Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends results in the literature for the N-Laplacian operator.


Overview.
Let Ω be a bounded domain in R N with N ≥ 2 and with Lipschitz boundary ∂Ω. We denote by ω N −1 the measure of the unit sphere in R N and N ′ = N/(N − 1). Since the time when the Trudinger-Moser inequality was first proved (cf. [7,23,27]) existence and multiplicity of solutions for various nonlinear problems with exponential nonlinearity were investigated. For instance, Adimurthi [1] proved the existence of a positive solution to the quasi-linear elliptic problem where ∆ N u := div(|∇u| N −2 ∇u) is the N -Laplacian operator for 0 < λ < λ 1 (N ), being λ 1 (N ) > 0 the first eigenvalue of ∆ N with Dirichlet boundary conditions, see also [10]. The case N = 2 was investigated in [8,9], where the existence of a nontrivial solution was found for λ ≥ λ 1 . Recently, in [28] it was proved that problem (1.1) admits a nontrivial weak solution whenever λ > 0 is not an eigenvalue of −∆ N in Ω with Dirichlet boundary conditions. In addition in [28] a bifurcation result for higher (nonlinear) eigenvalues (which are suitably defined via the cohomological index) is also obtained, yielding in turn multiplicity results. The issue of Trudinger-Moser type embeddings for fractional spaces is rather delicate and only quite recently, Parini and Ruf [25] (see also the refinement obtained in [17]) provided a partial result in the Sobolev-Slobodeckij space We also refer the reader to [18,19,21,24] for results in a different functional framework, namely the Bessel potential spaces H s,p . In fact, they proved that the supremum α N,s (Ω) of α ≥ 0 with (1.2) sup u∈W s,N/s 0 (Ω), [u] s,N/s ≤1ˆΩ e α |u| N/(N−s) dx < +∞, is positive and finite. Furthermore, they proved the existence of α * N,s (Ω) ≥ α N,s (Ω) such that the supremum in (1.2) is +∞ for α > α * N,s (Ω). On the other hand it still remains unknown whether α N,s (Ω) = α * N,s (Ω). The case N = 1 and s = 1/2 was earlier considered in [16] (see also [14]), where the authors study the existence of weak solutions to the problem where C s > 0 is a suitable normalization constant. We also mention [11,12] for other investigations in the one dimensional case on the whole space R, facing the problem of the lack of compactness. In particular in [12], the existence of ground state solutions for the problem was proved, where f is a TrudingerMoser critical growth nonlinearity.
To the authors' knowledge, in the framework of the Sobolev-Slobodeckij spaces W s,N/s 0 (Ω), fractional counterparts of the local quasilinear N -Laplacian problem (1.1) were not previously tackled in the literature. This is precisely the goal of this manuscript.
1.2. The main result. Let N ≥ 1 and s ∈ (0, 1). In the following, the standard norm for the L p space will always be denoted by | · | p . For λ > 0, we consider the quasilinear problem where (−∆) s N/s is the nonlinear nonlocal operator defined on smooth functions by We refer the interested reader to [22] and the references therein for an overview on recent progresses on existence, nonexistence and regularity results for equations involving the fractional p-laplacian operator (−∆) s p , p > 1. The standard sequence of eigenvalues for (−∆) s N/s via the Krasnoselskii genus does not furnish enough information on the structure of sublevels and thus the eigenvalues will be introduced via the cohomological index. Assume that λ k ≤ λ < λ k+1 = · · · = λ k+m < λ k+m+1 for some k, m ≥ 1 and then problem (1.3) has m distinct pairs of nontrivial solutions ± u λ j , j = 1, . . . , m such that u λ j → 0 as λ ր λ k+1 . In particular, if This result, which follows from the results in Section 5, is nontrivial since the classical linking arguments of [8,9] cannot be used in the quasi-linear setting. Instead the abstract machinery developed in [28] will be applied. We also would like to stress that, since the Trudinger-Moser embedding (1.2) still holds with nonoptimal exponent (contrary to the local case), it is not clear how to prove Brezis-Nirenberg type results, namely that problem (1.3) admits a nontrivial weak solution whenever λ > 0 is not an eigenvalue of (−∆) s N/s .

Preliminaries
As anticipated in the introduction, we work in the fractional Sobolev space W A function u ∈ W s,N/s 0 As proved in [15, Proposition 2.12], a weak solution turns into a poinwise solution if u ∈ C 1,γ loc for some γ ∈ (0, 1) sufficiently close to 1. The integral on the right-hand side is well-defined in view of [25,Proposition 3.2] and the Hölder inequality. Weak solutions coincide with critical points of the C 1 functional We recall that W s,N/s 0 (Ω) is uniformly convex, and hence reflexive. Indeed, for u ∈ W s,N/s 0 Then the mapping u → u is a linear isometry from W s,N/s 0 (Ω) to L N/s (R 2N ), so the uniform convexity of L N/s (R 2N ) gives the conclusion.
We also have the following Brézis-Lieb lemma in W s,N/s 0 (Ω).
The main result of this section is the following theorem, which is due to P.L. Lions [20] in the local case s = 1.
(Ω) with u j = 1 for all j ∈ N and converging a.e. to a nonzero function u, then . The first integral on the right-hand side is finite, and the second integral equalŝ Then α q u j − u N/(N −s) ≤ β and hence the last integral is less than or equal tô for all sufficiently large j, which is bounded since β < α N,s (Ω) and v j = 1.
We close this preliminary section with a technical lemma.

Palais-Smale condition
Proof. For any M > 0, writê By Lemma 2.3 (i) and (3.1), we havê and the desired conclusion follows by letting j → ∞ first and then M → ∞.
We are now ready to prove Theorem 3.1.
Now suppose that 0 < c < (s/N ) α N,s (Ω) (N −s)/s . We claim that the weak limit u is nonzero. Suppose u = 0. Then where 1/p + 1/q = 1. The first integral on the right-hand side converges to zero since u = 0, while the second integral is bounded for j ≥ j 0 since q |u j | N/(N −s) = q α | u j | N/(N −s) with q α < α N,s (Ω) and Then u j → 0 by (3.3), and hence c = 0 by (3.2) and (3.5), a contradiction. So u is nonzero.
For v ∈ C ∞ 0 (Ω), an argument similar to that in the proof of Lemma 3.2 using the estimate Then this holds for all v ∈ W s,N/s 0 (Ω) by density, and taking v = u gives Next we claim that We have where u j = u j / u j . By Then u j N/(N −s) ≤ α − 2ε for all j ≥ j 0 for some j 0 , and The last expression goes to zero as M → ∞ uniformly in j since u j is bounded and (3.9) holds, so (

Eigenvalue problem
The asymptotic problem associated with (1.3) as u goes to zero is the eigenvalue problem The weak formulation of this problem can be written as the operator equation (Ω) such that u j ⇀ u and A(u j ), u j − u → 0 as j → ∞ has a subsequence that converges strongly to u (see e.g. [26,Proposition 1.3]). Moreover, B is a compact operator since the embedding  [26,Proposition 3.53]). The main result of this section is the following.
Theorem 4.1. If λ k < λ k+1 , then the sublevel set Ψ λ k contains a compact symmetric subset of index k.
First a couple of lemmas.
Lemma 4.2. The operator A is strictly monotone, i.e., for some α, β ≥ 0, not both zero, and then the equality in the first inequality gives Since u and v vanish a.e. in R N \ Ω, it follows that αu = βv a.e. in Ω.  Proof. The existence follows from a standard minimization argument and the uniqueness from Lemma 4.2. Clearly, J is homogeneous of degree (N − s)/s. To see that it is continuous, let w j → w in L N/s (Ω) and let u j = J(w j ), so Testing with v = u j gives by the Hölder inequality, which together with the imbedding W s,N/s 0 (Ω) ֒→ L N/s (Ω) shows that (u j ) is bounded. Therefore, a renamed subsequence of (u j ) converges to some u weakly, strongly in L N/s (Ω) and a.e. in Ω. Then u is a weak solution of problem (4.4) as in the proof of Theorem 3.1, so u = J(w). Testing (4.6) with u j − u gives We are now ready to prove Theorem 4.1.
In particular, we have the following existence result. .
We only give the proof of Theorem 5.2. The proof of Theorem 5.1 is similar and simpler. The proof will be based on an abstract critical point theorem proved in Yang and Perera [28] that generalizes Bartolo et al. [3,Theorem 2.4].
Let Φ be an even C 1 -functional on a Banach space W . Let A * denote the class of symmetric subsets of W , let r > 0, let S r = {u ∈ W : u = r}, let 0 < b ≤ ∞, and let Γ denote the group of odd homeomorphisms of W that are the identity outside Φ −1 (0, b). The pseudo-index of M ∈ A * related to i, S r , and Γ is defined by Benci [4]).
Theorem 5.4 ( [28, Theorem 2.4]). Let K 0 and B 0 be symmetric subsets of M = {u ∈ W : u = 1} such that K 0 is compact, B 0 is closed, and for some k ≥ 0 and m ≥ 1. Assume that there exists R > r such that .
If, in addition, Φ satisfies the (PS) c condition for all c ∈ (0, b), then each c * j is a critical value of Φ and there are m distinct pairs of associated critical points. We are now ready to prove Theorem 5.2. We take B 0 := Ψ λ k+1 , so that i(M \ B 0 ) = k by (4.3). Let R > r > 0 and let K, B, and X be as in Theorem 5.4. By Lemma 2.3 (iv), To prove that u λ j → 0 as λ ր λ k+1 , it suffices to show that for every sequence ν n ր λ k+1 , a subsequence of v n := u νn j converges to zero. We have