Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-\Delta)^s u = |u|^{2^*_s-2-\varepsilon}u&\text{in } B_R, \\ u = 0&\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-\Delta)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a"tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.


Introduction
Let s ∈ (0, 1), let n ∈ N be such that n > 2s and let Ω ⊂ R n be a bounded smooth domain. Consider the following non-local elliptic problem (−∆) s u = f (u) in Ω, u = 0 in R n \ Ω, (1.1) where (−∆) s is the s-Laplacian, f (u) = |u| 2 * s −2−ε u or f (u) = εu + |u| 2 * s −2 u for n > 6s, ε > 0 is a small parameter and 2 * s := 2n n−2s is the critical exponent for the fractional Sobolev embedding. In the recent paper [7] the authors studied the asymptotic properties of least energy positive solutions to Problem (1.1), i.e. positive solutions u ε such that u ε 2 s → S n 2s s , as ε → 0 + , where · s is the standard seminorm in H s (R n ) and S s is the best fractional Sobolev constant. They proved, in the case of the spectral fractional Laplacian, that such solutions concentrate and blow-up at some point x 0 ∈ Ω, providing also information about the blow-up speed with respect to ε. Their result is hence the fractional counterpart of the classical results of Han and Rey (see [12,23]) for the Laplacian.
Motivated by that, it is natural to ask whether is possible or not to extend to the fractional framework analogous results about the asymptotic behavior of least energy sign-changing solutions to almost critical and critical semilinear elliptic problems for the Laplacian (see [2,3,14,15,16,22]).
At first glance the answer seems to be positive, but differently from the case of constantsign solutions, several difficulties arise when studying the qualitative properties of sign-changing solutions. Indeed, in view of the non-local interactions between the nodal components, we cannot take benefit from the fractional moving plane method (see [5]), and the strong maximum principle does not work properly (see [8,Sect. 1]). Moreover, when considering least energy sign-changing solutions, i.e. sign-changing solutions u ε to (1.1) such that u ε 2 s → 2S n 2s s , as ε → 0 + , we cannot establish by mere energetic arguments, neither by a Morse-index approach, the number of nodal components. In the local case it is well known that they possess exactly two nodal regions, since each nodal component carries the energy S n 2 1 (see [2,3]). In the fractional case we can only say that both the positive and the negative part globally carry the same energy S n 2s s , when ε → 0 + , but this does not hold true in general for each individual nodal component and causes many troubles when performing the asymptotic analysis.
In our contribution [8] we tackled the case of least energy radial sign-changing solutions to Problem (1.1) in a ball, when f (u) = εu + |u| 2 * s −2 u is the critical nonlinearity and n > 6s. In the spirit of the pioneering papers [10,11], we showed that these solutions change sign at most twice and exactly once when s is close to 1. Moreover, when s > 1 2 , we proved that they behave like a tower of two bubbles as ε → 0 + , namely, the positive and the negative part blowup and concentrate at the same point (which is the center of the ball) with different speeds. Nevertheless, we needed to assume that these solutions change sign exactly once to determine which one between the positive and the negative part blew-up faster (see [8,Sect.1]).
We point out that for 2s < n ≤ 6s, according to a classical result of Atkinson, Brezis, and Peletier (see [1]), radial sign-changing solutions in a ball may not exist when ε > 0 is close to zero, while they do exist for n > 6s (see [8,Theorem 3.7]).
In this paper we consider slightly subcritical nonlinearities f (u) = |u| 2 * s −2−ε u, and we extend the results of [8] to all s ∈ (0, 1) without any extra assumption. The same proofs work also in the case of critical nonlinearities with minor modifications. The main result of our paper is the following: Theorem 1.1. Let s ∈ (0, 1) and let n > 2s. Let (u ε ) ε be a family of least energy radial sign-changing solutions to

2)
where B R is the euclidean ball of radius R > 0 centered at the origin. Assume without loss of generality that u ε (0) > 0 and set M ± ε := |u ± ε | ∞ . Then, as ε → 0 + it holds that: converges in C 0,α loc (R n ), for some α ∈ (0, 1), to the fractional standard bubble U 0,µ in R n centered at the origin and such that U 0,µ (0) = 1, (v) if s ∈ ( 1 2 , 1) then |y ε | → 0, where y ε ∈ B R is any point such that |u ε (y ε )| = M − ε . Theorem 1.1 establishes the first existence result of sign-changing bubble-tower solutions for non-local semilinear elliptic problems driven by the s-Laplacian, when s > 1 2 . For s ∈ (0, 1 2 ] we still get that the positive and the negative part blow-up with different speeds, but for the negative part we cannot provide any information about its concentration point. From a technical point of view (see the proof of Lemma 4.3) this is due to the fractional Strauss inequality for radial functions, namely sup where K n,s is an explicit positive constant depending only on n, s. Indeed, as pointed out in [6, Remark 2, Remark 4], (1.3) does not hold when s ∈ (0, 1 2 ]. We also stress that in view of the non-local nature of our problem the positive and negative parts are are not, in general, sub or super solutions to Problem (1.1) in their domain of definition, so it seems quite hard to overcome this difficulty by applying scaling arguments to u + ε , u − ε separately. On the other hand, as proved in [3] for the Laplacian, if the blow-up speeds of u + ε , u − ε are comparable then they must concentrate at two separate points. Therefore, in view of (ii), we believe that also for s ∈ (0, 1 2 ] the negative part concentrates at the center of the ball. We plan to investigate this question in separate paper. In addition, we think that, as done in [22] for the Laplacian, by using a Lyapunov-Schmidt reduction method it should be possible to construct sign-changing bubble-tower solutions in general bounded domains, for all s ∈ (0, 1).
We point out that, thanks to (ii) and (iii), any global maximum point is close to the origin, when ε > 0 is sufficiently small. Moreover, in Lemma 4.6 we specify that any such a point belongs to the nodal component containing the origin and blows-up faster than any other extremal value achieved in the other nodal components, independently on the number of sign-changes. In the local case, by using ODE techniques, it is well known that the global maximum point is the origin and the absolute values of the extrema are ordered in a radially decreasing way. Our result allows to recover these properties, at least asymptotically, via PDE-only arguments.
In the second part of this work we study the nodal set of least energy radial sign-changing solutions to (1.2). We remark that, if u ε is a nodal solution to (1.2) and u ε ≥ 0 in a subdomain D ⊂ B R , the fractional strong maximum principle does not ensure, in general, that u ε > 0 in D (see [4,Remark 4.2] and [8,Sect. 1]). In addition [19,Theorem 1.4] only grants that u ε does not vanish in a set of positive measure. Nevertheless, combining the results of [8] with a new argument based on energy and regularity estimates, we show that for any s ∈ (0, 1) least energy radial sign-changing solutions to (1.2) vanish only where a change of sign occurs (see Lemma 4.5, Lemma 5.2).
Finally, in Theorem 5.8 we prove that for any s 0 ∈ (0, 1), if there exists a L 2 (B R )-continuous family A = {u ε,s } s∈[s0,1) of least energy nodal radial solutions to (1.2), then every element of the family changes sign exactly once, provided that ε > 0 is small enough. The key ingredients of the proof are the estimates contained in [24,Theorem 1.2], and the continuity of the map is the infimum of the energy over the nodal Nehari set, which is a new result of its own interest (see Proposition 5.6).
The outline of the paper is the following: in Section 2 we fix the notation and we recall some known results about the existence of sign-changing solutions to (1.2), in Section 3 we study the asymptotic behavior, as ε → 0 + , of the energy levels C M(Ω) (s, ε) in generic bounded domains. In Section 4 we prove Theorem 1.1. Finally in Section 5 we analyze the nodal set of least energy radial sign-changing solutions to (1.2) and we prove Theorem 5.8.

Notation and preliminary results
In this section we recall some definitions and known facts that will be used in this work.
2.1. Functional setting, standard bubbles. In this paper (−∆) s stands for the (restricted) s-Laplacian operator, which is formally defined as where the constant C n,s is given by Let s ∈ (0, 1) and let n > 2s. For a given smooth bounded domain Ω ⊂ R n , we consider as a working functional space the Sobolev space |u(x) − u(y)| 2 |x − y| n+2s dx dy, and whose associated scalar product is The Sobolev space D s (R n ) is defined as the completion of C ∞ 0 (R n ) with respect to the above norm. By the fractional Sobolev embedding theorem it holds that D s (R n ) ֒→ L 2 * s (R n ) and The previous embeddings are continuous, and the second one is compact when p ∈ [1, 2 * s ). The best Sobolev constant is characterized as where | · | p denotes the usual L p -norm, for p ∈ [1, ∞]. To simplify the notation we will not specify the domain of integration in | · | p , but it will be always clear from the context that it is either R n , or a fixed bounded domain Ω, or a family of bounded domains when considering rescaled functions. The value of S s is explicitly known (see [9]), it depends continuously on s ∈ [0, 1], and it is achieved exactly by the family then the functions also known as "standard fractional bubbles", satisfy for all µ > 0, x 0 ∈ R n and

Existence of constant-sign and sign-changing solutions.
Let Ω ⊂ R n be a smooth bounded domain and consider the problem where ε ∈ (0, 2 * s − 2). Weak solutions to (2.4) correspond to critical points of the functional The Nehari manifold and the nodal Nehari set are, respectively, defined by Since we deal with subcritical nonlinearities, by standard variational methods we know that there exists a minimizer u ε ∈ N s,ε (Ω) of I s,ε , and we set Moreover, the minimizer is a weak solution to (2.4) and it is of constant sign. We also remark that, equivalently, constant-sign weak solutions to (2.4) can be found as minimizers to and the following relation holds (2.5) In the case of sign-changing solutions, as proved in [28], there exists a minimizer of the energy over the nodal Nehari set, and it is a weak solution to (2.4). We refer to such solutions as least energy sign-changing (or nodal) solutions and we set Let us now turn our attention to the radial case. Taking Ω = B R , where B R = B R (0) denotes the ball in R n of radius R > 0 centered at the origin, we set N r s,ε (B R ) := {u ∈ X s 0 (B R ) ; u ∈ N s,ε (B R ) and u is radially symmetric}, M r s,ε (B R ) := {u ∈ X s 0 (B R ) ; u ∈ M s,ε (B R ) and u is radially symmetric}. As a consequence of the fractional moving plane method (see [5]), positive solutions of (2.4) in B R are radially symmetric and radially decreasing. In particular, it holds that Concerning the case of nodal solutions, arguing as in [28] we obtain least energy radial signchanging solutions as minimizers of the energy over the radial nodal Nehari set, and as before we denote We point out that it is not known whether or not C M r (BR) (s, ε) coincide with C M(BR) (s, ε), but they have the same limit when ε → 0 + (see Lemma 3.3).
3. Asymptotic analysis of the energy levels as ε → 0 + In this section we study the asymptotic behavior as ε → 0 + of the energy levels C N (Ω) (s, ε), C M(Ω) (s, ε) defined in Sect. 2. We begin with the following technical result.
Lemma 3.1. Let s ∈ (0, 1) and n > 2s. Let Ω ⊂ R n be a domain, let x 0 ∈ Ω and ρ > 0 be such where U x0,µ is defined by (2.2), then the following estimates hold: where the constants C are positive and depend only on n, s, x 0 , µ and ρ. Moreover, for any where the appearing constants are positive and depend only on n, s, x 0 and ρ. Let 0 < s 0 < s 1 ≤ 1 and let n > 2s 1 . Then, if s ∈ [s 0 , s 1 ) and ε ∈ 0, 2s0 n−2s0 , both τ 0 and the above constants C can be taken in such a way that they depend on n, µ, ρ, s 0 , s 1 , but not on s, τ and ε.
Proof. Inequalities (3.2) are proved in [26], [27] and hold true for all sufficiently small τ > 0 with constants C independent on τ . Concerning the dependence of the constants C on the other parameters we refer to [8,Remark 2.2]. Let us focus on the proof of (3.3). Taking if necessary a smaller τ 0 > 0 so that τ 0 < min{1, 2ρ}, we find that, when where the constants C > 0 depend on n, s, µ, but not on τ nor on ε. Furthermore, since 0 < ε < 2s n−2s we have for some constant C > 0 independent on τ and ε. Recalling the definition of b n,s , one can see that all the previous constants can be taken in a uniform way with respect to s ∈ [s 0 , s 1 ) when n > 2s 1 and ε ∈ 0, 2s0 n−2s0 . Hence the right-hand side inequalities in (3.3) are proved. In order to prove the the left-hand side inequalities it suffice to notice that, thanks to our Then, using also (3.2), we find that s − Cτ n , for some constant C > 0 which depends only on n, s and ρ, but not on τ , ε, and which is uniform with respect to s ∈ [s 0 , s 1 ). The proof is complete.
As a consequence we obtain the following uniform asymptotic result on C N (Ω) (s, ε).
In the next result we describe the asymptotic behavior of C M(Ω) (s, ε), as ε → 0 + . Differently from the case of critical nonlinearities (see [8,Lemma 3.6]), there are some difficulties in proving uniform energy estimates from above which are directly related to C N (Ω) (s, ε). To overcome these difficulties we provide a uniform upper bound in terms of 2s n S n 2s s instead, which is obtained by using as competitors for the energy superpositions of standard bubbles centered at the same point and with different concentration speeds. Lemma 3.3. Let s ∈ (0, 1), n > 2s and let Ω ⊂ R n be a smooth bounded domain. We have Moreover, let 0 < s 0 < s 1 ≤ 1 and n > 2s 1 . Then there existsε =ε(s 0 , s 1 ) ∈ (0, 2 * s0 − 2) such that for every ε ∈ (0,ε) where the function g 2 does not depend on s and g 2 (ε) → 0 as ε → 0 + . The same result holds for M r s,ε (B R ). Proof. Let us fix s ∈ (0, 1), n > 2s and let Ω ⊂ R n be a smooth bounded domain. We claim that (3.14) As an immediate consequence, from Lemma 3.2, we get that 2s n S n 2s To prove (3.14) it suffices to notice that, given u ∈ M s,ε (Ω), then for every α, β > 0 it holds This follows from the explicit computation of I s,ε (αu + −βu − ), taking into account that (u + , u − ) s < 0 and that sup t≥0 Hence, choosing u ∈ M s,ε (Ω) such that I s,ε (u) = C M(Ω) (s, ε) and α, β in such a way that αu + , βu − ∈ N s,ε (Ω) (which is always possible), we obtain the desired result.
To conclude the proof of (3.12) we need to prove the lim sup inequality. To this end, we consider u s τ ′ and u s τ ′′ of the form (3.1), sharing all the parameters µ, ϕ, ρ, x 0 , apart from τ . To simplify the notation, we assume without loss of generality that 0 ∈ Ω and we take x 0 = 0. Moreover, we choose ρ and µ as in Lemma 3.1 so that (3.2), (3.3) hold true whenever ε is small enough. Finally, for the concentration parameters, we take τ ′ , τ ′′ of the form Notice that δ = 2 * s − 1, when ε is small enough, and that it can be taken in a uniform way with respect to s when s ∈ [s 0 , s 1 ).
Arguing as in [8, Theorem 3.5, Step 2], we infer that To conclude we need to estimate the right-hand side of (3.16). The first crucial fact is that, in (3.16), it is sufficient to consider only linear combinations αu s τ ′ − βu s τ ′′ with α, β in a compact subset of R + ∪ {0}. More precisely, we prove that there existsC > 0 independent on ε (and depending only on s 0 , s 1 when s ∈ [s 0 , s 1 )) such that, for any α, β ≥ 0 satisfying α + β ≥C, it holds Indeed, by a straightforward computation and using Lemma 3.1 we have for some constant C independent on both ε, τ ′ , τ ′′ and s, when s ∈ [s 0 , s 1 ). On the other hand, arguing exactly as in [8, Lemma 3.6] and using again Lemma 3.1, we infer that for any θ ∈ (0, 1) Now, thanks to our choice of µ we have Hence, recalling that τ ′′ = ε 2 n−2s and taking θ = C ′ ε 2 * s −ε , where C ′ will be chosen later, from Lemma 3.1 we obtain that for any ε > 0 small enough, where C does not depend on ε, nor on s when s ∈ [s 0 , s 1 ). Therefore, Then, exploiting the properties of the function t → t t and thanks to the definition of δ, we find C > 0 such that for all sufficiently small ε > 0 Finally, thanks to (3.18) and (3.21) we infer that which implies that there existsC > 0, not depending on ε, such that if (α + β) ≥C then I s,ε (αu s τ ′ − βu s τ ′′ ) ≤ 0, as claimed. We observe thatC can be taken in a uniform way with respect to s, when s ∈ [s 0 , s 1 ).
It remains to treat the case α + β ≤C. To this end we begin with a preliminary estimate on the scalar product between two bubbles. A careful analysis of the argument carried out in [27,Proposition 21] shows that where the constant C does not depend on τ ′ nor on τ ′′ , and it is uniformly bounded with respect to s ∈ [s 0 , s 1 ). Performing a change of variables, and recalling that U s 0,µ solves (2.3), we get that where we used that |U s 0,µ | ∞ = 1, in view of our choice of µ, and where the constant C > 0 does not depend on τ ′ nor on τ ′′ and it is uniformly bounded with respect to s ∈ [s 0 , s 1 ). Summing up, and recalling the definition of τ ′ and τ ′′ , we obtain where we used that |u s τ ′′ | ∞ = ε −1 . Even in this case all the appearing constants are independent on ε, and they are uniformly bounded with respect to s when s ∈ [s 0 , s 1 ). Then, using again the elementary estimate sup t≥0 where all the constants C > 0, and thus g, do not depend on s, when s ∈ [s 0 , s 1 ). In particular, g satisfies g(ε) → 0 as ε → 0 + . At the end, putting together ( For the proof of the second part, fixing 0 < s 0 < s 1 ≤ 1, then, thanks to Lemma 3.2 and the definition of g, we deduce that inequalities (3.14) and (3.24) are uniform with respect to s when s ∈ [s 0 , s 1 ). At the end, arguing as in Lemma 3.2 we obtain (3.13), for some function g 2 independent on s and such that g 2 (ε) → 0, as ε → 0 + .
In the radial case the proof is identical. Indeed, since in the construction we take standard bubbles centered at the same point, then the functions αu s τ ′ −βu s τ ′′ are radial and thus admissible competitors. The proof is then complete.

Asymptotic analysis of least energy radial sign-changing solutions
In this section we study the asymptotic behavior of least energy radial nodal solutions to (1.2), as ε → 0 + . Theorem 1.1 will be a consequence of the results contained in this section. We begin by a couple of preliminary known results. and set M s,ε,± := |u ± s,ε | ∞ . As ε → 0 + we have: The same results hold for a family (u s,ε ) ⊂ M r s,ε (B R ) of radial solutions to Problem (1.2) such that I s,ε (u s,ε ) = C M r (BR) (s, ε). Moreover, for every 0 < s 0 < s 1 ≤ 1 and n > 2s 1 , the limits (i) − (iii) are uniform with respect to s ∈ [s 0 , s 1 ).
Proof. It suffices to argue as in [8,Lemma 4.3], with some minor modifications.
The following estimate will play a central role in this paper.
Then v ∈ C 0,s (R n ) and v C 0,s (R n ) ≤ C|g| L ∞ (BR) where the constant C > 0 depends only on n, s 0 , s 1 and R 0 , but neither on s nor on R.
Proof. The estimate is a consequence of results contained in [24]. Concerning the dependence on the parameters s 0 , s 1 , it can be deduced from a careful analysis of the proof in [24] (see also [8,Proposition 2.3]). As for the dependence of the constant C on the domain, it turns out that C depends only on the radii coming from the outer and inner ball conditions for B R . Hence, it is clear that C can be chosen in a uniform way with respect to R if we assume that R ≥ R 0 , for some R 0 > 0.
In the next result we study the asymptotic behavior of the rescaled solutions defined in (4.1).
Proof. As seen in the proof of Lemma 4.3, the functionsũ s,ε weakly satisfy (4.2) and by construction it holds that |ũ s,ε | ∞ ≤ 1. Then, since M βs,ε s,ε R → +∞, thanks to Proposition 4.2 and a standard argument, up to a subsequence, we havẽ u s,ε →ũ s in C 0,α loc (R n ), for someũ s ∈ C 0,α loc (R n ), α ∈ (0, s). We point out thatũ s ≡ 0. Indeed, let x ε ∈ B R be such that |u s,ε (x ε )| = M s,ε . By construction we have |ũ s,ε (M βs,ε s,ε x ε )| = 1, and thanks to Lemma 4.3 we infer that the point M βs,ε s,ε x ε stays in a compact subset of R n . Therefore, from the C 0,α locconvergence ofũ s,ε in R n , we get thatũ s is non trivial. Now we show thatũ s ∈ D s (R n ). In fact, by Lemma 4.1-(i) and since M s,ε → +∞, we infer that s , as ε → 0 + , and in particular, up to a subsequence,ũ s,ε ⇀ v for some v ∈ D s (R n ). Then, sinceũ s,ε →ũ s in C 0,α loc (R n ), we get that v =ũ s and we are done. In addition, applying Fatou's Lemma we also deduce that ũ s 2 Let us prove now thatũ s is a weak solution to (4.5). Indeed, for every ϕ ∈ C ∞ c (R n ), sincẽ u s,ε is a weak solution to (4.2) we have where ε is small enough so that supp ϕ ⊂ B M βs,ε s,ε R . Sinceũ s,ε →ũ s for a.e. x ∈ R n , using the well known relations (see e.g. [8]) and and thanks to Lebesgue's dominated convergence theorem, passing to the limit as ε → 0 + in (4.7) we infer that Now, sinceũ s ∈ D s (R n ) we are allowed to use again (4.8), obtaining thatũ s weakly satisfies We prove now thatũ s is of constant sign. To this end, assume by contradiction thatũ s is signchanging. Then, usingũ ± s ∈ D s (R n ) as test functions in (4.9) and recalling that (ũ + s ,ũ − s ) s < 0, we get that ũ ± s 2

Hence, by the Sobolev inequality we infer that
and thus 2S n 2s s . Finally, usingũ s as a test function in (4.9) we have ũ s 2 s = |ũ s | 2 * s 2 * s , and we obtain that 2S n 2s s < ũ s 2 s , which contradicts (4.6). At the end we notice that, sinceũ s is a pointwise limit of radial functions, it is radial too. Moreover, sinceũ s is of constant sign, assuming without loss of generality thatũ s ≥ 0, we easily deduce, by the fractional strong maximum principle and the fractional moving plane method (see [5]), thatũ s is also decreasing along the radii and thusũ s achieves its maximum at the origin. The proof is complete.
Proof. We prove directly the second part of the Lemma. Assume by contradiction that there exist 0 < s 0 < s 1 ≤ 1, three sequences ε k → 0 + , C k → 0 + , (s k ) k ∈ [s 0 , s 1 ), and a sequence of nodal radial least energy solutions u k := u s k ,ε k such that |u k (0)| < C k . Up to a subsequence, we can always assume that s k → σ, with σ ∈ [s 0 , s 1 ]. Now, only two possibilities can occur: setting M k := |u k | ∞ , either (M k ) k is a bounded sequence or there exists a subsequence such that M k → +∞.
Assume that (M k ) k is bounded. We first observe that (M k ) k is bounded away from zero, otherwise we could find a subsequence such that M k → 0, but this would contradict Lemma 4.1. Therefore, up to a subsequence we can assume that M k → l, for some real number l > 0. Adapting the arguments of Lemma 4.4 and using Lemma 3.3 we readily infer that, up to a subsequence, u k ⇀ u in X s0 0 (B R ) and u k → u in C 0,α (R n ), for some α ∈ (0, s 0 ). Furthermore we have u ≡ 0 and it holds that Using that u k → u in L 2 (B R ), thanks to the fractional Sobolev embedding and Fatou's Lemma we find where the last inequality is a consequence of the second part of Lemma 3.3, while the equalities are due to the interpretation via the Fourier transform of the fractional Laplacian (see e.g. [13]). From this discussion it follows that u is a non trivial weak solution of This readily contradicts the Pohozaev identity when σ = 1. If σ < 1, the fractional Pohozaev identity only implies the nonexistence of constant-sign solutions to (4.12) (see [25]). In order to obtain a contradiction we show that u is of constant sign. Indeed, arguing as in the proof of Lemma 4.4, we have that any sign-changing solution u to (4.12) must satisfy u 2 σ > 2σ n S n 2σ σ . Hence, thanks to (4.11) it follows that u is of constant sign and we get the desired contradiction.
In the next lemma we show, independently on the number of sign-changes, that M s,ε is achieved in the nodal component containing the origin and blows up faster than every other extremal value achieved in the other components. Before stating the result we introduce some notation. Assuming without loss of generality that u s,ε (0) > 0, thanks to Lemma 4.5, for all sufficiently small ε > 0 the following quantities are well defined: In other words, r 1 ε is the first nodal radius, M + s,ε is the maximum of the solution in the first nodal component, whileM s,ε is the absolute maximum achieved in the other nodal components.

Characterization of the nodal set
In this section we study the nodal set of least energy radial sign-changing solutions to Problem (1.2). We begin with a couple of known preliminary results, which provide, respectively, an upper bound on the number of sign changes and a characterization of the nodal set.
Proof. It suffices to argue as in [8,Theorem 5.1] first, and then as in [8,Theorem 5.2], taking into account Lemma 3.3 and Lemma 4.1. In particularε > 0 is given by Lemma 3.3.
Proof. It suffices to takeε s := min{ε s ,ε s }, where ε s ,ε s are given by Lemma 4.5 and Lemma 5.1, respectively. Then, the results follows immediately by adapting the arguments of [8,Theorem 1.2].
In the next Lemma we prove the upper semi-continuity of the map s → C M r (BR) (s, ε). Proof. Let us fix s 0 , s 1 , n and ε as in the statement. Let (s k ) k ⊂ [s 0 , s 1 ) be a sequence such that s k → σ ∈ [s 0 , s 1 ], and consider a radial solution u σ,ε of (1.2) which realizes C M r (BR) (σ, ε). Assume that σ < 1. We aim to construct a sequence of almost minimizers of C M r (s k , ε). We proceed in three different steps. We point out that when σ = 1 the proof is identical, taking into account the conventions (−∆) 1 u = −∆u, u 2 1 = |∇u| 2 2 , and that (u + , u − ) 1 ≡ 0 for all u ∈ H 1 0 (B R ).
Step 1. There exists a sequence (ϕ We first observe that, thanks to Lemma 5.2, the boundaries of supp (u ± σ,ε ) consist in a finite union of spheres. Therefore, adapting known density results (see e.g. [20]) we find two sequences of radial functions . Observe that, from the continuity of the scalar product, we have Now we recall that it is always possible to find α j > 0, β j > 0 such that α jφ [8,Remark 3.4]), which is equivalent to solving the following We claim that, definitely, 0 < α < α j < α and 0 < β < β j < β, for some positive constants α, α, β, β. Indeed, sinceφ ± j → u ± σ,ε , and u ± σ,ε are non trivial, then the quantities |φ ± j | σ are uniformly bounded and uniformly away from zero. Moreover, by the definition of the scalar product we always have (φ + j ,φ − j ) σ < 0. Then, treating (5.1) as an algebraic system in α j , β j having as coefficients |φ ± j | it is easy to verify that, up to a sequence, it cannot happen that α j → +∞ or α j → 0 + , and the same holds for β j . The claim is thus proved.
Let (s k ) k ⊂ (0, 1) be a sequence such that s k → σ. Let us fix a small number τ > 0. Thanks to Step 1, there exists a function ϕ τ ∈ C ∞ c (B R ) ∩ M r σ,ε (B R ) such that On the other hand, thanks to Step 2 there existk =k(τ ) > 0 and a sequence of functions (ϕ k ) k such that ϕ k ∈ C ∞ c (B R ) ∩ M r s k ,ε (B R ) and As a consequence, we get that Therefore, since u σ,ε is a minimizer and ϕ k ∈ M r s k ,ε (B R ), we infer that for all k ≥k(τ ) Taking the lim sup as k → +∞ we get that and since τ > 0 is arbitrary we obtain the desired result. The proof is then complete.
In the next result we prove a uniform bound with respect to s for the L ∞ -norm of the solutions.
Lemma 5.4. Let 0 < s 0 < s 1 ≤ 1, n > 2s 1 and ε ∈ (0,ε), whereε is given by Lemma 3.3. Then there exists C > 0, depending on ε but not on s, such that for every least energy radial sign-changing solution u s,ε ∈ M r s,ε (B R ) of (2.4). Proof. Let us fix s 0 , s 1 , n and ε as in the statement. The first inequality is trivial. As for the second one, it can be proved in two different ways. Indeed, from [21, Theorem 3.2] there exists M ∈ C(R + ) such that |u s,ε | ∞ ≤ M (|u s,ε | 2 * s ). A careful analysis of the proof shows that the function M can be chosen in such a way that M depends only on n, R, s 0 , s 1 and ε, but not on s. Since u s,ε ∈ M r s,ε (B R ) ⊂ N s,ε (B R ) and u s,ε is a least energy sign-changing solution to (1.2), we infer that Thus, thanks to the fractional Sobolev embedding and Lemma 3.3 we deduce that |u s,ε | 2 * s ≤ C 1 , for some constant C 1 > 0 independent on s. Similarly, using that 2C N (BR) (s, ε) ≤ C M r (BR) (s, ε) and Lemma 3.2 we obtain that |u s,ε | 2 * s ≥ C 0 > 0, where C 0 does not depend on s, and the desired result easily follows.
Alternatively, we can argue as follows: fix s 0 , s 1 , n and ε as in the statement. Since u s,ε is a least energy sign-changing solution to (1.2) with u s,ε ∈ M r s,ε (B R ) ⊂ N s,ε (B R ), and since Lemma 3.3 holds, by (5.2) we get that the quantity |u s,ε | and a sequence (u s k ,ε ) k such that δ s k := |u s k ,ε | ∞ → +∞, as k → +∞. Up to a subsequence, s k → σ ∈ [s 0 , s 1 ], as k → +∞. Let us consider the rescaled functions Arguing exactly as in Lemma 4.3 we see that v k → v in C 0,α loc (R n ), for some α ∈ (0, s 0 ), where v ≡ 0. On the other hand, by Fatou's Lemma we have Hence v ≡ 0 and we get a contradiction. The proof is complete.
In the next result we study the asymptotic behavior of the solutions as s goes to some limit value.
Proof. It suffices to argue as in [8,Theorem 6.7], taking into account Lemma 3.3 and Lemma 5.4.
As a corollary of the previous results we obtain the continuity of the map s → C M r (BR) (s, ε).
The following Lemma grants that every least energy nodal radial solution in a ball changes sign exactly once, when s is close to one.
Proof. We begin by recalling that, in the local case, when n ≥ 3 there exists ε 1 > 0 such that, for every ε ∈ (0, ε 1 ), least energy radial sing-changing solutions to change sign exactly once (see e.g. [3]). Now, let us fix s 0 ∈ (0, 1) and define ε 0 := min{ε, ε 1 }, whereε is given by Lemma 5.2 for s 0 and s 1 = 1. Let us fix ε ∈ (0, ε 0 ) and assume by contradiction that there exist (s k ) k ⊂ [s 0 , 1) such that s k → 1 − and a sequence (u s k ,ε ) k of least energy radial sign-changing solutions in B R which change sign exactly twice for any k (these functions change sign at most twice in view of Lemma 5.2). Then, by Proposition 5.6 we have that u s k ,ε → u 1,ε in C 0,α loc (R n ), for some α ∈ (0, s 0 ), and that u 1,ε is a least energy sing-changing solution to (5.4). In particular, in view of our choice of ε, u 1,ε changes sign exactly once.
On the other hand, arguing as in the proof of [8, Theorem 1.3], we infer that the number of sign changes is preserved when passing to the limit as s → 1 − and thus u 1,ε has to change sign twice. This gives a contradiction and concludes the proof.
Finally, we can state and prove Theorem 5.8. We first recall that, when speaking of a L 2 (B R )continuous family A = {v s,ε } s∈[s0,1) of least energy nodal radial solutions to Problem (1.2), we mean a map Φ : [s 0 , 1) → L 2 (B R ) such that Φ is continuous and Φ(s) = v s,ε ∈ M r s,ε (B R ) is a least energy radial sign-changing solution to Problem (1.2) for any s ∈ [s 0 , 1).
In view of the previous disccusion S ε is not empty. We claim that S ε is closed. Indeed, let (s k ) k ⊂ S ε be a sequence such that s k → σ, for some σ ∈ [s 0 , s 1 ], and consider the associated sequence (v s k ,ε ) k ⊂ A. By Lemma 5.5 and thanks to Proposition 5.6, up to a subsequence, we have v s k ,ε → u ε in C 0,α (B R ) for some α ∈ (0, s 0 ), where u ε ∈ X σ 0 (B R ) is a least energy nodal radial solution of (1.2) with s = σ. In particular, v s k ,ε → u ε in L 2 (B R ) and, since we are assuming that A is L 2 (B R )-continuous, it holds that u ε = v σ,ε ∈ A. Now, taking into account Lemma 5.2, since v s k ,ε → u ε in C 0,α (B R ) and v s k ,ε changes sign once for all k, we infer that the only possibility is that v σ,ε changes sign only once. Hence σ ∈ S ε , and the claim is proved.
We claim that S s0,ε is open. To prove the claim we show that the complementary set S c ε is closed. By definition and thanks to Lemma 5.2 we have S c ε = {s ∈ [s 0 , s 1 ] ; v s,ε changes sign exactly twice}.