Local versus nonlocal elliptic equations: short-long range field interactions

1 Introduction and main results

Consider the following class of mixed local–nonlocal elliptic equations

where Ω is a bounded smooth domain of RN,N>2,s∈(0,1),ϵ>0 is a real parameter and

is the fractional Laplacian operator. Here c_N,s is a positive normalizing constant and PV stands for the Cauchy Principal Value.

Integro-differential equations of the form (1.1) arise naturally in the study of stochastic processes with jumps.

The generator of an N-dimensional Lévy process has the following general structure:

where ν is the Lévy measure and satisfies ∫RNmin{1,|y|2}dν(y)<+∞,andχB1 is the usual characteristic function of the unit ball B₁ of ℝ^N. The first term of (1.2) on the right-hand side corresponds to the diffusion, the second one to the drift, and the third one to the jump part.

In the particular case when ν = 0, namely when there are no jumps, 𝓛 turns into a classical and extensively studied second-order differential operator. On the other hand, the case when the process has no diffusion and no drift has attracted a lot of attention recently (we refer to the book [18] for the development of the existence theory for several nonlinear nonlocal problems; see also the survey [21] for related regularity properties of solutions). In particular, one of the most prominent operators belonging to this class is the fractional Laplacian (−Δ)^s, which turns out to be the infinitesimal generator of isotropic s-stable Lévy processes and plays an important role in analysis and probability theory. The issue of the existence of solutions to fractional Laplacian problems, as well as the study of qualitative properties of such solutions, has been addressed by many authors and the literature devoted to the field grows up continuously (we mention, in particular, the following recent contributions [1, 4, 6, 7, 11, 17, 23, 25, 26] and the references therein). Finally, it is well-known, e.g. from [13, 19], that Markov processes with both diffusion and jump components are suitable to modeling many situations in finance and control theory, see also [8, 9, 10, 29].

For the above reasons, the study of the operator 𝓛 with diffusion, drift and jump components appears quite intriguing and, as far as we know, there are just a few contributions in this direction, mainly relying on a combination of probabilistic and analytic techniques. For instance in [12], under the assumption that the diffusion is uniformly elliptic and suitable conditions on ν, the author established a Harnack-type inequality and a regularity result for functions that are nonnegative in ℝ^N and harmonic in some domain. In [14, 15] the author derived interior Schauder estimates for both classical and weak solutions of the equation 𝓛 u = f.

Here we focus on the case 𝓛 = ϵ Δ –(–Δ)s, obtained from (1.2) by setting a_ij = ϵΔ_ij, ϵ > 0, b_j = 0 and ν given by the symmetric kernel |x - y|^-N-2s. We study this problem from a purely analytical point of view. Precisely, the paper is divided into two parts. Firstly, for fixed ϵ > 0 (without loss of generality, we may assume ϵ = 1), we provide some basic results related to (1.1), including spectral properties, maximum principles, existence, nonexistence, symmetry and regularity of weak solutions. With these preliminary tools, in the second part we will address further issues related to the asymptotic profile, as ϵ → 0 as well as $\epsilon\to +\infty$, of ground states of the model problem

If Ω is a bounded star-shaped domain, it was proved in [22] that (1.3) has no nontrivial bounded solutions if p≥2∗:=2NN−2 . When ϵ = 0, problem (1.3) possesses a positive solution for 2<p<2s∗:=2NN−2s,N>2s , and, if Ω is a bounded, C^1,1 and star-shaped domain, do not exist positive bounded solution for p=2s∗ and there are no nontrivial bounded solution for p>2s∗ , see [23, 24]. It is natural to investigate the behavior of the ground state u_ϵ in (1.3), as ϵ → 0 (corresponding to the fractional case), which may converge, blow up or vanish, depending on the range of p.

On the other hand, in the case ϵ → + ∞, setting vϵ(x)=ϵ−1p−2uϵ(x) , the function v_ϵ satisfies

whose natural limit problem is

By Sobolev embedding theorems and regularity theory, classical solutions to (1.5) do exist for 2 < p < 2^* with N > 2, and thus one may suspect in this case that u_ϵ(x) blows up almost everywhere in Ω, as ϵ → + ∞.

We mention that the operator 𝓛 can be given a local realization by means of a Caffarelli-Silvestre-type extension [6]. However, if one proceeds as in [5] by directly extending u ∈ H^s(Ω) to the cylinder C_Ω := Ω × (0, +∞), i.e. considering C_Ω as the domain of the extension w of u, with w = 0 on the lateral boundary ∂_L C_Ω of C_Ω, then one would end up with the fractional Laplacian operator defined by spectral decomposition, which on bounded domains does not coincide with (−Δ)^s. The idea is then to start from the half-space R+N+1 and to require the trace of the extension to vanish a.e. outside Ω. This can be achieved as follows.

Define the space

where R+N+1:=RN×(0,+∞) , equipped with the norm

It is known from [6] that, for all u∈Hs(RN) there exists a unique function E(u)∈ Y^s, called the harmonic extension of u to R+N+1 , satisfying

and (–Δ)^s u corresponds to the Dirichlet-to-Neumann map

for a suitable normalizing constant k_s > 0. Next define the following subspace

whose elements have traces over ℝ^N which vanish outside Ω. We say that w∈Y0s is a weak solution to

provided

for all φ∈Y0s . It is clear that if w∈Y0s solves (1.7) in the weak sense, then its trace over ℝ^N weakly solves

Before presenting our main results, let us introduce the functional framework. Set Q := R N × R N ∖ C Ω × C Ω , with CΩ:=RN∖Ω . We denote by X^s(Ω) the linear space of Lebesgue measurable functions u:RN→R such that the restriction of u to Ω belongs to H¹(Ω) and

A norm in X^s(Ω) is defined as follows

Let us also define the set

For u∈X0s , we can write the norm (1.10) as

and, by Poincaré's inequality, this norm in X0s is also equivalent to

We cliam that (X0s,∥⋅∥s) is an Hilbert space with scalar product

For the proof, see Lemma 12.

Throughout this paper, we will use the symbol ∥⋅∥p,p∈[1,+∞] to denote the usual L^p-norm in ℝ^N, while the letters C, C_i will represent generic positive constants which may change from line to line.

2 Proofs of theorems 1–4 and 7–9

Let us prove first the following preliminary result

Proof of Theorem 1. The proof is modeled on the one of the nonlocal operator −LK studied in [18]. However, for reader's convenience, we give the detailed proof.

For simplicity set I(u):=∥u∥s2 and M:={u∈X0s:∥u∥2=1} .

(1) Let {u_j} be a minimizing sequence for I, that is,

Then {I(u_j)} is bounded, i.e. {u_j} is bounded in X0s . Up to a subsequence, still denoted by {u_j}, u_j ⇀ u^* in X0s for some u∗∈X0s and u_j → u^* in L²(Ω). Since ∥u_j∥₂ = 1, we get ∥u^*∥₂ = 1, that is, u^* ∈ 𝓜. On the other hand,

and this implies that I(u∗)=infu∈MI(u) .

Let t ∈ (-1,1), v∈X0s,Ct=∥u∗+tv∥2 and u_t = (u^* + tv)/C_t. Then u_t ∈ 𝓜 and one has

By the very definition of u^* we get

Now, assume that λ₁ is attained at some e1∈X0s with ∥e₁∥₂ = 1. By (2.6), we get I(e₁) = λ₁. To prove that e₁ ≥ 0, we first show that if e is an eigenfunction related to λ₁ with ∥e∥₂ = 1, then both e and |e| realize the minimum in (1.14). In fact, if x1∈{x∈RN:e(x)>0} and x2∈{x∈RN:e(x)<0} , then

Thus, if both sets {x∈RN:e(x)>0}and{x∈RN:e(x)<0} have positive measure, we have I(|e|) < I(e). Note that |e|∈X0s and ∥|e|∥₂ = ∥e∥₂ = 1. Thus, since e₁ is a minimizer, we get I(|e|) = I(e) = I(e₁) = λ₁. This also implies that e ≥ 0 or e ≤ 0 a.e. in Ω. By replacing e₁ by |e₁|, we may conclude that e₁ ≥ 0.

(2) Suppose by contradiction that there exists another eigenfunction e˜1∈X0s,e˜1≠e1 , corresponding to λ₁. Then, by (1) we may assume that e˜1≥0 . Setting u1:=e˜1∥e˜1∥2 and v₁ := e₁ - u₁, we claim that v₁ = 0. In fact, direct calculations show that v₁ is also an eigenfunction related to λ₁, and so from (1) we deduce v₁ ≥ 0 or v₁ ≤ 0 a.e. in Ω, that is, e12≥u12ore12≤u12 a.e. in Ω. However, since ∥e1∥22−∥u1∥22=0 , we get e12=u12 and hence v₁ = 0 a.e. in Ω. This also implies that v₁ = 0 a.e. in ℝ^N and therefore, if w∈X0s is an eigenfunction related to λ₁, then w = ± ∥w∥₂ e₁ a.e. in ℝ^N.

(3) Since 𝓜_k is a weakly closed subspace of X0s , arguing as in step (1), we deduce that λ_k+1 is attained at some ek+1∈X0s . On the other hand, being Mk+1⊆Mk , we get 0 < λ₁ ≤ λ₂ ≤ … ≤ λ_k ≤ … Now, we claim that λ₁ ≠ λ₂. If not, e₂ ∈ 𝓜₂ would be also an eigenfunction related to λ₁. By (2), e₂ = ±∥e₂∥₂e₁ a.e. in ℝ^N. Thus, we would get 0=〈e1,e2〉s=±∥e2∥2∥e1∥s2 , which yields e₁ ≡ 0 a.e. in ℝ^N, a contradiction.

Analogously to the proof of (2.6), we have

We claim that equality (2.7) holds for any w∈X0s , i.e., that e_k+1 is an eigenfunction related to λ_k+1. Let us consider the decomposition X0s=span{e1,⋯,ek}⊕(span{e1,⋯,ek})⊥=span{e1,⋯,ek}⊕Mk+1 , and write any given v∈X0s as v = v₁ + v₂, with v1=∑j=1kcjej,cj∈R,andv2∈Mk+1 . Testing (2.7) with w = v₂ we then obtain

which proves the claim.

Next we show that λ_k → +∞ as k → +∞. Suppose, by contradiction, that λ_k → C ∈ ℝ⁺ as k → +∞. Since ∥ek∥s2=λk , up a subsequence, e_kj → e in L²(Ω) as k_j → +∞. On the other hand, being 〈 e_ki, e_kj〉_s = 0 for k_i ≠ k_j, we get ∫_Ω e_ki e_kjdx = 0 for k_i ≠ k_j and thus

which is a contradiction, as

Finally, let us show that any eigenvalue of (1.13) can be written in the form (1.15). Arguing again by contradiction, assume that there exists an eigenvalue λ ∉ {λ_k} and let e be an eigenfunction corresponding to λ with ∥e∥₂ = 1. Then λ=∥e∥s2≥∥e1∥s2=λ1 and this implies the existence of k ∈ ℕ such that λ_k < λ < λ_k+1. Thus, e ∉ 𝓜_k+1 and so there exists j ∈ {1,2,⋯, k} such that 〈 e, e_j〉_s ≠ 0, a contradiction.

(4) Let us prove that {e_k} is a basis of X0s by a standard Fourier analysis technique. Given u∈X0s , we define

where e˜i=ei∥ei∥s . Letting v_j = u - u_j, we get

that is,

Now, for l > j, we get

which implies that {v_j} is a Cauchy sequence in ℝ, and thus there exists v∗∈X0s such that vj→v∗inX0s as j → +∞. For j ≥ i we also have

Passing to the limit, we get 〈v∗,e˜i〉s=0 for any i ∈ ℕ. This means that v^* = 0 and therefore u__j = u - v_j → u in X0sasj→+∞.So,u=∑i=1∞〈u,e˜i〉se˜i , as we wanted.

In order to show that {e_i} is a basis for L²(Ω), too, chosen v ∈ L²(Ω), we let vj∈C02(Ω) such that ∥v−vj∥L2(Ω)<1j . Since {e_i} is a basis of X0s and X0s⊂L2(Ω) , there exist k_j ∈ ℕ and a function w_j ∈ span {e₁,…, e_kj} such that ∥vj−wj∥s≤1j . Thus, by Sobolev embeddings,

which yields the claim.

(5) X0s=span{ek,⋯,ek+h}⊕(span{ek,⋯,ek+h})⊥ . So, any eigenfunction u∈X0s∖{0} corresponding to λ_k admits the representation u = u₁ + u₂, with u1=∑i=kk+hciei∈span{ek,⋯,ek+h} for some c_i ∈ ℝ, and u₂ ∈ (span{e_k,⋯,e_k+h})^⊥. Obviously, 〈 u₁, u₂〉_s = 0, which yields

By definition, u₁ is also an eigenfunction corresponding to λ_k. Thus,

and

On the other hand, we have

Recalling that u and u₁ are eigenfunctions corresponding to λ_k, we deduce that u₂ enjoys the same property and thus, u2∈(span{e1,⋯,ek+h})⊥=Mk+h+1 .

We claim that u₂ ≡ 0. If not, we would have

Collecting (2.8), (2.9) and (2.10), we would obtain

clearly a contradiction. Therefore, u = u₁ ∈ span{e_k,⋯,e_k+h} and this completes the proof.  □

Proof of Theorem 2. Inspired by [7], assume that u(x) < 0 in Ω. Owing to the lower semi-continuity of u on Ωˉ , there exists some x0∈Ωˉ such that

Moreover, since u ≥ 0 in ℝ^N∖Ω, we deduce that x₀ is an interior point of Ω. Taking into account the fact that Δ u(x₀) ≥ 0, we get

which contradicts the assumptions. Moreover, if u(x₀) = 0 at some point x₀ ∈ Ω, then

which forces u(x) ≡ 0 in ℝ^N.  □