Bounded Solutions for Nonlocal Boundary Value Problems on Lipschitz Manifolds with Boundary

2.1 General Assumptions

1. Let 𝐚:ℝ→ℝ be a measurable function satisfying the following conditions:

   1. 𝐚∈C⁢(ℝ) is non-decreasing and odd.

   2. There exist two constants c𝐚,C𝐚>0 such that, for all t∈ℝ,

      (2.1) 𝐚 ⁢ ( t ) ⁢ t ≥ c 𝐚 ⁢ | t | p   and   | 𝐚 ⁢ ( t ) | ≤ C 𝐚 ⁢ | t | p - 1 ,

      for some p∈(1,∞).

Let Ω⊂ℝN be a bounded open set with a Lipschitz continuous boundary ∂⁡Ω and 0<s<1. We consider the operator AΩ,p,s formally given by

A Ω , p , s ⁢ u ⁢ ( x ) = 2 ⁢ lim ε ↓ 0 ⁡ ∫ { y ∈ Ω : | x - y | > ε } K Ω ⁢ ( x , y ) ⁢ 𝐚 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 y , x ∈ Ω ,

provided that the limit exists. We assume the following assumptions. The kernel KΩ:ℝN×ℝN→ℝ is symmetric and there exist two constants 0<cΩ≤CΩ depending only on Ω and KΩ such that

1. Analogously, let 𝐛:ℝ→ℝ be a measurable function satisfying the following conditions:

   1. 𝐛∈C⁢(ℝ) is non-decreasing and odd.

   2. There exist two constants c𝐛,C𝐛>0 such that, for all t∈ℝ,

      (2.3) 𝐛 ⁢ ( t ) ⁢ t ≥ c 𝐛 ⁢ | t | q   and   | 𝐛 ⁢ ( t ) | ≤ C 𝐛 ⁢ | t | q - 1 ,

      for some q∈(1,∞).

For 0<l<1 and q∈(1,∞), we introduce the boundary operator B∂⁡Ω,q,l formally given by

B ∂ ⁡ Ω , q , l ⁢ u ⁢ ( x ) = 2 ⁢ lim ε ↓ 0 ⁡ ∫ { y ∈ ∂ ⁡ Ω : | x - y | > ε } K ∂ ⁡ Ω ⁢ ( x , y ) ⁢ 𝐛 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 σ y , x ∈ ∂ ⁡ Ω ,

where σ denotes the usual Lebesgue surface measure on ∂⁡Ω. The kernel K∂⁡Ω:ℝN-1×ℝN-1→ℝ is symmetric and there exist two constants 0<c∂⁡Ω≤C∂⁡Ω depending only on ∂⁡Ω and K∂⁡Ω such that

We wish to investigate the existence of bounded weak solutions to the nonlocal problem

subject to the boundary condition

for some α,β>0, where DA,B is the corresponding coupling operator between (2.5) and (2.6), i.e.,

〈 A Ω , p , s ⁢ u , v 〉 Ω = ∫ Ω ∫ Ω K Ω ⁢ ( x , y ) ⁢ 𝐚 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ ( v ⁢ ( x ) - v ⁢ ( y ) ) ⁢ 𝑑 y ⁢ 𝑑 x - 〈 D A , B ⁢ u , v 〉 ∂ ⁡ Ω ,

where 〈⋅,⋅〉S, S∈{Ω,∂⁡Ω}, denotes the duality between an appropriate reflexive Banach space and its dual defined on S (see Section 3 below for more details). Here, δ∈{0,1} and we recall that s,l∈(0,1).

1. We assume the following conditions on the nonlinearities fΩ and f∂⁡Ω:

   1. fΩ:Ω×ℝ→ℝ and f∂⁡Ω:∂⁡Ω×ℝ→ℝ are measurable.

   2. fΩ⁢(x,⋅)∈C⁢(ℝ) for a.e. x∈Ω; f∂⁡Ω⁢(x,⋅)∈C⁢(ℝ) for σ-a.e. x∈∂⁡Ω.

   3. There exist some constants α0,β0,η0>0, α1,β1≥0, and a positive function η:ℝ→ℝ+ such that for a.e. x∈S, S∈{Ω,∂⁡Ω}, and for all t∈ℝ with |t|≥t0, for some t0>0,

      (2.7) f Ω ⁢ ( x , t ) ⁢ t ≥ - α 0 ⁢ η ⁢ ( | t | ) - α 1   and   f ∂ ⁡ Ω ⁢ ( x , t ) ⁢ t ≥ - β 0 ⁢ η ⁢ ( | t | ) - β 1 ,

      as long as

      0 < η ⁢ ( | t | ) | t | ≤ η 0 = η 0 ⁢ ( t 0 )   for all ⁢ | t | ≥ t 0 .

2.2 The Functional Setup

For a bounded open set Ω⊂ℝN with boundary ∂⁡Ω and for s∈(0,1), p∈[1,∞), we denote by

W s , p ⁢ ( Ω ) := { u ∈ L p ⁢ ( Ω ) : ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y < ∞ }

the fractional order Sobolev space endowed with the norm

∥ u ∥ W s , p ⁢ ( Ω ) := ( ∫ Ω | u | p ⁢ 𝑑 x + ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y ) 1 p .

We let

W 0 s , p ⁢ ( Ω ) = 𝒟 ⁢ ( Ω ) ¯ W s , p ⁢ ( Ω )

and

W 0 s , p ⁢ ( Ω ¯ ) = { u ∈ W s , p ⁢ ( ℝ N ) : u = 0 ⁢ a.e. on ⁢ ℝ N ∖ Ω } .

By definition, W0s,p⁢(Ω¯) is always a subspace of Ws,p⁢(Ω) but in general there is no obvious inclusion between W0s,p⁢(Ω¯) and W0s,p⁢(Ω). It has been shown in [15, Theorem 1.4.2.2] that if Ω is an open set with continuous boundary, then 𝒟⁢(Ω) is dense in W0s,p⁢(Ω¯). But if Ω is a bounded open set with Lipschitz continuous boundary, then when s≠1p we have W0s,p⁢(Ω¯)=W0s,p⁢(Ω). We refer to [15, Chapter 1] for further details on this topic.

Next, for every bounded open set Ω⊂ℝN, 0<s<1, 1<p<∞ we have the following Sobolev embedding (see, e.g., [7]):

All the embeddings in (2.8) also hold if W0s,p⁢(Ω) is replaced by the space W0s,p⁢(Ω¯).

The following result is contained in [2, Corollary 2.8 and Remark 2.3] for the case p=2 and in [15, 31] for general p∈(1,∞).

In view of Theorem 2.1, we have that if 0<s≤1p and Ω has a Lipschitz continuous boundary, then every u∈Ws,p⁢(Ω) is zero σ-a.e. on ∂⁡Ω, where we recall that σ denotes the usual Lebesgue surface measure on ∂⁡Ω. Therefore, to talk about traces of functions in Ws,p⁢(Ω) that are not necessarily null on ∂⁡Ω, it is not a restriction to assume that 1p<s<1.

The following Sobolev embedding result can be found in [3, 7, 15].

Next, for l∈(0,1) and q∈[1,∞), we denote by

W l , q ⁢ ( ∂ ⁡ Ω ) := { u ∈ L q ⁢ ( ∂ ⁡ Ω ) : ∫ ∂ ⁡ Ω ∫ ∂ ⁡ Ω | u ⁢ ( x ) - u ⁢ ( y ) | q | x - y | N - 1 + q ⁢ l ⁢ 𝑑 σ x ⁢ 𝑑 σ y < ∞ }

the fractional order Sobolev space endowed with the norm

∥ u ∥ W l , q ⁢ ( ∂ ⁡ Ω ) := ( ∫ ∂ ⁡ Ω | u | q ⁢ 𝑑 σ + ∫ ∂ ⁡ Ω ∫ ∂ ⁡ Ω | u ⁢ ( x ) - u ⁢ ( y ) | q | x - y | N - 1 + l ⁢ q ⁢ 𝑑 σ x ⁢ 𝑑 σ y ) 1 q .

Comparing with Proposition 2.2, we have the following embedding result.

For p,q∈(1,∞), 1p<s<1, 0<l<1 and δ∈{0,1}, we endow the Banach space

𝕎 δ ⁢ l s , p , q ⁢ ( Ω ¯ ) := { U = ( u , u | ∂ ⁡ Ω ) : u ∈ W s , p ⁢ ( Ω ) , δ ⁢ u | ∂ ⁡ Ω ∈ W l , q ⁢ ( ∂ ⁡ Ω ) }

with the norm

∥ U ∥ 𝕎 l s , p , q ⁢ ( Ω ¯ ) = ∥ u ∥ W s , p ⁢ ( Ω ) + ∥ u ∥ W l , q ⁢ ( ∂ ⁡ Ω )

if δ=1, and

𝕎 0 s , p , q ⁢ ( Ω ¯ ) = 𝕎 s , p ⁢ ( Ω ¯ ) := { U = ( u , u | ∂ ⁡ Ω ) : u ∈ W s , p ⁢ ( Ω ) }

with the norm

∥ U ∥ 𝕎 s , p ⁢ ( Ω ¯ ) p = ∥ u ∥ W s , p ⁢ ( Ω ) p + ∥ u ∥ W s - 1 p , p ⁢ ( ∂ ⁡ Ω ) p

if δ=0. Furthermore, it is well known that

∥ U ∥ 𝕎 s , p ⁢ ( Ω ¯ ) p = ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ∂ ⁡ Ω | u | p ⁢ 𝑑 σ

defines an equivalent norm on 𝕎s,p⁢(Ω¯) (see, e.g., [30, Theorem 2.3]).

For r,q∈[1,∞] with 1≤r,q<∞ or r=q=∞ we endow the Banach space

𝕏 r , q ( Ω ¯ ) := L r ( Ω ) × L q ( ∂ Ω ) = { ( f , g ) , f ∈ L r ( Ω ) , g ∈ L q ( ∂ Ω ) }

with the norm

We will simply write 𝕏r⁢(Ω¯):=𝕏r,r⁢(Ω¯). If Ω has a Lipschitz continuous boundary, by virtue of Proposition 2.2 and Proposition 2.3 we can summarize the corresponding continuous embeddings according to the following cases:

for δ=1, and

for δ=0.

For more information on fractional order Sobolev spaces we refer to [1, 7, 15, 19, 31] and their references.