Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

Abdellaziz Harrabi; Belgacem Rahal

doi:10.1515/anona-2016-0168

Artikel Open Access

Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

Abdellaziz Harrabi und Belgacem Rahal

Veröffentlicht/Copyright: 20. Dezember 2016

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Advances in Nonlinear Analysis Band 8 Heft 1

Abstract

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

Keywords: Liouville theorem; stable solutions; nonlinear boundary value condition; stability outside a compact set; monotonicity formula

MSC 2010: 35J55; 35J65; 35B33; 35B65.

1 Introduction and main results

In this paper, we study the nonexistence of stable solutions for the following nonlinear Neumann mixed boundary value problem:

(1.1) { - Δ ⁢ u = | u | p - 1 ⁢ u in ⁢ ℝ + n , ∂ ⁡ u ∂ ⁡ ν = | u | q - 1 ⁢ u on ⁢ Γ 1 , ∂ ⁡ u ∂ ⁡ ν = 0 on ⁢ Γ 0 ,

as well as the nonlinear Dirichlet–Neumann mixed boundary value problem

(1.2) { - Δ ⁢ u = | u | p - 1 ⁢ u in ⁢ ℝ + n , ∂ ⁡ u ∂ ⁡ ν = | u | q - 1 ⁢ u on ⁢ Γ 1 , u = 0 on ⁢ Γ 0 ,

where n ≥ 1 ,

ℝ + n = { x = ( x 1 , … , x n ) ∈ ℝ n : x n > 0 } ,

Γ 1 = { x = ( x 1 , … , x n ) ∈ ℝ n : x n = 0 , x 1 < 0 } ,

Γ 0 = { x = ( x 1 , … , x n ) ∈ ℝ n : x n = 0 , x 1 > 0 } ,

p > 1 and q > 1 .

The main results of this paper will be collected in Theorems 1.5 and 1.9 below and they will be concerned with Liouville-type results for suitable solutions of (1.1) and (1.2).

We define two critical exponents which play an important role in the sequel, namely the classical Sobolev exponent

p s ⁢ ( n ) = { + ∞ if n ≤ 2 , n + 2 n - 2 if n ≥ 3 ,

and the Joseph–Lundgren exponent

p c ⁢ ( n ) = { + ∞ if n ≤ 10 , ( n - 2 ) 2 - 4 ⁢ n + 8 ⁢ n - 1 ( n - 2 ) ⁢ ( n - 10 ) if n ≥ 11 .

Let us recall that Liouville-type theorems and properties of the subcritical case has been extensively studied by many authors. The first Liouville theorem was proved by Gidas and Spruck in [4], in which they proved that, for 1 < p < p s ⁢ ( n ) , the following equation does not possess positive solutions:

(1.3) - Δ ⁢ u = | u | p - 1 ⁢ u in ⁢ ℝ n ,

Moreover, it was also proved that the exponent p s ⁢ ( n ) is optimal, in the sense that problem (1.3), indeed, possesses a positive solution for p ≥ p s ⁢ ( n ) and n ≥ 3 . So the exponent p s ⁢ ( n ) is usually called the critical exponent for problem (1.3). Soon afterward, similar results were established in [5] for positive solutions of the subcritical problem in the upper half-space ℝ + n :

(1.4) { - Δ ⁢ u = | u | p - 1 ⁢ u in ℝ + n , u = 0 on ∂ ⁡ ℝ + n .

Later, Chen and Li [2] obtained similar nonexistence results for the above two equations by using the moving plane method.

On the other hand, we note that the above-mentioned results only claim that the above equations do not possess positive solutions. A natural question is to understand more about the sign-changing solutions. In [1], Bahri and Lions proved that when p < p s ⁢ ( n ) , no sign-changing solution with finite Morse index exists for (1.3) and (1.4). To prove this result, they first deduced some integrable conditions on the solution based on finite Morse index; then they used the Pohozaev identity to prove the nonexistence result. After this work, there were many extensions on similar problems. For example, Harrabi, Rebhi and Selmi extended these results to more general nonlinear problems in [7, 8], see also [6]. The finite Morse index solutions to the corresponding nonlinear problems (1.3) and (1.4) have been completely classified by Farina [3]. One main result of [3] is that nontrivial finite Morse index solutions to (1.3) exist if and only if p ≥ p c ⁢ ( n ) and n ≥ 11 , or p = p s ⁢ ( n ) and n ≥ 3 .

On the other hand, elliptic equations with nonlinear boundary value conditions and finite Morse index of the form

(1.5) { - Δ ⁢ u = | u | p - 1 ⁢ u in ℝ + n , ∂ ⁡ u ∂ ⁡ ν = | u | q - 1 ⁢ u on ∂ ⁡ ℝ + n ,

was examined in [11]. It was shown that there is no nontrivial bounded solution of (1.5) with finite Morse index, provided

(1.6) 1 < p ≤ p s ⁢ ( n ) , 1 < q ≤ n n - 2 , ( p , q ) ≠ ( p s ⁢ ( n ) , n n - 2 ) , n ≥ 3 .

Recently, a question was raised as to whether or not problems (1.1) and (1.2) admit sign-changing solutions. A partial answer came from [10] by assuming additionally that solutions have finite Morse indices. Now, we state this result as follows.

Theorem 1.1 ([10]).

If p and q satisfy (1.6), then problems (1.1) and (1.2) do not possess nontrivial bounded solutions with finite Morse index.

The aim of this paper is to study the nonexistence result for C 2 -solutions for problems (1.1) and (1.2) belonging to one of the following classes: stable solutions and solutions which are stable outside a compact set. In order to prove our results, first we deduce suitable a priori estimates for stable solutions of equations (1.1) and (1.2), which are enough for the subcritical case p < p s ⁢ ( n ) . Next, in the supercritical case, i.e., p > n + 2 n - 2 , motivated by the monotonicity formula, we will prove the nonexistence of nontrivial solutions which are stable outside a compact set. Furthermore, our approach permits to generalize also the results in [11].

In order to state our results, we need to recall the following definition.

Definition 1.2.

We say that a solution u of (1.1), belonging to C 2 ⁢ ( ℝ + n ¯ ) ,

is stable if

Q u ( ψ ) := ∫ ℝ + n | ∇ ψ | 2 - q ∫ Γ 1 | u | q - 1 ψ 2 - p ∫ ℝ + n | u | p - 1 ψ 2 ≥ 0 for all ψ ∈ C c 1 ( ℝ + n ¯ ) ,
has Morse index equal to K ≥ 1 if K is the maximal dimension of a subspace X K of C c 1 ⁢ ( ℝ + n ¯ ) such that Q u ⁢ ( ψ ) < 0 for any ψ ∈ X K ∖ { 0 } ,
is stable outside a compact set 𝒦 ⊂ ℝ + n if Q u ⁢ ( ψ ) ≥ 0 for any ψ ∈ C c 1 ⁢ ( ℝ + n ¯ ∖ 𝒦 ) .

Similarly, if we say that a solution u of (1.2) belongs to C 2 ⁢ ( ℝ + n ¯ ) is stable (respectively, stable outside a compact set 𝒦 ), if Q u ⁢ ( ψ ) ≥ 0 for all ψ ∈ C c 1 ⁢ ( ℝ + n ∪ Γ 1 ) (respectively, ψ ∈ C c 1 ⁢ ( ℝ + n ∪ Γ 1 ∖ 𝒦 ) ).

Remark 1.3.

Clearly, a solution is stable if and only if its Morse index is equal to zero.
It is well know that any finite Morse index solution u is stable outside a compact set 𝒦 ⊂ Ω . Indeed, there exist K ≥ 1 and X K := Span ⁢ { ϕ 1 , … , ϕ K } ⊂ C c 1 ⁢ ( Ω ) such that Q u ⁢ ( ϕ ) < 0 for any ϕ ∈ X K ∖ { 0 } . Hence, Q u ⁢ ( ψ ) ≥ 0 for every ψ ∈ C c 1 ⁢ ( Ω ∖ 𝒦 ) , where 𝒦 := ⋃ j = 1 K supp ⁡ ( ϕ j ) , Ω = ℝ + n ¯ or Ω = ℝ + n ∪ Γ 1 .

In the following, we state Liouville-type results for solutions u ∈ C 2 ⁢ ( ℝ + n ¯ ) of (1.1) and (1.2). In what follows, we divide our study to stable solutions and solutions which are stable outside a compact set.

1.1 Stable solutions

To state the following result we need to introduce some notation. We set H ⁢ ( t ) = 2 ⁢ t - 1 + 2 ⁢ t ⁢ ( t - 1 ) , and denote by B R the open ball centered at the origin and with radius R.

Proposition 1.4.

Let u ∈ C 2 ⁢ ( R + n ¯ ) be a stable solution of (1.1) or (1.2). Then, for any α ∈ [ 1 , H ⁢ ( min ⁡ ( p , q ) ) ) , there exists a constant C > 0 such that

(1.7) ∫ B R ∩ ℝ + n ( | u | p + α + | ∇ ( | u | α - 1 2 u ) | 2 ) + ∫ Γ 1 ∩ B R | u | q + α ≤ C R n - 2 ⁢ p + α p - 1 for all R > 0 .

Proposition 1.4 provides an important estimate on the integrability of u and ∇ ⁡ u . As we will see, our nonexistence results will follow by showing that the right-hand side of (1.7) vanishes under the right assumptions on p when R → + ∞ . More precisely, as a corollary of Proposition 1.4, we can state our first Liouville-type theorem.

Theorem 1.5.

Let u ∈ C 2 ⁢ ( R + n ¯ ) be a stable solution of (1.1) or (1.2).

If 1 < q < p < n + 2 ⁢ H ⁢ ( q ) n - 2 , then u ≡ 0 .
If p ≤ q and 1 < p < p c ⁢ ( n ) , then u ≡ 0 .

1.2 Solutions which are stable outside a compact set

Next we consider the case of solutions of (1.1) and (1.2), which are stable outside a compact set. Recall that Wang and Zheng in [10] have classified all bounded finite Morse index solutions of (1.1) and (1.2) for ( p , q ) satisfying (1.6). The main goal of this paper is to classify all (positive or sign-changing) solutions of (1.1) and (1.2) which are stable outside a compact set in the supercritical case, under some assumptions on the exponents p and q. To this end, we first introduce the following proposition.

Proposition 1.6.

Let u ∈ C 2 ⁢ ( R + n ¯ ) be a solution of (1.1) or (1.2) that is stable outside a compact set. Then, for any α ∈ [ 1 , H ⁢ ( min ⁡ ( p , q ) ) ) , there exists a constant C > 0 such that

∫ B R ∩ ℝ + n ( | u | p + α + | ∇ ( | u | α - 1 2 u ) | 2 ) + ∫ Γ 1 ∩ B R | u | q + α ≤ C ( 1 + R n - 2 ⁢ p + α p - 1 ) for all R > 0 .

Thanks to Proposition 1.6, we obtain the following corollary.

Corollary 1.7.

Let n ≥ 3 , and let u ∈ C 2 ⁢ ( R + n ¯ ) be a solution of (1.1) or (1.2) that is stable outside a compact set. If 1 < q < p < n + 2 ⁢ H ⁢ ( q ) n - 2 . Then, there exists α ∈ [ 1 , H ⁢ ( q ) ) such that

∫ ℝ + n ( | u | p + α + | ∇ ( | u | α - 1 2 u ) | 2 ) + ∫ Γ 1 | u | q + α < ∞ .

When attempting to prove the nonexistence of nontrivial stable solutions outside a compact set of (1.1) or (1.2), in the supercritical case when q ≥ p , we need first to establish the following version of monotonicity formula.

1.3 Monotonicity formula of equation (1.1) and (1.2)

The monotonicity formula is a powerful tool to understand supercritical elliptic equations or systems. This approach has been used successfully for the Lane–Emden equation in [9]. Let us first describe the monotonicity formula, which plays a central role in this work. Equation (1.1) or (1.2) has two important features. It is variational, with the energy functional given by

∫ ( 1 2 | ∇ u | 2 - 1 p + 1 | u | p + 1 ) - 1 q + 1 ∫ | u | q + 1 .

Under the scaling transformation

u λ ⁢ ( x ) = λ 2 p - 1 ⁢ u ⁢ ( λ ⁢ x ) ,

this suggests that the variations of the rescaled energy

∫ B 1 ∩ ℝ + n ( 1 2 | ∇ u λ | 2 - 1 p + 1 | u λ | p + 1 ) - 1 q + 1 λ 1 - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 ,

with respect to the scaling parameter λ, are meaningful.

Proposition 1.8.

Let u ∈ C 2 ⁢ ( R + n ¯ ) be a solution of equation (1.1) or (1.2) and λ > 0 a constant. Let also

(1.8) E ( u , λ ) = ∫ B 1 ∩ ℝ + n ( 1 2 | ∇ u λ | 2 - 1 p + 1 | u λ | p + 1 ) - λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 + 1 p - 1 ∫ ∂ ⁡ B 1 ∩ ℝ + n | u λ | 2 .

Then

(1.9) d ⁢ E d ⁢ λ = λ ∫ ∂ ⁡ B 1 ∩ ℝ + n ( d ⁢ u λ d ⁢ λ ) 2 d σ + 2 ⁢ q - p - 1 ( p - 1 ) ⁢ ( q + 1 ) λ - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 d x ′ for all p , q > 1 .

Furthermore, E is a nondecreasing function of λ if 2 ⁢ q - p - 1 ≥ 0 .

Now, from the above monotonicity formula, we classify solutions which are stable outside a compact set. To do so, we use again the L p + 1 norm estimates established in Proposition 1.6, and then we show that the blow-down limit u ∞ ⁢ ( x ) = lim λ → ∞ ⁡ λ 2 p - 1 ⁢ u ⁢ ( λ ⁢ x ) exists. Then, by the work of Farina [3], we derive that u ∞ ≡ 0 . Thanks to this, we deduce that lim λ → + ∞ ⁡ E ⁢ ( u , λ ) = 0 . In addition, since u is C 2 , one easily verifies that E ⁢ ( u , 0 ) = 0 . And so, E ⁢ ( u , λ ) ≡ 0 , since E is nondecreasing, which means that d ⁢ E d ⁢ λ ≡ 0 . Thanks to the boundary condition, we readily deduce that such solutions are trivial if p s ⁢ ( n ) < p < p c ⁢ ( n ) and n ≥ 3 .

Theorem 1.9.

Let n ≥ 3 , and let u ∈ C 2 ⁢ ( R + n ¯ ) be a solution of (1.1) or (1.2) that is stable outside a compact set. If p ≤ q and p s ⁢ ( n ) < p < p c ⁢ ( n ) , then u ≡ 0 .

This paper is organized as follows. In Section 2, we give the proof of Proposition 1.4 and Theorem 1.5. Section 3 is devoted to the proof of Propositions 1.6 and 1.8. Finally, in Section 4, we prove Theorem 1.9.

2 The Liouville theorem for stable solutions: proof of Theorem 1.5

In this section we prove all the results concerning the classification of stable solutions, i.e., Proposition 1.4 and Theorem 1.5.

Proof of Proposition 1.4.

The proof follows the main lines of the demonstration of [3, Proposition 4], with small modifications. We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. For any R > 0 , we consider the function ϕ R ∈ C c 2 ⁢ ( ℝ n ) , defined by ϕ R ⁢ ( x ) = h ⁢ ( | x | R ) , x ∈ ℝ n , where h ∈ C c 2 ⁢ ( ℝ ) , 0 ≤ h ≤ 1 , h ≡ 1 in [ - 1 , 1 ] , and h ≡ 0 in ℝ - [ - 2 , 2 ] . The function | u | α - 1 2 ⁢ u ⁢ ϕ R belongs to C c 1 ⁢ ( ℝ + n ¯ ) , and thus it can be used as a test function in the quadratic form Q u . Hence, the stability assumption on u gives

(2.1) p ∫ ℝ + n | u | p + α ϕ R 2 + q ∫ Γ 1 | u | q + α ϕ R 2 ≤ ∫ ℝ + n | ∇ ( | u | α - 1 2 u ϕ R ) | 2 .

A direct calculation shows that, for the right-hand side of (2.1), we have

∫ ℝ + n | ∇ ( | u | α - 1 2 u ϕ R ) | 2 = ∫ ℝ + n ( | u | α + 1 | ∇ ϕ R | 2 + ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 + 1 2 ∇ ϕ R 2 ∇ ( | u | α + 1 ) )

= ∫ ℝ + n | u | α + 1 ( | ∇ ϕ R | 2 - 1 2 Δ ϕ R 2 ) + 1 2 ∫ ∂ ⁡ ℝ + n | u | α + 1 ∂ ⁡ ϕ R 2 ∂ ⁡ ν + ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 .

Since

(2.2) ∂ ⁡ ϕ R ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ ℝ + n ,

it follows that

∫ ℝ + n | ∇ ( | u | α - 1 2 u ϕ R ) | 2 = ∫ ℝ + n ( | u | α + 1 | ∇ ϕ R | 2 + ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 + 1 2 ∇ ϕ R 2 ∇ ( | u | α + 1 ) )

(2.3) = ∫ ℝ + n | u | α + 1 ( | ∇ ϕ R | 2 - 1 2 Δ ϕ R 2 ) + ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 .

From (2.1) and (2.3), we obtain

(2.4) p ∫ ℝ + n | u | p + α ϕ R 2 + q ∫ Γ 1 | u | q + α ϕ R 2 ≤ ∫ ℝ + n | u | α + 1 ( | ∇ ϕ R | 2 - 1 2 Δ ϕ R 2 ) + ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 .

Now, multiply equation (1.1) by | u | α - 1 ⁢ u ⁢ ϕ R 2 , and then integrate by parts to find

α ∫ ℝ + n | ∇ u | 2 | u | α - 1 ϕ R 2 + ∫ ℝ + n ∇ u ∇ ( ϕ R 2 ) | u | α - 1 u - ∫ Γ 1 | u | q + α ϕ R 2 = ∫ ℝ + n | u | p + α ϕ R 2 .

Therefore,

∫ ℝ + n | u | p + α ϕ R 2 = 4 ⁢ α ( α + 1 ) 2 ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 + 1 α + 1 ∫ ℝ + n ∇ ( | u | α + 1 ) ∇ ( ϕ R 2 ) - ∫ Γ 1 | u | q + α ϕ R 2

= 4 ⁢ α ( α + 1 ) 2 ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 - 1 α + 1 ∫ ℝ + n | u | α + 1 Δ ( ϕ R 2 ) + 1 α + 1 ∫ ∂ ⁡ ℝ + n | u | α + 1 ∂ ⁡ ϕ R 2 ∂ ⁡ ν - ∫ Γ 1 | u | q + α ϕ R 2 .

Using (2.2), we obtain

∫ ℝ + n | u | p + α ϕ R 2 = 4 ⁢ α ( α + 1 ) 2 ∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 - 1 α + 1 ∫ ℝ + n | u | α + 1 Δ ( ϕ R 2 ) - ∫ Γ 1 | u | q + α ϕ R 2 .

By multiplying the latter identity by the factor ( α + 1 ) 2 4 ⁢ α , we derive

∫ ℝ + n ϕ R 2 | ∇ ( | u | α - 1 2 u ) | 2 = ( α + 1 ) 2 4 ⁢ α ∫ ℝ + n | u | p + α ϕ R 2 + ( α + 1 ) 2 4 ⁢ α ∫ Γ 1 | u | q + α ϕ R 2 + ( α + 1 ) 4 ⁢ α ∫ ℝ + n | u | α + 1 Δ ( ϕ R 2 ) .

Putting this back into (2.4) gives

( p - ( α + 1 ) 2 4 ⁢ α ) ∫ ℝ + n | u | p + α ϕ R 2 + ( q - ( α + 1 ) 2 4 ⁢ α ) ∫ Γ 1 | u | q + α ϕ R 2 ≤ ∫ ℝ + n | u | α + 1 ( | ∇ ϕ R | 2 + 1 - α 4 ⁢ α Δ ϕ R 2 ) .

For any α ∈ [ 1 , H ⁢ ( min ⁡ ( p , q ) ) ) , we obtain that p - ( α + 1 ) 2 4 ⁢ α > 0 and q - ( α + 1 ) 2 4 ⁢ α > 0 , hence

∫ ℝ + n | u | p + α ϕ R 2 + ∫ Γ 1 | u | q + α ϕ R 2 ≤ C ( p , q , α ) ∫ ℝ + n | u | α + 1 ( | ∇ ϕ R | 2 + | Δ ϕ R 2 | ) .

Now, we replace ϕ R by ϕ R m in the latter inequality and, for any m > 1 , we get

∫ B 2 ⁢ R ∩ ℝ + n | u | p + α ϕ R 2 ⁢ m + ∫ Γ 1 ∩ B 2 ⁢ R | u | q + α ϕ R 2 ⁢ m ≤ C ∫ B 2 ⁢ R ∩ ℝ + n | u | α + 1 ϕ R 2 ⁢ m - 2 ( | ∇ ϕ R | 2 + | Δ ϕ R | )

(2.5) ≤ C ( p , q , α , m ) R - 2 ∫ B 2 ⁢ R ∩ ℝ + n | u | α + 1 ϕ R 2 ⁢ m - 2

and

(2.6) ∫ B 2 ⁢ R ∩ ℝ + n | ∇ ( | u | α - 1 2 u ) | 2 ϕ R 2 ⁢ m ≤ C ( p , q , α , m ) R - 2 ∫ B 2 ⁢ R ∩ ℝ + n | u | α + 1 ϕ R 2 ⁢ m - 2 .

An application of Young’s inequality yields

(2.7) C ( p , q , α , m ) R - 2 ∫ B 2 ⁢ R ∩ ℝ + n | u | α + 1 ϕ R 2 ⁢ m - 2 ≤ C R n - 2 ⁢ p + α p - 1 + α + 1 p + α ∫ B 2 ⁢ R ∩ ℝ + n | u | p + α ϕ R ( 2 ⁢ m - 2 ) ⁢ p + α α + 1 .

If we take m = p + α p - 1 then 2 ⁢ m = ( 2 ⁢ m - 2 ) ⁢ p + α α + 1 and from (2.5)–(2.7), we obtain

∫ B 2 ⁢ R ∩ ℝ + n | u | p + α ϕ R 2 ⁢ m + ∫ B 2 ⁢ R ∩ ℝ + n | ∇ ( | u | α - 1 2 u ) | 2 ϕ R 2 ⁢ m + ∫ Γ 1 ∩ B 2 ⁢ R | u | q + α ϕ R 2 ⁢ m ≤ C R n - 2 ⁢ p + α p - 1 .

This implies

∫ B R ∩ ℝ + n | u | p + α + ∫ B R ∩ ℝ + n | ∇ ( | u | α - 1 2 u ) | 2 + ∫ Γ 1 ∩ B R | u | q + α ≤ C R n - 2 ⁢ p + α p - 1 .

This finishes the proof of Proposition 1.4. ∎

Proof of Theorem 1.5.

(1) By Proposition 1.4, there exists C > 0 such that

(2.8) ∫ B R ∩ ℝ + n | u | p + α ≤ C R n - 2 ⁢ p + α p - 1 .

Under the assumptions of Theorem 1.5, we can always choose α ∈ [ 1 , H ⁢ ( q ) ) such that n - 2 ⁢ p + α p - 1 < 0 . Therefore, by letting R → + ∞ in (2.8), we deduce

∫ ℝ + n | u | p + α = 0 ,

which yields u ≡ 0 in ℝ + n .

(2) By Proposition 1.4, for every α ∈ [ 1 , H ⁢ ( p ) ) , there exists constant C > 0 such that for every R > 0 ,

∫ B R ∩ ℝ + n | u | p + α ≤ C R n - 2 ⁢ p + α p - 1 .

As in Farina’s work we readily deduce, by letting R → + ∞ , that there is no nontrivial stable solution of (1.1) and (1.2), in the special case 1 < p < p c ⁢ ( n ) and q ≥ p . ∎

3 Proof of Propositions 1.6 and 1.8

In this section we are concerned with the proof of Propositions 1.6 and 1.8.

Proof of Proposition 1.6.

We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. We begin by defining some smooth compactly supported functions which will be used several times in the sequel. More precisely, we choose ϕ a , R ∈ C c 2 ⁢ ( ℝ n ) satisfying 0 ≤ ϕ a , R ≤ 1 everywhere on ℝ n and

ϕ a , R ⁢ ( x ) = { 0 for ⁢ | x | < a ⁢ or ⁢ | x | > 2 ⁢ R , 1 for ⁢ 2 ⁢ a < | x | < R

such that | ∇ ⁡ ϕ a , R | ≤ C ⁢ R - 1 and | Δ ⁢ ϕ a , R | ≤ C ⁢ R - 2 for R < | x | < 2 ⁢ R . We can proceed as in the proof of Proposition 1.4. Only some minor modifications are needed: the function | u | α - 1 2 ⁢ u ⁢ ϕ a , R belongs to C c 1 ⁢ ( ℝ + n ¯ ) , and thus it can be used as a test function in the quadratic form Q u . By the stability assumption on u, there exists a 0 > 0 such that Q u ⁢ ( | u | α - 1 2 ⁢ u ⁢ ϕ a 0 , R ) ≥ 0 for any R > 2 ⁢ a 0 . The rest of the proof is unchanged, thus we omit the details. The proof of Proposition 1.6 is thereby completed. ∎

Proof of Proposition 1.8.

For λ > 0 , define the function u λ by

u λ ⁢ ( x ) = λ 2 p - 1 ⁢ u ⁢ ( λ ⁢ x ) for ⁢ x ∈ ℝ + n .

Since u is a solution of (1.1), it follows that u λ satisfies

(3.1) { - Δ ⁢ u λ = | u λ | p - 1 ⁢ u λ in ℝ + n , ∂ ⁡ u λ ∂ ⁡ ν = λ 1 - 2 ⁢ q - 1 p - 1 ⁢ | u λ | q - 1 ⁢ u λ on Γ 1 , ∂ ⁡ u λ ∂ ⁡ ν = 0 on Γ 0 .

Take

(3.2) E ~ ⁢ ( u , λ ) = ∫ B 1 ∩ ℝ + n ( 1 2 ⁢ | ∇ ⁡ u λ | 2 - 1 p + 1 ⁢ | u λ | p + 1 ) ,

hence

(3.3) d d ⁢ λ ⁢ E ~ ⁢ ( u , λ ) = ∫ B 1 ∩ ℝ + n ( ∇ ⁡ u λ ⁢ ∇ ⁡ d ⁢ u λ d ⁢ λ - | u λ | p - 1 ⁢ u λ ⁢ d ⁢ u λ d ⁢ λ ) .

Integrating by parts, we get

d d ⁢ λ E ~ ( u , λ ) = ∫ ∂ ⁡ B 1 ∩ ℝ + n ∂ ⁡ u λ ∂ ⁡ r d ⁢ u λ d ⁢ λ + λ 1 - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B 1 | u λ | q - 1 u λ d ⁢ u λ d ⁢ λ

(3.4) = ∫ ∂ ⁡ B 1 ∩ ℝ + n ∂ ⁡ u λ ∂ ⁡ r ⁢ d ⁢ u λ d ⁢ λ + λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ⁢ ∫ Γ 1 ∩ B 1 d d ⁢ λ ⁢ ( | u λ | q + 1 ) .

In what follows, we express all derivatives of u λ in the r = | x | variable in terms of derivatives in the λ variable. In the definition of u λ , directly differentiating in λ gives

(3.5) λ ⁢ d ⁢ u λ d ⁢ λ = 2 p - 1 ⁢ u λ + r ⁢ ∂ ⁡ u λ ∂ ⁡ r

and

(3.6) λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ∫ Γ 1 ∩ B 1 d ⁢ | u λ | q + 1 d ⁢ λ = d d ⁢ λ ( λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 ) - ( p + 1 - 2 ⁢ q ) ⁢ λ - 2 ⁢ q - 1 p - 1 ( p - 1 ) ⁢ ( q + 1 ) ∫ Γ 1 ∩ B 1 | u λ | q + 1 .

From (3.4), (3.5) and (3.6), we obtain

d d ⁢ λ E ~ ( u , λ ) = λ ∫ ∂ ⁡ B 1 ∩ ℝ + n ( d ⁢ u λ d ⁢ λ ) 2 - 1 p - 1 ∫ ∂ ⁡ B 1 ∩ ℝ + n d ⁢ ( u λ ) 2 d ⁢ λ + d d ⁢ λ ( λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 )

(3.7) + 2 ⁢ q - p - 1 ( p - 1 ) ⁢ ( q + 1 ) λ - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B 1 | u λ | q + 1 .

Exploiting (3.2) and (3.7), we get (1.8) and (1.9).

For problem (1.2), the proof can be obtained similarly, with only some minor modifications. Since u is a solution of (1.2), we have that u λ satisfies

{ - Δ ⁢ u λ = | u λ | p - 1 ⁢ u λ in ℝ + n , ∂ ⁡ u λ ∂ ⁡ ν = λ 1 - 2 ⁢ q - 1 p - 1 ⁢ | u λ | q - 1 ⁢ u λ on Γ 1 , u λ = 0 on Γ 0 .

From (3.3), we get

d d ⁢ λ ⁢ E ~ ⁢ ( u , λ ) = ∫ ∂ ⁡ B 1 ∩ ℝ + n ∂ ⁡ u λ ∂ ⁡ r ⁢ d ⁢ u λ d ⁢ λ + ∫ Γ 1 ∩ B 1 ∪ Γ 0 ∩ B 1 ∂ ⁡ u λ ∂ ⁡ ν ⁢ d ⁢ u λ d ⁢ λ

= ∫ ∂ ⁡ B 1 ∩ ℝ + n ∂ ⁡ u λ ∂ ⁡ r ⁢ d ⁢ u λ d ⁢ λ + λ 1 - 2 ⁢ q - 1 p - 1 q + 1 ⁢ ∫ Γ 1 ∩ B 1 d d ⁢ λ ⁢ ( | u λ | q + 1 ) .

The last line comes from the fact that u λ ≡ 0 in Γ 0 ∩ B 1 for any λ > 0 , hence d ⁢ u λ d ⁢ λ = 0 in Γ 0 ∩ B 1 . The rest of the proof is unchanged, thus we omit the details.

Now, since 2 ⁢ q - p - 1 ≥ 0 , we have that E is a nondecreasing function of λ. This completes the proof of Proposition 1.8. ∎

4 The Liouville theorem for solutions which are stable outside a compact set: proof of Theorem 1.9

Let u be a smooth solution of (1.1) which is stable outside a compact set, q ≥ p and p s ⁢ ( n ) < p < p c ⁢ ( n ) , and n ≥ 3 . From Proposition 1.6 (applied to u on a ball of radius λ ⁢ R ), we know that for a given R > 0 ,

∫ B R ∩ ℝ + n ( | ∇ ⁡ u λ | 2 + | u λ | p + 1 ) ⁢ 𝑑 x ≤ C + C ⁢ R n - 2 ⁢ p + 1 p - 1

and

λ 1 - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B R | u λ | q + 1 ≤ C + C R n - 2 ⁢ p + 1 p - 1 .

So, ( u λ ) λ ≥ 1 is uniformly bounded in H 1 ∩ L p + 1 ⁢ ( B R ∩ ℝ + n ) for any R > 0 , and ( λ p + 1 - 2 ⁢ q ( p - 1 ) ⁢ ( q + 1 ) ⁢ u λ ) λ ≥ 1 is uniformly bounded in L q + 1 ⁢ ( Γ 1 ∩ B R ) for any R > 0 . In particular, a sequence ( u λ j ) converges weakly to some function u ∞ in H 1 ∩ L p + 1 ⁢ ( B R ∩ ℝ + n ) , for every R > 0 , as λ j → + ∞ . Note also that u λ satisfies the equation (3.1). Taking limits in the sense of distributions, it follows that

{ - Δ ⁢ u ∞ = | u ∞ | p - 1 ⁢ u ∞ in 𝒟 ′ ⁢ ( ℝ + n ) , ∂ ⁡ u ∞ ∂ ⁡ ν = 0 on ∂ ⁡ ℝ + n .

Applying [3, Theorem 9], we get u ∞ ≡ 0 .

Let ζ ∈ C c 1 ⁢ ( Ω ~ ) , where Ω ~ = ℝ + n ¯ ∖ B R 0 for R 0 sufficiently large. Denote ∂ ⁡ Ω ~ := Ω ~ ∩ Γ 1 . Multiply equation (1.1) by p ⁢ u ⁢ ζ 2 and then integrate by parts to find

p ∫ Ω ~ ∇ u ⋅ ∇ ( u ζ 2 ) - p ∫ ∂ ⁡ Ω ~ | u | q + 1 ζ 2 = p ∫ Ω ~ | u | p + 1 ζ 2 .

Therefore,

p ∫ Ω ~ ( | ∇ ( u ζ ) | 2 - u 2 | ∇ ζ | 2 ) - p ∫ ∂ ⁡ Ω ~ | u | q + 1 ζ 2 = p ∫ Ω ~ | u | p + 1 ζ 2 .

Since u is stable outside compact, it follows that

( p - 1 ) ∫ Ω ~ | ∇ ( u ζ ) | 2 + ( q - p ) ∫ ∂ ⁡ Ω ~ | u | q + 1 ζ 2 ≤ p ∫ Ω ~ u 2 | ∇ ζ | 2 .

Since q ≥ p , we have

( p - 1 ) ∫ Ω ~ | ∇ ( u ζ ) | 2 ≤ p ∫ Ω ~ u 2 | ∇ ζ | 2 .

Choose now ζ ⁢ ( x ) = ζ 0 ⁢ ( | x | λ ) , where ζ 0 ≡ 0 in B ε / 2 ∩ ℝ + n , ζ 0 ≡ 1 in B 1 ∖ B ε ∩ ℝ + n and ζ 0 ≡ 0 outside B 2 ∩ ℝ + n . Then, for λ > R 0 ε ,

∫ B λ ∖ B ε ⁢ λ ∩ ℝ + n | ∇ u | 2 ≤ C λ - 2 ∫ B 2 ⁢ λ ∩ ℝ + n u 2 .

Scaling back yields

∫ B 1 ∖ B ε ∩ ℝ + n | ∇ u λ | 2 ≤ C ∫ B 2 ∩ ℝ + n | u λ | 2 ,

and so

E 2 ⁢ ( u λ ; 1 ) = ∫ B 1 ∩ ℝ + n ( 1 2 ⁢ | ∇ ⁡ u λ | 2 - 1 p + 1 ⁢ | u λ | p + 1 )

= ∫ B ε ∩ ℝ + n ( 1 2 ⁢ | ∇ ⁡ u λ | 2 - 1 p + 1 ⁢ | u λ | p + 1 ) + ∫ B 1 ∖ B ε ∩ ℝ + n ( 1 2 ⁢ | ∇ ⁡ u λ | 2 - 1 p + 1 ⁢ | u λ | p + 1 )

≤ C ⁢ ε n - 2 ⁢ p + 1 p - 1 ⁢ E 2 ⁢ ( u λ ; ε ) + ∫ B 1 ∖ B ε ∩ ℝ + n ( 1 2 ⁢ | ∇ ⁡ u λ | 2 - 1 p + 1 ⁢ | u λ | p + 1 )

≤ C ( ε n - 2 ⁢ p + 1 p - 1 + ∫ B 2 ∩ ℝ + n | u λ | 2 ) .

Recalling that p > n + 2 n - 2 , i.e., n - 2 ⁢ p + 1 p - 1 > 0 , ( u λ ) converges strongly to u ∞ = 0 in L p + 1 ⁢ ( B R ∩ ℝ + n ) , thus also in L 2 ⁢ ( B R ∩ ℝ + n ) . We conclude, after letting λ → + ∞ and then ε → 0 , that

lim λ → + ∞ ⁡ E 2 ⁢ ( u ; λ ) = 0 .

We claim that the same holds true for E. To see this, simply observe that since E is nondecreasing,

E ⁢ ( u λ , 1 ) = E ⁢ ( u , λ ) ≤ 1 λ ⁢ ∫ λ 2 ⁢ λ E ⁢ ( u , t ) ⁢ 𝑑 t

= 1 λ ∫ λ 2 ⁢ λ E 2 ( u , t ) d t + 1 p - 1 λ - 1 ∫ λ 2 ⁢ λ ∫ ∂ ⁡ B 1 ∩ ℝ + n | u t | 2 - 1 q + 1 λ - 1 ∫ λ 2 ⁢ λ t 1 - 2 ⁢ q - 1 p - 1 ∫ Γ 1 ∩ B 1 | u t | q + 1 d σ

≤ sup t ≥ λ E 2 ( u , t ) + C ∫ B 2 ∩ ℝ + n | u λ | 2 .

Thanks to this, we deduce that

lim λ → + ∞ ⁡ E ⁢ ( u , λ ) = lim λ → + ∞ ⁡ E ⁢ ( u λ , 1 ) = 0 .

In addition, since u is C 2 , one easily verifies that E ⁢ ( u , 0 ) = 0 . So, E ⁢ ( u , λ ) ≡ 0 , since E is nondecreasing, and d ⁢ E d ⁢ λ = 0 , which means that u is homogeneous. Thanks to the boundary condition, we readily deduce that u ≡ 0 .

For problem (1.2), the proof can be obtained similarly, with some minor modifications. With the current definition of the stability of (1.2), we require that the support is compact in ℝ + n ∩ Γ 1 and prevent from taking a function non-radial on Γ 1 and also non-zero on Γ 1 . But in fact, the test function is u ⁢ ξ , which vanishes on Γ 0 . We can take this test function because we have Q ⁢ ( u ⁢ ξ ) ≥ 0 , by density, i.e., valid for v ∈ H 1 ⁢ ( ℝ + n ) with v = 0 on Γ 0 . The rest of the proof is unchanged, thus we omit the details. ∎

References

[1] A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), 1205–1215. 10.1002/cpa.3160450908Suche in Google Scholar

[2] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622. 10.1215/S0012-7094-91-06325-8Suche in Google Scholar

[3] A. Farina, On the classification of solutions of the Lane–Emden equation on unbounded domains of ℝ n , J. Math. Pures Appl. (9) 87 (2007), 537–561. 10.1016/j.matpur.2007.03.001Suche in Google Scholar

[4] B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901. 10.1080/03605308108820196Suche in Google Scholar

[5] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598. 10.1002/cpa.3160340406Suche in Google Scholar

[6] A. Harrabi, M. Ahmedou, S. Rebhi and A. Selmi, A priori estimates for superlinear and subcritical elliptic equations: The Neumann boundary condition case, Manuscripta Math. 137 (2012), 525–544. 10.1007/s00229-011-0488-zSuche in Google Scholar

[7] A. Harrabi, S. Rebhi and S. Selmi, Solutions of superlinear equations and their Morse indices I, Duke Math. J. 94 (1998), 141–157. Suche in Google Scholar

[8] A. Harrabi, S. Rebhi and S. Selmi, Solutions of superlinear equations and their Morse indices II, Duke Math. J. 94 (1998), 159–179. 10.1215/S0012-7094-98-09408-XSuche in Google Scholar

[9] F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math. 79 (1993), no. 2, 161–172. 10.1007/BF02568335Suche in Google Scholar

[10] X. Wang and X. Zheng, Liouville theorem for elliptic equations with mixed boundary value conditions and finite Morse indices, J. Inequal. Appl. 2015 (2015), 10.1186/s13660-015-0867-1. 10.1186/s13660-015-0867-1Suche in Google Scholar

[11] X. Yu, Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices, J. Math. Anal. Appl. 421 (2015), 436–443. 10.1016/j.jmaa.2014.07.010Suche in Google Scholar

Received: 2016-07-25

Revised: 2016-10-17

Accepted: 2016-10-18

Published Online: 2016-12-20

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/anona-2016-0168

Schlagwörter für diesen Artikel

Liouville theorem; stable solutions; nonlinear boundary value condition; stability outside a compact set; monotonicity formula

Creative Commons

BY 4.0