Stability of variational eigenvalues for the fractional p-Laplacian

By virtue of $\Gamma-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.

For p = 2, this definition coincides (up to a normalization constant depending on N and s, see [8]) with the linear fractional Laplacian (−∆) s , defined by where F is the Fourier transform operator and M s is the multiplication by |ξ| 2 s . Many efforts have been devoted to the study of problems involving the fractional p−Laplacian operator, among which we mention eigenvalue problems [7,21,27,30], regularity theory [14,26,28,29] and existence of solutions within the framework of Morse theory [25]. For the motivations that lead to the study of such operators, we refer the reader to the contribution [9] of Caffarelli. For completeness, we also mention that other types of nonlocal quasilinear operators, defined by means of extension properties, can be found in the literature (see [37]).
In this paper, we are concerned with Dirichlet eigenvalues of (−∆ p ) s on the set Ω. These are the real (positive) numbers λ admitting nontrivial solutions to the following problem It is known that it is possible to construct an infinite sequence of such eigenvalues diverging to +∞. This is done by means of variational methods similar to the so-called Courant minimax principle, that we briefly recall below. Then our main concern is the study of the singular limit of these variational eigenvalues as s ր 1, in which case the limiting problem of (1.2) is formally given by where ∆ p u = div(|∇u| p−2 ∇u) is the familiar p−Laplace operator. In order to neatly present the subject, we first need some definitions. The natural setting for equations involving the operator (−∆ p ) s is the space W s,p 0 (R N ), defined as the completion of C ∞ 0 (R N ) with respect to the standard Gagliardo semi-norm (1.4) [u] W s,p (R N ) := ˆR NˆRN |u(x) − u(y)| p |x − y| N +s p dx dy 1 p . Furthermore, in order to take the Dirichlet condition u = 0 in R N \ Ω into account, we consider the space W s,p 0 (Ω) = u : R N → R : [u] W s,p (R N ) < +∞ and u = 0 in R N \ Ω , endowed with (1.4). Since Ω is Lipschitz, the latter coincides with the space used in [6,7] and defined as the completion of C ∞ 0 (Ω) with respect to [ · ] W s,p (R N ) (see Proposition B.1 below). Then equation (1.2) has to be intended in the following weak sense: |u(x) − u(y)| p−2 (u(x) − u(y)) (ϕ(x) − ϕ(y)) |x − y| N +s p dx dy = λˆΩ |u| p−2 u ϕ dx, for every ϕ ∈ W s,p 0 (Ω). Let us introduce In the previous formulas, we noted for 0 < s ≤ 1 (1.6) W s m,p (Ω) = {K ⊂ S s,p (Ω) : K symmetric and compact, i(K) ≥ m} , and i(K) denotes the Krasnosel'skiȋ genus of K. We recall that for every nonempty and symmetric subset A ⊂ X of a Banach space, its Krasnosel'skiȋ genus is defined by [u] p W s,p (R N ) , mountain pass level, where u 1 is a minimizer associated with λ s 1,p (Ω) and Σ(u 1 , −u 1 ) is the set of continuous paths on S s,p (Ω) connecting u 1 and −u 1 (see [11,Corollary 3.2] for the local case, [7,Theorem 5.3] for the nonlocal one). Remark 1.1. For the limit problem (1.3), the continuity with respect to p of the (variational) eigenvalues λ 1 m,p has been first studied by Lindqvist [31] and Huang [24] in the case of the first and second eigenvalue, respectively. Then the problem has been tackled in more generality in [10,32,35]. We also cite the recent paper [13] where some generalizations (presence of weights, unbounded sets) have been considered.

Main result.
In order to motivate the investigation pursued in the present paper, it is useful to observe that based upon the results by Bourgain, Brezis and Mironescu [3,4], we have that if u ∈ W 1,p 0 (Ω) (1. It is not difficult to see that, due to symmetry reasons, the definition of K(p, N ) is indeed independent of the direction e ∈ S N −1 .
Formula (1.8) naturally leads to argue that the nonlocal variational eigenvalues λ s m,p could converge (once properly renormalized) to the local ones λ 1 m,p . This is the content of the main result of the paper. Observe that we can also assure convergence of the eigenfunctions in suitable (fractional) Sobolev norms. Moreover, if u s is an eigenfunction of (1.2) corresponding to the variational eigenvalue λ s m,p (Ω) and such that u s L p (Ω) = 1, then there exists a sequence {u s k } k∈N ⊂ {u s } s∈(0,1) such that for every p ≤ q < ∞ and every 0 < t < p q , where u is an eigenfunction of (1.3) corresponding to the variational eigenvalue λ 1 m,p (Ω) and such that u L p (Ω) = 1. Remark 1.3 (The case p = 2). To the best of our knowledge this result is new already in the linear case p = 2, namely for the fractional Laplacian operator (−∆) s . In the theory of stochastic partial differential equations this corresponds to the case of a stable Lévy process. The kernel corresponding to (−∆) s determines the probability distribution of jumps in the value of the stock price, assigning less probability to big jumps as s increases to 1. Therefore, since the parameter s has to be determined through empirical data, the stability of the spectrum with respect to s allows for more reliable models of random jump-diffusions, see [2] for more details.
It is also useful to recall that for p = 2, problems (1.2) and (1.3) admit only a discrete set of eigenvalues, whose associated eigenfunctions give an Hilbertian basis of L 2 (Ω) (once properly renormalized). Then we have that these eigenvalues coincide with those defined by (1.5), see Theorem A.2 below.
One of the main ingredients of the proof of Theorem 1.2 is a Γ−convergence result for Gagliardo semi-norms, proven in Theorem 3.1 below. Namely, by defining the family of functionals E s,p : we prove that for s k ր 1 we have where Γ − lim denotes the Γ−limit of functionals, with respect to the norm topology of L p (Ω). We refer to [12] for the relevant definitions and facts needed about Γ−convergence.  [20]. As a byproduct of the method, we obtain a variational characterization of the constant K(p, N ) appearing in the limit (see Lemma 3.9 below), which is quite typical of the blow-up procedure.
Remark 1.5. In Theorem 1.2 the variational eigenvalues are defined by means of the Krasnosel'skiȋ genus, but the same result still holds by replacing it with a general index i having the following properties: is defined whenever K = ∅ is a compact and symmetric subset of a topological vector space, such that 0 ∈ K; (ii) if X is a topological vector space and ∅ = K ⊆ X \ {0} is compact and symmetric, then there exists U ⊂ X \ {0} open set such that K ⊆ U and i( K) ≤ i(K) for any compact, symmetric and nonempty K ⊆ U ; (iii) if X, Y are two topological vector spaces, ∅ = K ⊆ X \ {0} is compact and symmetric and π : K → Y \ {0} is continuous and odd, then i(π(K)) ≥ i(K 1.3. Plan of the paper. In Section 2, we collect various preliminary results, such as sharp functional inequalities and convergence properties in the singular limit s ր 1. We point out that even if most of the results of this section are well-known, we need to prove them in order to carefully trace the sharp dependence on the parameter s in all the estimates. In Section 3, we prove the Γ−convergence (1.12). For completeness, we also include a convergence result for dual norms, in the spirit of Bourgain-Brezis-Mironescu's result. Then the main result Theorem 1.2 is proven in Section 4. Two appendices close the paper and contribute to make it self-contained.

Preliminaries
2.1. Some functional inequalities. We start with an interpolation inequality.
Proof. We first consider the case t ∈ (0, 1). Let u ∈ C ∞ 0 (R N ), then we have The first integral is estimated bŷ where θ ∈ [0, 1) is determined by scale invariance and is given precisely by (2.2). In conclusion, For the other term, for every ℓ > β we havê We choose (2.6)ˆR for some C = C(N, p) > 0. On the other hand, we have

Thus we getˆ{
where C = C(N, p, q) > 0. By combining (2.4) and (2.7), we get , possibly with a different C = C(N, p, q) > 0. We now use the previous inequality with u χ (x) = u(x χ −1/q ) and optimize in χ > 0. We get still for some constant C = C(N, p, q) > 0. Observe that by hypothesis on s we have α − β > 0. The left-hand side of (2.8) is maximal for Thus we get In order to prove (2.3), it is sufficient to repeat the previous proof, this time replacing the choice (2.5) by and then using thatˆR in place of (2.6), which follows from basic calculus and invariance by translations of the L p norm.
For u ∈ C ∞ 0 (Ω), by inequality (2.9) we get If we now apply the Poincaré inequality for W 1,p 0 (Ω) on the right-hand side, we obtain inequality (2.10) for functions in C ∞ 0 (Ω). By using density of C ∞ 0 (Ω) in the space W 1,p 0 (Ω), we get the desired conclusion.
In what follows, we define the sharp Sobolev constant We need the following result by Maz'ya and Shaponishkova, see [ Theorem 2.4 (Hardy inequality for convex sets). Let 1 < p < ∞ and s ∈ (0, 1) with s p > 1. Then for any convex domain Ω ⊂ R N and every u ∈ C ∞ 0 (Ω) we have with δ Ω (x) = dist(x, ∂Ω) and C N,p > 0 a costant depending on N and p only.
The optimal constant D N,p,s is given by We claim that Indeed, by concavity of the map τ → τ (s p−1)/p we have Thus we get On the other hand, from the definition of Γ we also have By using these estimates, we get (2.13).
The next result is a Poincaré inequality for Gagliardo semi-norms. This is classical, but as always we have to carefully trace the sharp dependence on s of the constants concerned.
for a constant C = C(N, p) > 0. Moreover, if Ω is convex and s p > 1, then possibly with a different constant C = C(N, p) > 0, still independent of s.

Proof.
Since Ω is bounded, we have Ω ⊂ B R (x 0 ), with 2 R = diam(Ω) and x 0 ∈ Ω. Let h ∈ R N be such that |h| > 2 R, so that Then for every u ∈ C ∞ 0 (Ω) we havê where in the last estimate we used [6, Lemma A.1]. By taking the infimum on the admissible h, we get (2.14).
Let us now suppose that Ω is convex and s p > 1. In order to prove (2.15), we proceed as in the proof of [6, Proposition B.1]. For every u ∈ C ∞ 0 (Ω) we have In order to estimate the last term, we first observe that, if δ Ω (x) = dist(x, ∂Ω), we get where we also used that s p > 1. We can now use Hardy inequality (2.12), so to obtain By also using that 1 − s < 1, we finally get By combining this, (2.14) and observing that p/(s p − 1) > 1, we get (2.15).
More precisely, we will need the following extension. The main difference with Theorem 2.7 is that functions are now taken in W 1,p 0 (Ω) and the seminorm on Ω is replaced by the seminorm on the whole R N .
Proof. Let u ∈ W 1,p 0 (Ω), we observe that u ∈ W s,p 0 (R N ) for all s ∈ (0, 1) thanks to Corollary 2.2. Furthermore, since Ω is a bounded set, by virtue of Theorem 2.7 we have Let us first prove that (2.17) holds for u ∈ C ∞ 0 (Ω). Recalling that u = 0 outside Ω, we have It follows that and the claim is proved for u ∈ C ∞ 0 (Ω). Assume now that u ∈ W 1,p 0 (Ω). Then there exists a sequence {φ j } j∈N ⊂ C ∞ 0 (Ω) such that ∇φ j − ∇u L p (Ω) → 0 as j goes to ∞. In turn, by inequality (2.10) we have with C independent of s and j. Thus for every ε > 0, there exists j 0 ∈ N independent of s such that and consequently for every j ≥ j 0 . Then for every j ≥ j 0 for every j ≥ j 0 . By using the first part of the proof we thus get for every j ≥ j 0 If we now use (2.18) and exploit the arbitrariness of ε > 0, we get (2.17) for a general u ∈ W 1,p 0 (Ω).

Dual spaces.
Let Ω ⊂ R N be as always an open and bounded set with Lipschitz boundary. Let 1 < p < ∞ and s ∈ (0, 1), we set p ′ = p/(p − 1) and which is equipped with the natural dual norm .
The symbol ·, · denotes the relevant duality product. For s = 1, the space W −1,p ′ (Ω) and the corresponding dual norm are defined accordingly.
The following is the dual version of (2.10).
Proof. By (2.10) we have By taking the supremum over u, the conclusion follows from the definition of dual norm.
If F ∈ W −s,p ′ (Ω), by a simple homogeneity argument (i.e. replacing u by t u and then optimizing in t) we have Thus in particular we get In the local case s = 1, with similar computations we get

2.4.
A bit of regularity. We conclude this section with a regularity result. This is not new, but once again our main concern is the dependence on s of the constants entering in the estimates below. We also need to pay particular attention to the case p = N , which is borderline in the limit as s goes 1.
Theorem 2.10. Let 1 < p < ∞ and s ∈ (0, 1). If u ∈ W s,p 0 (Ω) is an eigenfunction of (−∆ p ) s with eigenvalue λ, then we have: Proof. In the case 1 < p < N , we have of course s p < N as well. Then by appealing to [ where T p,s is the sharp Sobolev constant (2.11). By using Theorem 2.3, we obtain Then from the previous we get (2.22), once it is noticed that (1 − (s p)/N ) 1−(s p)/N is bounded from below by a universal constant c > 0, for 0 < s p < N .
Let us now consider the case p = N and 3 4 ≤ s.
By proceeding as in the second part of the proof of 2 [6, Theorem 3.3] and using the same notation, after a Moser's iteration we end up with where C = C(N ) > 0 and the geometric constant α s N (Ω) is given by Observe that this is not 0, since W s,N 0 (Ω) ֒→ L 2 N (Ω) as soon as which is verified. In order to estimate α s N (Ω) from below, we observe that by choosing q = p = N , t = s and β = 1/2 in (2.1), we get (C denotes a constant depending on N only, varying from a line to another) where in the second inequality we used Poincaré inequality (2.14). We can now use Sobolev inequality in the left-hand side, i.e.
[u] N where we used the definition (2.11). Then by joining the two previous estimates, appealing to the definition of α s N (Ω) and recalling that s ≥ 3/4, we get where C = C(N ) > 0. By inserting this estimate in (2.25), we get the conclusion in this case as well.
For s p > N , we already know that W s,p 0 (Ω) ֒→ C 0,s−N/p , but of course we need to estimate the embedding constant in terms of s. We take x 0 ∈ R N and R > 0. We consider the ball B R (x 0 ) having radius R and centered at x 0 , then we havê We now observe that so that by exchanging the order of integration in the last integral for C = C(N ) > 0. If we now divide by R N and use once again [6, Lemma A.1], we get By arbitrariness of R and x 0 , we obtain that u is in C 0,s−N/p (R N ) by Campanato's Theorem (see [22,Theorem 2.9]), with the estimate where C = C(N, p) > 0. The last estimate is true for every u ∈ W s,p 0 (Ω). On the other hand, if u ∈ W s,p 0 (Ω) is an eigenfunction with eigenvalue λ, then by the equation we also have By inserting this estimate in (2.26), we get Finally, the estimate (2.24) follows from (2.27) by taking y ∈ R N \ Ω.
Remark 2.11. Though we will not need it here, for the conformal case s p = N a global L ∞ estimate can be found in [6, Theorem 3.3].

A Γ−convergence result
In this section we will prove the following result.  where E s k ,p and E 1,p are the functionals defined by (1.10) and (1.11).
This Γ−convergence result will follow from Propositions 3.3 and 3.11 below. Before proceeding further with the proof of this result, let us highlight that by combining Theorem 3.1 and [12, Proposition 6.25], we get the following.   (Ω) such that Proof. If u ∈ W 1,p 0 (Ω), there is nothing to prove, thus let us take u ∈ W 1,p 0 (Ω). If we take the constant sequence u k = u and then apply the modification of Bourgain-Brezis-Mironescu result of Proposition 2.8, we obtain concluding the proof.
In order to prove the Γ − lim inf inequality, we need to find a different characterization of the constant K(p, N ). The rest of this subsection is devoted to this issue.
In what follows, we note by Q = (−1/2, 1/2) N the open N −dimensional cube of side length 1. Given a ∈ R N , we define the linear function Ψ a (x) = a, x . For every a ∈ S N −1 , we define the constant We will show in Lemma 3.9 that indeed this quantity does not depend on the direction a.
This proves (3.5), since ∇Ψ a has unit norm in L p (Q).
We are going to prove that indeed K(p, N ) = Θ(p, N ; a) for every a ∈ S N −1 . To this aim, we first need a couple of technical results. In what follows, by W s,p 0 (Q) we note the completion of C ∞ 0 (Q) with respect to the semi-norm Lemma 3.6. For every 1 < p < ∞ and every a ∈ S N −1 we have . N ; a), we already know that
In order to prove the reverse inequality, let us take a sequence {s k } k∈N such that 0 < s k < 1 and s k ր 1. Then we take {v k } k∈N such that Without loss of generality, we can assume that s k p > 1, so that for the space W s k ,p 0 (Ω) we have the Poincaré inequality (2.15). We introduce a smooth cut-off function η ∈ C ∞ 0 (Q) such that for some parameter 0 < τ < 1. Then we define the sequence {w k } k∈N by We observe that by construction we have w k − Ψ a ∈ W s,p 0 (Q). Moreover we havê thus w k still converges in L p (Q) to Ψ a . We now have to estimate the Gagliardo semi-norm of w k . To this aim, we first observe that (3.7) Let us set Then by definition of w k , (3.7) and Minkowski inequality we have By using the properties of η, we have obtained We have to estimate the last two integrals. By recalling that Ψ a (x) = a, x , we have 3 For the other integral, we have with C = C(N, p) > 0. By collecting all these estimates and using them in (3.8), we get By arbitrariness of 0 < τ < 1, this finally proves the desired result.
Before proceeding further, we need to know that linear functions are (s, p)−harmonic.
Proof. We first observe that the double integral is well-defined and absolutely convergent. Indeed,ˆR |x − y| N +s p ϕ(x) dx dy. 3 We use that for x ∈ Q For the first term we havê For the second one, by observing that the integral in the x variable is equivantely performed on K := spt(ϕ) ⋐ Q, we get In order to prove (3.9), for every ε > 0 we havê where we used that by symmetrŷ Moreover, we have (we still denote K = spt(ϕ)) By arbitrariness of ε we get the conclusion.
Lemma 3.8. Let a ∈ S N −1 . For every 1 < p < ∞ and s ∈ (0, 1) such that s p > 1, let u s be the unique solution of Then, u s converges to Ψ a in L p (Ω) as s goes to 1. Moreover, we have Proof. Since we are interested in the limit as s goes to 1, without loss of generality we can further assume that s > (p − 1)/p as well, i.e.
The existence of a (unique, by strict convexity) solution u s follows by the Direct Methods, since coercivity of the functional v → [v] p W s,p (Q) can be inferred thanks to Poincaré inequality (2.15) (here we use the assumption s p > 1). We take ϕ ∈ W s,p 0 (Q), by minimality of u s there holds Still by minimality of u s , we also get since Ψ a is admissible for problem (3.10). On the other hand, the linear function Ψ a is "almost" a solution of (3.10), thanks to Lemma 3.7. Indeed from (3.9), for every ϕ ∈ C ∞ 0 (Q) we get Thus we obtain where as before we set δ Q (x) = dist(x, ∂Q). Hence, Since we are assuming s > 1/p, we can apply Hardy inequality (2.12) to the last term and obtain (3.14) for some constant C = C(N, p) > 0 (observe that we used that 1 − s < 1). From (3.12) and (3.14) we finally obtain for every ϕ ∈ C ∞ 0 (Q) By density, the previous estimate is still true for every ϕ ∈ W s,p 0 (Q), thus we can use (3.15) with ϕ = Ψ a − u s . We distinguish two cases.
Case p ≥ 2. We use the basic inequality (|s| p−2 s − |t| p−2 t)(s − t) ≥ 2 2−p |s − t| p in order to obtain from (3.15) for a constant C = C(N, p) > 0. This implies Case 1 < p < 2. We use the inequality where we used Hölder inequality with exponents 2/p and 2/(2 − p), relation (3.15) and the subadditivity of the function t → t p/2 . The previous estimate and (3.13) imply Since 2 − p < 1, after a simplification we get It thus follows again Observe that as a byproduct of (3.16) and (3.17), we also get This shows that lim

thus (3.11) is proved.
Finally, since u s − u ∈ W s,p 0 (Q), Q is a convex set and we are assuming s p > 1, we can use Poincaré inequality (2.15) in conjuction with (3.16) or (3.17). In both cases we have where C = C(N, p) > 0. This concludes the proof.
Finally, we can prove an equivalent characterization of K(p, N ). In particular, Θ(p, N ; a) does not depend on the direction a.
Proof. By (3.5) we know that K(p, N ) ≥ Θ(p, N ; a). In order to prove the reverse inequality, we define the linear function Ψ a (x) = a, x . Let v s ∈ W s,p (Q) be a sequence converging to Ψ a in L p (Q) and such that v s − Ψ a ∈ W s,p 0 (Q). We consider the function u s defined in Lemma 3.8, then from (3.11) we get By appealing to (3.6), we get Θ(p, N ; a) = K(p, N ).

3.2.
The Γ − lim inf inequality. At first, we need a technical result which will be used various times. ( there exist an increasing sequence {s k } k∈N ⊂ (s 0 , 1) converging to 1 and a function u ∈ W 1,p 0 (Ω) such that lim k→∞ u s k − u L p (Ω) = 0.
Since s > s 0 , from the previous estimate, we can also infer for every s 0 < s < 1.
Estimates (3.19) and (3.21) and the fact that u s ≡ 0 in R N \ Ω enables us to use the Riesz-Fréchet-Kolmogorov Compactness Theorem for L p . Thus, there exists a sequence {u s k } k∈N and u ∈ L p (R N ) such that lim k→∞ u s k − u L p (Ω) = 0.
In order to conclude, we need to prove that u ∈ W 1,p 0 (Ω). Up to a subsequence, we can suppose that u s k converges almost everywhere. This implies that u ≡ 0 in R N \ Ω. Moreover, thanks to Fatou Lemma we can pass to the limit in (3.20) and obtain This implies that the distributional gradient of u is in L p (R N ). Thus u ∈ W 1,p (R N ) and it vanishes almost everywhere in R N \Ω. Since Ω is Lipschitz, this finally implies that u ∈ W 1,p 0 (Ω) by [5,Propostion IX.18].
The following result will complete the proof of Theorem 3.1.
Proposition 3.11 (Γ − lim inf inequality). Given {s k } k∈N ⊂ R an increasing sequence converging to 1 and {u k } k∈N ⊂ L p (Ω) converging to u in L p (Ω), we have Proof. The proof follows that of [1, Lemma 7]. We start by observing that if there is nothing to prove. Thus, let us suppose that for some uniform constant L > 0. By Lemma 3.10, we get that u ∈ W 1,p 0 (Ω). We now continue the proof of (3.22). For every measurable set A ⊂ Ω we define the absolutely continuous measure µ(A) :=ˆA |∇u| p dy, and we observe that, by Lebesgue's Theorem For x ∈ Ω, set C r (x) := x + r Q. We claim that In order to prove (3.24), for every measurable function v, we introduce the notation v r,x (y) : We keep on using the notation Ψ a (x) = a, x , for any given vector a ∈ R N . Then we will prove (3.24) at any point x ∈ Ω such that and such that (3.23) holds. We recall that (3.25) is true at almost every x ∈ Ω by [16, Theorem 2, page 230]. Therefore, to prove (3.24) it will be sufficient to show that To this aim, let r j ց 0 be a sequence such that For any j ∈ N we can choose k = k(j) so large that (Cr j (x)) < 1/j. Then, by using i), the definitions of α k and (u k ) r,x and ii) we have On the other hand by iii) we have Thus by triangle inequality we get that (u k(j) ) r j ,x converges to Ψ a in L p (Q), with direction a = ∇u(x). This in turn implies The conclusion is exactly as in [1,Lemma 7]. Let us consider for ε > 0 the following family of closed cubes By observing that α C r (x) = α(C r (x)) and µ C r (x) = µ(C r (x)), and using (3.24), we get that F is a fine Morse cover (see [19,Definition 1.142]) of µ−almost all of Ω, then we can apply a suitable version of Besicovitch Covering Theorem (see [19,Corollary 1.149]) and extract a countable subfamily of disjoint cubes {C i } i∈I ⊂ F such that µ(Ω\∪ i∈I C i ) = 0. This yields . By the arbitrariness of ε we get This concludes the proof.

3.3.
A comment on dual norms. By using Theorem 3.1, we can prove a dual version of the Bourgain-Brezis-Mironescu result. The result of this section is not needed for the proof of Theorem 1.2 and is placed here for completeness.
Proof. We are going to use the variational characterization (2.20) for dual norms. By Corollary 3.2, the family of functionals We now observe that the functionals (3.27) are equi-coercive on L p (Ω). Indeed, if u ∈ L p (Ω) is such that this of course implies that u ∈ W s,p 0 (Ω). Moreover, by Young inequality and (2.19) we have for a constant C = C(N, p, Ω) > 0 independent of s (provided s is sufficiently close to 1). Thus from (3.28) we get Remark 3.13. We recall the following dual characterization of · W −s,p ′ (Ω) from [6, Section 8] where R * s,p is the adjoint of the linear continuous operator R s,p : W s,p for every u ∈ W s,p 0 (Ω).
Formula (3.29) is the nonlocal analog of the well-known duality formula Then we end this section with the following curious convergence result.

4.1.
Convergence of the variational eigenvalues. By Theorem 3.1, we already know that For every 1 < p < ∞, let us define the functional g p : We now observe that for every increasing sequence k n and for any sequence {u n } n∈N ⊂ L p (Ω) such that there exists a subsequence {u n j } j∈N such that lim j→∞ g p (u n j ) = g p (u).
Indeed, this is a consequence of Lemma 3.10. Then the functionals E s,p and g p satisfy all the assumptions in [13,Corollary 4.4], which implies where K m,p (Ω) = K ⊂ {u : g p (u) = 1} : K compact and symmetric, i(K) ≥ m .
In order to conclude, we only need to show that the minimax values with respect to the W s,p 0 (Ω)−topology are equal to those with respect to the weaker topology L p (Ω). Observe now that, for every b ∈ R, the restriction of g p to {u ∈ L p (Ω) : E s,p (u) ≤ b} is continuous: for s = 1 this is classical, while for 0 < s < 1 we can appeal for example to [ where we recall that W s m,p (Ω) has been defined in (1.6). By using ( up to choosing 1 − s sufficiently small. By appealing again to Lemma 3.10, this in turn implies that there exists a sequence {s k } k∈N with s k ր 1 such that the corresponding sequence of eigenfunctions {u s k } k∈N converges strongly in L p to a function u ∈ W 1,p 0 (Ω). By strong convergence, we still have u L p (Ω) = 1.
In order to prove that u is an eigenfunction of the local problem, let us notice that each u s k weakly solves Thus it is the unique minimizer of the following strictly convex problem Observe that the sequence {F s k } k∈N converges strongly in L p ′ (Ω) to the function (4.5) F = −K(p, N ) λ 1 m,p (Ω) |u| p−2 u, thanks to the strong convergence of {u s k } k∈N and to the first part of the proof. By appealing to the Γ−convergence result of Corollary 3.2, we thus get that u is a solution (indeed the unique, again by strict convexity) of the limit problem min v∈L p (Ω) with F ∈ L p ′ (Ω) defined in (4.5). As a solution of this problem, u has to satisfy the relevant Euler-Lagrange equation, i.e. u weakly solves This proves that the renormalized eigenfunctions {u s k } k∈N converges strongly in L p (Ω) to an eigenfunction u corresponding to λ 1 m,p (Ω) having unit norm. In order to improve the convergence in W t,q 0 (Ω) for every p ≤ q < ∞ and every t < p/q, it is now sufficient to use the interpolation inequality of Proposition 2.1 with r = +∞ so that α = s k p/q. Observe that since s k is converging to 1, if t < p/q we can always suppose that t < s k p q , up to choosing k large enough. This yields for a constant 4 C = C(N, p, q, t) > 0 which varies from a line to another as desired.
Remark 4.1 (Pushing the convergence further). In the previous result, we used that the initial convergence in L p norm can be "boosted" by combining suitable interpolation inequalities and regularity estimates exhibiting the correct scaling in s. Thus, should one obtain that eigenfunctions are more regular with good a priori estimates, the previous convergence result could still be improved. Though it is known that eigenfunctions are continuous for every 1 < p < ∞ and 0 < s < 1 (see [26,28]), unfortunately the above mentioned results do not provide estimates with an explicit dependence on s and thus we can not directly use them.
In the case p = 2, regularity estimates of this type can be found in [8,Lemma 4.4] for bounded solutions of the equation in the whole space where f is a (smooth) nonlinearity. For such an equation, the authors prove Schauder-type estimates for the solutions, with constants independent of s (provided s > s 0 > 0).

Appendix A. Courant vs. Ljusternik-Schnirelmann
Here we prove that for p = 2 the variational eigenvalues defined by the Ljusternik-Schnirelman procedure (1.5) coincide with the usual eigenvalues coming from Spectral Theory (see Theorem A.2 below). Thus in particular for p = 2 definition (1.5) give all the eigenvalues. This fact seems to belong to the folklore of Nonlinear Analysis, but since we have not been able to find a reference in the literature, we decided to include this Appendix.
Let H 1 ⊂ H 2 be two separable infinite dimensional Hilbert spaces, endowed with scalar products ·, · H i and norms On the space H 1 is defined a symmetric bilinear form Q : H 1 × H 1 → R. We assume the following: 1. the inclusion I : H 1 → H 2 is a continuous and compact linear operator; 2. Q is continuous and coercive, i.e. for C ≥ 1 1 Thus Q defines a scalar product on H 1 , whose associated norm is equivalent to · H 1 . We set S = {u ∈ H 1 : u H 2 = 1}. Then the restriction of the functional u → Q[u, u] to S has countably many critical values 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ m ≤ · · · ր +∞, with associated a sequence of critical points {ϕ n } n∈N ⊂ S defining a Hilbertian basis of H 2 . The hypotheses above guarantee that R is a well-defined compact, positive and self-adjoint linear operator. Then discreteness of the spectrum follows from the Spectral Theorem, see for example [ A minimizer for the previous problem is given by (A.1) F m := Span ϕ 1 , . . . , ϕ m .