2.15) |D^αPⁱ| ≤τ, |α|=k, i= 1, . . . , N.

Now let ψ ∈ [D(Rⁿ)]^N. It is clear that there exist m, R₁ such that suppψ ⊂ B(0, m)⊂O_x0Rfor allx⁰∈Ω^′ and 0< R≤R₁.

Puttingϕ=ψ in (2.5), we have (2.16)

N

X

i=1

X

|α|≤k

Z

B(0,m)

R^k−|α|s aⁱ_α(x^s+Rsy, γ₁(us(x^s+Rsy)), γ₂(us(x^s+Rsy)))·

·D^αψⁱ(y)dy=

=

N

X

i=1

X

|α|≤k

Z

B(0,m)

R_s^k−|α|sf_αⁱ(x^s+R_sy)D^αψⁱ(y)dy.

For|α|< k,i= 1, . . . , N

|R^k−|α|s aⁱ_αD^αψⁱ| ≤c₇ R^k−|α|s →0, if s→ ∞.

Using the Lebesgue’s dominated convergence theorem, we have

(2.17) lim

s→∞

Z

B(0,m)

R^k−|α|s aⁱ_αD^αψⁱdy= 0.

From the imbeddingH^k,∞(Ω)֒→H^k,p(Ω)֒→֒→C^k−1(Ω) it follows that γ₁(us(x^s+Rsy))→γ₁(P(x)), in B(0, m), s→ ∞,

γ₂(us(x^s+Rsy)) =γ₂(u_sx^s_Rs(y))→γ₂(P(y)) a.e. in B(0, m), s→ ∞.

Then fori= 1, . . . , N,|α|< k, the continuity ofaⁱ_α and Lebesgue’s theorem imply

(2.18)

s→∞lim Z

B(0,m)

aⁱ_α(x^s+R_sy, γ₁(us(x^s+R_sy)), γ₂(us(x^s+R_sy)))·

·D^αψⁱ(y)dy=

= Z

B(0,m)

aⁱ_α(x, γ₁(P(x)), γ₂(P(y)))D^αψⁱ(y)dy.

Now fori= 1, . . . , N, if|α|=k, thenpα=pandH^1,p(Ω)֒→֒→C(Ω). Using this fact and Lebesgue’s theorem, we have (k≥1)

(2.19) lim

s→∞

Z

B(0,m)

sf_αⁱ(x^s+R_sy)D^αψⁱ(y)dy=f_αⁱ(x) Z

B(0,m)

D^αψⁱ(y)dy= 0.

If|α|< k, pα> nthen using the same argument we have

(2.20) lim

s→∞

Z

B(0,m)

R^k−|α|s sf_αⁱ(x^s+Rsy)D^αψⁱ(y)dy= 0.

If|α|< k, pα=n, thenH^1,pα(Ω)֒→L^q(Ω) for allq≥1.

Choosingqso thatk− |α| −ⁿ_q >0 and using H¨older inequality, we obtain

(2.21) Z

B(0,m)

R^k−|α|s sf_αⁱ(x^s+Rsy)D^αψⁱ(y)dy

≤c₈R^k−|α|−

n

s qk^sf_αⁱk_L^q_(Ω)≤

≤c₉R^k−|α|−

n

s q ·G→0 for s→ ∞.

If|α|< k, p_α< n, thenH^1,pα(Ω)֒→L^qα(Ω), where _q¹_α =_p¹_α −_n¹. Using H¨older inequality, we have

(2.22)

Z

B(0,m)

R^k−|α|s sf_αⁱ(x^s+Rsy)D^αψⁱ(y)dy ≤

≤c₁₀R(k−|α|+1)(1−ⁿ_p)

s · k^sf_αⁱk_Lqα(Ω)≤

≤c₁₁R(k−|α|+1)(1−ⁿ_p)

s ·G→0 for s→ ∞.

These facts imply that

N

X

i=1

X

|α|=k

Z

Rⁿ

aⁱ_α(x, γ₁(P(x)), γ₂(P(y)))D^αψⁱ(y)dy= 0 for all ψ∈[D(Rⁿ)]^N.

From (2.15) and the condition (L) it follows thatP ∈P_k^N. (2.3), (2.11) and (2.13) imply

ε≤lim inf

s→∞ R⁻ⁿ_s

N

X

i=1

X

|α|=k

Z

B(x^s,Rs)

|D^αuⁱ_s(x)−(D^αuⁱ_s)_x^s_,Rs|²dx≤

≤lim inf

s→∞

N

X

i=1

X

|α|=k

Z

B(0,1)

|D^αu_sxsRs(y)−D^αPⁱ|²dy= 0.

From this it follows that (2.10) holds uniformly with respect tox⁰∈Ω^′andu∈[M].

Now from (0.2), we obtain an equation in variations which has the following form

(2.23)

N

X

i,j=1 n

X

s,t=1

X

|α|≤k

|β|=k

Z

Ω

δ_st∂aⁱ_α

∂η_β^j(x, γ(u))D^β(∂u^j

∂xt)D^αϕⁱ_sdx=

=

N

X

i=1 n

X

s=1

X

|α|≤k

Z

Ω

h∂f_αⁱ

∂x_s −∂aⁱ_α

∂x_s(x, γ(u))−

−

N

X

j=1

X

|β|<k

∂aⁱ_α

∂η^j_β(x, γ(u))D^β(∂u^j

∂xs)i

D^αϕⁱ_sdx, whereδst— the symbol of Kronecker delta.

Putting (2.24) gsⁱ

α=∂f_αⁱ

∂xs −∂aⁱ_α

∂xs(x, γ(u))−

N

X

j=1

X

|β|<k

∂aⁱ_α

∂η_β^j(x, γ(u))D^β(∂u^j

∂xs) and using the fact thatu∈[M], we obtain the assertion:

There exists a constantG^′>0 such that (2.25)

N

X

i=1 n

X

s=1

X

|α|≤k

kgsⁱ

αk_Lpα(Ω)≤G^′. Fori, j= 1, . . . , N;s, t= 1, . . . , n; |α| ≤k,|β|=k, we define

As^ij

αβt(x, κ) =δst∂aⁱ_α

∂η^j_β(x, γ1(u), κ), x∈Ω, κ∈RN n̺(n,k−1), κ∈R^{N σ(n,k)}.

It is clear that (u∈[C^k−1(Ω)]^N)

(2.26) As^ij

αβt∈C(Ω×RN n̺(n,k−1)).

PuttingU_t^j = ^∂u_∂x^j

t forj = 1, . . . , N, t= 1, . . . , n in (2.23), we obtain a quasilinear system

(2.27)

N

X

i,j=1 n

X

s,t=1

X

|α|≤k

|β|=k

Z

Ω

As^ij

αβt(x, γ₁(U))D^βU_t^jD^αϕⁱ_sdx=

=

N

X

i=1 n

X

s=1

X

|α|≤k

Z

Ω

gsⁱ

αD^αϕⁱ_sdx.

It is a matter of routine calculation to verify the assumptions of Lemma 1.6 for the system (2.27). (These assumptions hold uniformly with respect to allu∈[M].)

Now from Lemma 1.6 it follows that there exists a constantc >0 such that for u∈[M], u∈[C^k,µ(Ω^′)]^N andkuk_Ck,µ(Ω^′)]^N ≤c.

By the standard method from [2], we shall prove

Theorem 2.28. Suppose that the system(0.1)has the property of regularity(R).

Then Liouville’s property(L)holds.

Proof: Letx⁰ ∈Ω, ξ∈R^{N ̺(n,k−1)}and ube a solution (inRⁿ) to the system

(2.29)

N

X

i=1

X

|α|=k

Z

Rⁿ

aⁱ_α(x⁰, ξ, γ₂(u(x)))D^αϕⁱ(x)dx= 0, ϕ∈[D(Rⁿ)]^N,

such that forM >0

(2.30) |D^αuⁱ| ≤M, |α|=k, i= 1, . . . , N.

ForR >0, we define

uⁱ_R(y) = uⁱ(Ry)

R^k , i= 1, . . . , N.

Putting ϕ(_R^x) as a test function in (2.29) and using transformation x = Ry, we obtain

N

X

i=1

X

|α|=k

Z

Rⁿ

aⁱ_α(x⁰, ξ, γ₂(u_R(y)))D^αϕⁱ(y)dy= 0.

(2.30) and the property (R) imply

(2.31) |D^αuⁱ_R(y)−D^αuⁱ_R(0)| ≤c|y|^µ,

|α|=k, i= 1, . . . , N, R >0, y∈B(0, η), η >0, µ∈(0,1).

Now let us choosex∈Rⁿ. Then there existsR₀ >0 such thaty_R=_R^x ∈B(0, η) for allR≥R₀. Using (2.31), we obtain

|D^αuⁱ(x)−D^αuⁱ(0)| ≤c|x|^µ

R^µ, |α|=k, R≥R₀, i= 1, . . . , N.

ForRtending to infinity, we have thatu∈P_k^N.

References

[1] Balanda L., Viszus E.,On Liouville theorem and H¨older continuity of weak solutions to some quasilinear elliptic systems of higher order, to appear.

[2] Giaquinta M., Neˇcas J.,On the regularity of weak solutions to non-linear elliptic systems of partial differential equations, J. Reine Angew. Math.316(1980), 140–159.

[3] Giaquinta M., Neˇcas J.,On the regularity of weak solutions to nonlinear elliptic systems via Liouville’s type property, Comment. Math. Univ. Carolinae20(1979), 111–122.

[4] Fuˇc´ık S., John O., Kufner A.,Function Spaces, Academia, Prague, 1977.

[5] Neˇcas J.,Introduction to the theory of non-linear elliptic equations, Teubner–Texte zur Ma- thematik, Leipzig, 1983.

[6] Neˇcas J.,Les m´ethodes directes en Th´eorie des ´equations elliptiques, Academia, Prague, 1967.

St´atn´ı v´yzkumn´y ´ustav pro stavbu stroj˚u, 190 11 Praha 9 – Bˇechovice, Czechoslovakia

Faculty of Mathematics and Physics, Comenius University, Mlynsk´a dolina, 842 15 Bratislava, Czechoslovakia

(Received May 31, 1991)