Abstract
We study the nodal set of stationary solutions to equations of the form \((\Delta )^s u = \lambda _+ (u_+)^{q1}  \lambda _ (u_)^{q1}\quad \text {in }B_1,\) where \(\lambda _+,\lambda _>0, q \in [1,2)\), and \(u_+\) and \(u_\) are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable twophase membrane problem \(q=1\) as well as sublinear equations for \(1<q<2\). We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blowup analysis of the nodal set. As in the local case \(s=1\), we prove that the admissible vanishing orders can not exceed the critical value \(k_q= 2s/(2 q)\). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that \(k_q< 1\), we prove a remarkable difference with the local case: solutions can only vanish with order \(k_q\) and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgrentype or a 2parameter family of Weisstype monotonicity formulas, according to the vanishing order of the solution.
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1 Introduction
The analysis of the nodal set of solutions of elliptic equations has been the subject of an intense study in the last decades, starting from the works [9, 19,20,21], with a special focus on the measure theoretical features of its singular part.
These works provide a fairly complete picture of the geometric structure of the nodal set in the case of solutions of linear equations and they easily extend to a wide class of superlinear equations of type \(\Delta u = f(u)\), provided that the nonlinearity is locally Lipschitz continuous, that \(f(0) = 0\) and that \(u \in L^\infty _{\tiny {{\text{ loc }}}}\). From a geometric point of view, the nodal set of a weak solution of class \(C^1\) splits into a regular part, which is locally a \(C^1\) graph, and a singular set which is a countable union of subsets of sufficiently smooth \((n2)\)dimensional manifolds. Moreover these equations satisfy the strong unique continuation principle and the solutions vanish with finite integer order (see e.g. [15, 16, 21]). A similar structure also holds under weaker assumptions, that is, for weak solutions of linear equations in divergence form with Lipschitz coefficients and bounded first and zero order terms (see [19]).
Instead, the picture change drastically if we switch to semilinear elliptic equations with nonLipschitz nonlinearities: given \(q \in [1,2)\), let us consider for example the class of equations
where \(\lambda _+,\lambda _>0, q \in [1,2), B_1\) is the unit ball in \(\mathbb {R}^n\) and \(u_+=\max (u,0)\) and \(u_=\max (u,0)\) are respectively the positive and negative part of u. Notice that the main feature of these equations stays in the fact that the right hand side is not locally Lipschitz continuous as function of u, and precisely has sublinear character for \(q \in (1,2)\) and discontinuous behaviour for \(q=1\). It is well known in the literature that in the case \(\lambda _+,\lambda _\le 0\), the features of the nodal set of solutions are substantially different in comparison with the linear case since dead cores appear and no unique continuation can be expected.
However, in the unstable setting the solutions resembles some features of the linear case. Indeed, recently in [33] have been proved the validity of the unique continuation principle for every \(q \in [1,2)\) by controlling the oscillation of the Almgrentype frequency formula for solutions with a dead core. On the other hand, in [25] has been shown that the strong unique continuation principle holds for every \(q \in (1,2)\), with an alternative approach based on Carleman’s estimate: in both papers it has been emphasized that the standard approaches are not applicable in the sublinear and discontinuous cases and have to be considerably adjusted. Finally, in [31] the authors investigate the geometric properties of the nodal set and the local behaviour of the solutions by proving the finiteness of the vanishing order at every point and by studying the regularity of the nodal set of any solution. More precisely, they show that the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most \((n2)\).
Ultimately, the main features of the nodal set are strictly related to those of the solutions to linear (or superlinear) equations, with a remarkable difference: the admissible vanishing orders can not exceed the critical value \(k_q=2/(2  q)\). Moreover, at this threshold, they proved the nonvalidity of any estimates of the \((n  1)\)dimensional measure of the nodal set of a solution in terms of the vanishing order.
The purpose of this paper is to study the structure of the nodal sets of nontrivial solutions to
where \(\lambda _+,\lambda _>0, q \in [1,2), s\in (0,1)\) and the fractional Laplacian is defined by
This study is driven by the wish to extend the previous theory to the fractional setting emphasizing the possible difference between the two type of operators due to the nonlocal attitude of the equations. Starting from the problem of unique continuation, many result have been achieved in the study of the nodal set of solution of nonlocal elliptic equation, in particular by using local realisation of the fractional powers of the Laplacian based on the extension technique popularized by the authors in [5]. Also in this setting, the key tools in proving unique continuation in the linear case are based on the validity of an Almgrentype monotonicity formula (see [11, 12, 17]), or Carleman estimates (see e.g. [23, 24]), which are not applicable in a standard way in our case.
In a slightly different direction, researcher also analyzed the structure of the nodal sets from the geometric point of view by classifying the possible local behaviour of solution near their nodal set: recently in [28] the authors provided a stratification result for the nodal set of linear equation by applying a geometrictheoretic analysis of the nodal set of solutions to degenerate or singular equations associated to the extension operator of the fractional Laplacian. In particular, they proved the existence of two stratified singular sets where the solution either resembles a classical harmonic function or a generic polynomial: in the first case, the stratification coincides with the one of the nodal set of solutions of local elliptic equations; in the second one a stratification still occurs but the bigger stratum is contained in a countable union of \((n1)\)dimensional \(C^{1,\alpha }\) manifolds, in contrast with the local case \(s=1\) (see [28, Section 8] for more detail in this direction).
On the other hand, the picture changes considerably in the case of solution with either sublinear \(q\in (1,2)\) or discontinuous \(q=1\) nonlinearity, as in (2). Indeed, it is clear that in the case \(\lambda _+,\lambda _\le 0\) (where the signs of the coefficients are opposite to ours), the features of the nodal set of solutions are substantially different in comparison with the linear case: dead cores appear and no unique continuation can be expected. In those scenarios one may try to describe the structure and the regularity of the free boundary \(\partial \{u=0\}\). When \(q \in (1,2)\) we refer to [36, 37] where the authors consider an AltPhillips type functional in the fractional setting for the case of nonnegative solutions \(u\ge 0\); while for the case \(q=1\) the equation is the so called two phase obstacle problem and we refer to [2, 3] and reference therein. Since in the fractional case minimisers of the twophase obstacle problem do not change sign, we refer to [18, 26] for some general result in the onephase setting.
In contrast, very little is known about the structure of the nodal sets in the case \(\lambda _+,\lambda _>0\). In [1] the authors considered the unstable twophase obstacle problem \(q=1\) and they proved that separation of phases does not occur in the unstable setting. Moreover, they characterized the local behaviour of minimisers near the freeboundary and they proved a bound on the Hausdorff dimension of the singular set.
In this paper we deal with the two phases problem (2), treating simultaneously the case \(q = 1\), which we call unstable two phase membrane problem and the case \(q\in (1,2)\), a prototype of sublinear equation. Notice that our results slightly extend the classification of blowup limit obtained for local minimisers in [1] to weak solution of (2) satisfying (4).
Statements of the main results. Let \(u\in H^s(\mathbb {R}^n)\) be a weak solution of (2) in the sense of distributions. Exploiting the local realisation of the fractional Laplacian deeply explained in [5], through the paper we will developed a local analysis of solution of the extended problem in \(\mathbb {R}^{n+1}_+\) (see Sect. 2 for more details). Hence, let us consider a weak solution \(u \in H^{1,a}(B^+_1)\) of
where \(a=12s \in (1,1)\),
and
where \(B_r(X_0)\) denote the ball of center \(X_0\) and radius r in \(\mathbb {R}^{n+1}\) (through the paper we will simply denote \(B^+_r(0)\) with \(B^+_r\)). Moreover, if \(X_0 \in \Sigma \), we will denote as \(S^{n1}_r(X_0)\) the boundary of \(\partial ^0 B^+_r(X_0)\) in \(\Sigma \), that is the \((n1)\)dimensional sphere of radius r centered at \(X_0\). From now on, we simply write “solution” instead of “weak stationary solution”, for the sake of brevity. Through the paper we will always denote with \(\Gamma (u) = \{(x,0):u(x,0)=0\}\) the restriction of the nodal set of u on \(\{y=0\}\).
According to Definition 2.1, through the paper we will consider solutions of (3) satisfying
for every \(X_0\in \partial ^0 B^+_1\) and \(r\in (0,\mathrm {dist}(X_0,\partial B_1))\) (see Sect. 2).
Remark 1.1
The validity of (4) plays a major role for the construction of monotonicity formulas and it is a necessary assumption for those point \(X_0 \in \Gamma (u)\) such that \({\mathcal {V}}(u,X_0)=2s/(2q)\), in the case \(q<2(1s)\) (see Remark 1.6). Nevertheless, it is a reasonable assumption which relax the hypothesis of being a minimal solution. Indeed, if u is a minimiser of
it can be proved by taking inner variations along directions \(\xi \in C^\infty _c(B_1; \mathbb {R}^{n+1})\) that u satisfies a more general class of Pohozaevtype identities (see [1] for \(q=1\)). Moreover, since for \(q<2(1s)\) there exist minimizers of (5) of type (12), one could only expect \(C^{1,\alpha }\)regularity of solutions of (3) at most for those \(q \in [1,2)\) such that \(q>2(1s)\). For the sake of completeness, we consider this notion of solutions in order to characterize the all possible behaviours for \(q \in [1,2)\).
Inspired by [31], we introduce two different notions of vanishing order, which will be proved a posteriori to be equal. Therefore, consider the norm
Through the paper, we often use \(\left\ {\cdot } \right\ _{X_0,r}\) to simplify the notation of the norm in \(H^{1,a}(B_r^+(X_0))\).
Definition 1.2
Let \(u \in H^{1,a}(B_1^+)\) be a solution of (3) satisfying (4) and \(X_0 \in \Gamma (u)\). The \(H^{1,a}\)vanishing order of u at \(X_0\) is defined as \({\mathcal {O}}(u,X_0) \in \mathbb {R}^+\), with the property that
Moreover, if such number does not exist, i.e.
we set \({\mathcal {O}}(u,X_0)=+\infty \).
The advantage of this formulation relays in the fact that we have better control of both the behaviour of the trace of solutions on \(\partial ^0 B^+_1\) and the character of the solution in the whole extended space. Instead, we recall here the classical definition of vanishing order, which will be used as well through the paper.
Definition 1.3
Let \(u \in H^{1,a}(B_1^+)\) be a solution of (3) satisfying (4) and \(X_0 \in \Gamma (u)\). The vanishing order of u at \(X_0\) is defined as \({\mathcal {V}}(u,X_0) \in \mathbb {R}^+\), with the property that
By (15) we will easily deduce that \({\mathcal {O}}(u,X_0) \le {\mathcal {V}}(u,X_0)\). The following result establishes the validity of the strong unique continuation principle for every \(q \in [1,2), \lambda _+>0,\lambda _\ge 0\) and \(s\in (0,1)\)
Theorem 1.4
Let \(q \in [1,2), \lambda _+>0,\lambda _\ge 0\) and \(u \in H^{1,a}(B_1^+)\) a solution of (3) satisfying (4) such that \(X_0 \in \Gamma (u)\). If \({\mathcal {V}}(u,X_0)=+\infty \), then necessarily \(u \equiv 0\); in particular, if for every \(\beta >0\) we have
it follows that \(u\equiv 0\).
This result implies the validity of the strong unique continuation principle also for the case \(\lambda _=0\), which resembles the result in the local setting. Moreover, in the case \(\lambda _+,\lambda _>0\), we can improve the previous result by characterizing all the admissible vanishing orders. Thus, let
be the critical exponent associated to weak solutions of (3) and \(\beta _q \in \mathbb {N}\) be the larger positive integer strictly smaller than \(k_q\), that is
Then, the admissible vanishing orders are all the positive integers smaller or equal than \(\beta _q\) and the critical value \(k_q\) itself.
Theorem 1.5
Let \(q \in [1,2), \lambda _+,\lambda _>0\) and \(u\in H^{1,a}(B^+_1)\) be a nontrivial solution of (3) satisfying (4) with \(X_0 \in \Gamma (u)\). Then
In particular, if \(k_q\le 1\) then \({\mathcal {V}}(u,X_0)=k_q\).
Remark 1.6
In the case \(s=1\), our result recovers the case considered in [31]. Nevertheless, Theorem 1.5 reveals a deep difference between the local and nonlocal equations for small value of \(s\in (0,1)\): while the vanishing orders of solution of (1) have a universal bound \(k_q=2/(2q)\), which is always greater or equal than 1 for \(q \in (0,2)\) (see [31] for the sublinear case \(q\in [1,2)\), [32] for the singular case \(q \in (0,1)\)), in the fractional setting this is not always true even in the sublinear case and it implies, for some values of \(s\in (0,1)\) and \(q\in [1,2)\), the occurrence of solutions which vanish only with order \(k_q< 1\). As we will see, this phenomena will also affect the structure and the regularity of the nodal set.
Now, using a blowup argument inspired by the one of [31], we proved the validity of a generalized Taylor expansion of the solutions near the nodal set: while in the linear (and superlinear) case solutions behave like homogeneous \(L_a\)harmonic polynomials of order \(k\ge 1\) symmetric with respect to \(\{y=0\}\) (see [28, Section 4] for a complete characterization of the class of symmetric \(L_a\)harmonic polynomials \(\mathfrak {sB}_k^a\)), in the sublinear setting this is not necessary the case.
Theorem 1.7
Let \(q \in [1,2),\lambda _+,\lambda _>0\) and \(u\in H^{1,a}(B^+_1)\) be a solution of (3) satisfying (4) with \(X_0 \in \Gamma (u)\). Then, the following alternative holds:

(1)
if \({\mathcal {V}}(u,X_0) \in \{n \in \mathbb {N}\setminus \{0\}:n\le \beta _q\} \), then there exists a \({\mathcal {V}}(u,X_0)\)homogeneous entire \(L_a\)harmonic function \(\varphi ^{X_0} \in \mathfrak {sB}_{{\mathcal {V}}(u,X_0)}^a(\mathbb {R}^{n+1})\) symmetric with respect to \(\{y=0\}\), such that
$$\begin{aligned} u(X)=\varphi ^{X_0}(XX_0) + O(\left{XX_0} \right^{{\mathcal {V}}(u,X_0)+1}); \end{aligned}$$(10) 
(2)
if \({\mathcal {V}}(u,X_0)=k_q\), then for every sequence \(r_k \searrow 0^+\) we have, up to a subsequence, that
$$\begin{aligned} \frac{u(X_0 + r_k X)}{\left\ {u} \right\ _{X_0,r_k}} \rightarrow {\overline{u}} \quad \text{ in } C^{0,\alpha }_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+}), \end{aligned}$$for every \(\alpha \in (0,\min (1,2s))\), where \({\overline{u}}\) is a \(k_q\)homogeneous nontrivial solution to
$$\begin{aligned} {\left\{ \begin{array}{ll} L_a {\overline{u}}=0 &{} \text{ in } \mathbb {R}^{n+1}_+ \\ \partial ^a_y {\overline{u}} = \mu \left( \lambda _+ ({\overline{u}}_+)^{q1}  \lambda _ ({\overline{u}}_)^{q1}\right) &{}\text{ on } \mathbb {R}^n \times \{0\}, \end{array}\right. } \end{aligned}$$(11)for some \(\mu \ge 0\). Moreover, the case \(\mu = 0\) is possible if and only if \(k_q\in \mathbb {N}\).
Remark 1.8
in the case of local diffusion \(s=1\) it is known that a growth estimate of the Laplacian of a function near its nodal set immediately implies the validity of a Taylor expansion of the function itself in terms of harmonic polynomials (see [6, Lemma 3.1]), whereas in the nonlocal setting the validity of a similar result is still unknown.
Therefore, our strategy is to take advantage of the bound on the vanishing order to ensure the validity of an asymptotic limit of the DirichlettoNeumann operator \(\partial ^a_y\) near the nodal set. Then, the expansion follows by a blowup analysis based on an application of an Almgren and Monneautype monotonicity formulas. Finally, in order to improve the convergence estimate of the remainder in the Taylor expansion (10), we apply a blowup analysis on the difference between the function and its tangent map: this result will be crucial to prove the \(C^{1,\alpha }\)regularity of the strata of the nodal set.
Thus, we think that this methodology could be use to extend the fundamental Lemma in [6, Lemma 3.1] to the fractional setting.
This result leads to a partial stratification of the nodal set and, via the dimension reduction principle due to Federer, to an estimate of the Hausdorff dimension of the nodal and singular set.
In the light of the previous results, let us define with \({\mathcal {R}}(u)\) and \({\mathcal {S}}(u)\) the regular and singular part of \(\Gamma (u)\) defined by
and with \({\mathcal {T}}(u)\) the “purely sublinear” part of the nodal set
While in the local case \(s=1\) the sets \({\mathcal {S}}(u)\cup {\mathcal {T}}(u)\) coincides with those points with vanishing gradient, in the fractional setting \(s \in (0,1)\) this is not always the case since the critical value is not necessary greater than 1. Indeed, this decomposition of \(\Gamma (u)\) seems more natural in the fractional setting: by Theorem 1.5, we already know that if \(k_q>1\) then
while if \(k_q\le 1\) we get
Indeed we will see that, for those value of \(s\in (0,1)\) and \(q\in [1,2)\) such that \(k_q>1\), near the points of the nodal set where the function vanishes with order strictly less then \(k_q\), the nodal set resembles the picture of the nodal set of sharmonic functions (see [28] for a deeper analysis of the singular set \({\mathcal {S}}(u)\)).
Theorem 1.9
Let \(q \in [1,2),\lambda _+,\lambda _>0\) and \(u \in H^{1,a}(B^+_1)\) be a solution of (3) satisfying (4). The nodal set \(\Gamma (u)\) splits as
where

(1)
the regular part \({\mathcal {R}}(u)\) is locally a \(C^{1,\alpha }\)regular \((n1)\)hypersurface on \(\mathbb {R}^{n}\);

(2)
the singular part \({\mathcal {S}}(u)\) is an \((n1)\)dimensional countably rectifiable set. Moreover, the decomposition
$$\begin{aligned} {\mathcal {S}}(u)= \bigcup _{j=0}^{n1} {\mathcal {S}}_j(u) \end{aligned}$$holds true, where each \({\mathcal {S}}_j(u)\) is contained in a countable union of jdimensional \(C^{1,\alpha }\) manifolds;

(3)
the sublinear part \({\mathcal {T}}(u)\) has Hausdorff dimension at most \((n1)\). Moreover, for \(k_q<1\) the nodal set coincides with the sublinear stratum and the Haudorff estimate is optimal in the sense that there exists a collection of 2dimensional \(k_q\)homogeneous solutions such that
$$\begin{aligned} u_1(x,0)=A_1\left( x_+^{k_q}  x_^{k_q}\right) \quad \text{ or }\quad u_2(x,0)=A_2\left{x} \right^{k_q}\quad \text{ for } \text{ every } x \in \mathbb {R}. \end{aligned}$$(12)
The result on \({\mathcal {T}}(u)\) is remarkably different to its local counterpart: while for \(s=1\) the bound \((n2)\) on the Hausdorff dimension is optimal, we believe that the result on the \((n1)\)dimension of \({\mathcal {T}}(u)\) in the case \(k_q< 1\) can be easily generalized to all \(s\in (0,1)\) and \(q \in [1,2)\), thanks to the characterization of \(L_a\)harmonic function in [28]. Moreover, we claim that a viscosity approach, based on an improvement of flatness, could give a regularity result for those points where the blowup limit behave like (12) (see Remark 7.1 for more detail in this direction), in the case \(k_q < 1\). At the moment, we leave it as an open problem.
Structure of the paper The paper is organized as follows. In Sect. 2 we recall some preliminary results on the functional setting and the regularity of solutions. Moreover, we introduce the notions of vanishing order used through the paper. Next, in Sect. 3, we prove the validity of the weak unique continuation principle and Theorem 1.4 by using a 2parameter Weisstype monotonicity formula which allows, in Sect. 4, to introduce a characterisation of the threshold \(k_q\).
Finally, in Sect. 5 we prove the first part of Theorem 1.7 and Theorem 1.9 by developing a blowup analysis based on the validity of two Almgrentype formulas for those points with vanishing order smaller than \(k_q\) and, in Sect. 6, we complete the proof of Theorem 1.7 by applying a blowup analysis on those points with vanishing order equal to \(k_q\). As byproduct, we will recover Theorem 1.5.
Finally, in Sect. 7 we prove the existence of \(k_q\)homogeneous solutions of the form (12), for those values of s and q so that \(k_q<1\). This result will lead to the Hausdorff estimate of \({\mathcal {T}}(u)\) in Theorem 1.9.
2 Preliminaries
In this Section we start by showing preliminary results related to the trace embedding of the \(H^{1,a}\)space and the regularity of solutions to our problem.
For \(a \in (1,1), \Omega \subseteq \mathbb {R}^{n+1}_+\) let us consider \(H^{1,a}(\Omega ) = H^1(\Omega , y^a \mathrm {d}X)\) the weighted Sobolev space associated to the Muckenhoupt \(A_2\) weights \(\omega (X)=y^a\) with \(X=(x,y)\) (see [10] for more details). Now, given a weak solution \(u \in H^s(\mathbb {R}^n)\) of (2), by the CaffarelliSilvestre extension there exists a unique function \(v \in H^{1,a}(\mathbb {R}^{n+1})\) such that
with \(a=12s \in (1,1)\) and \(L_a v = \mathrm {div}(y^a \nabla v)\). By a trace result of Nekvinda [22], it is known that the space \(H^s(\mathbb {R}^n)\) coincides with the trace on \(\mathbb {R}^n\times \{0\}\) of \(H^{1,a}(\mathbb {R}^{n+1})\). Thus, we obtain that, up to a normalization constant, the problem
is a localized version of (2) with \(u(x)=v(x,0)\) (see also [4, Theorem 3.1]). Instead of consider general weak solution in \(H^{1,a}(B_1^+)\) we need to assume the validity of some Pohozaevtype identities, which are usually associated to the concept of stationary solution with respect to domain variations.
Definition 2.1
We say that \(u\in H^{1,a}(B_1^+)\cap L^\infty (B_1^+)\) is a nontrivial weak stationary solution of (13) if for every \(\varphi \in C^\infty _c(B_1)\) we have
and
for every \(X_0\in \partial ^0 B^+_1\) and \(r\in (0,\mathrm {dist}(X_0,\partial B_1))\).
We remark that the existence of solutions follows by standard methods of the calculus of variations and a straightforward application of the following trace embedding.
Through the paper, for \(X_0 \in \partial ^0 B^+_1\) and \(r \in (0,1\left{X_0} \right)\), we will always consider the space \(H^{1,a}(B_r^+(X_0))\) as the completion of \(C^\infty (\overline{B_r^+(X_0)})\) with respect to the norm
By the trace embedding on \(\partial ^+ B^+_r(X_0)\) and the Poincaré inequality, this norm is equivalent to the classical one of [10]. From now on, we often use the notation \(\left\ {\cdot } \right\ _{X_0,r}\) to simplify the notation of the norm in \(H^{1,a}(B_r^+(X_0))\). The equivalence of this norm with the classic one is a consequence of the trace theory and the Poincaré inequality.
Lemma 2.2
Let \(u\in H^{1,a}(B^+)\) and \(q \in [1,2^\star ]\), where \(2^\star = 2n/(n2s) = 2n/(n+a1)\) is Sobolev’s exponent for the fractional Laplacian. There exists a constant \(C_1=C_1(n, p,a)\) such that
for every \(0< r < 1\). Namely, the space \(H^{1,a}(B^+_r(X_0))\) is continuously embedded in \(L^q(\partial ^0 B^+_r(X_0))\), for every \(r \in (0,1)\).
Proof
This result is a direct consequence of the characterization of the class of traces of \(H^{1,a}(B^+_r)\) with \(r\in (0,1)\) (see [22, Theorem 2.11] for the complete theory), and the Sobolev embedding in the context of fractional SobolevSlobodeckij spaces \(H^s(\partial ^0B^+_r)\). \(\square \)
Since \(\partial ^0 B^+_r\) is a Lipschitz domain with bounded boundary in \(\mathbb {R}^n\), the compact embedding in the fractional Sobolev spaces implies the following remark (see [8] for further details).
Lemma 2.3
Let \(q \in [1,2^\star )\), where \(2^\star = 2n/(n2s) = 2n/(n+a1)\). Then \(H^{1,a}(B^+_r(X_0))\) is compactly embedded in \(L^q(\partial ^0 B^+_r(X_0))\), for every \(r \in (0,1)\).
We conclude the Section by recalling a regularity result for solutions of (3) in the sense of Definition 2.1. As we will see through the paper, in order to develop a blowup analysis near nodal points it is enough to prove a \(\alpha \)Hölder regularity for small \(\alpha \in (0,1)\), since the classification of the admissible vanishing order is obtained by monotonicitytype formulas. Therefore, by [29, Theorem 1.5] (see also [29, 30] for more regularity results), we easily deduce the following result.
Proposition 2.4
Let \(u \in H^{1,a}(B^+_1)\) be a weak solution of (3) in the sense of Definition 2.1. Then, for any compact set \(K\subset B_1\) we get \(u \in C^{0,\alpha }(K \cap \overline{B^+_1})\), for every \(\alpha \in (0,\min \{1,2s\})\).
3 Strong unique continuation principle
This Section is devoted to the proof Theorem 1.4, that is the strong unique continuation principle for solution of (3). In order to achieve the main result we start our analysis by proving the weak unique continuation principle: if a solution u is identically zero in a neighborhood in \(\mathbb {R}^{n}\times \{0\}\) of a point \(X_0 \in \partial ^0 B^+_1\), then necessary \(u\equiv 0\) on \(\partial ^0 B^+_1\). Moreover, since \(q \in [1,2)\), it implies that \(u\equiv 0\) on \(B^+_1\) (see [28, Proposition 5.9]).
Our proof of the unique continuation is deeply based on the validity of an Almgrentype monotonicity formula. Indeed, let
then for \(X_0 \in \Gamma (u)\) and \(r \in (0,\text{ dist }(X_0,\partial ^+ B^+_1))\), we introduce the functionals
and the associated Almgrentype formula
Through the paper we will often abuse the notation E(u, r), H(u, r) and N(u, r) when it is not restrictive to assume that \(X_0=0\). By the GaussGreen formula we immediately obtain
while, by differentiating the functions \(r\mapsto H(X_0,u,r)\), we get
In the following Proposition we compute the derivative of the denominator of the Almgrentype quotient by taking care of the sublinear term on the boundary \(\partial ^0 B^+_r\).
Proposition 3.1
Let \(X_0 \in \Gamma (u)\) and \(r \in (0,\mathrm {dist}(X_0,\partial B_1))\). Then, it holds
where \(C^s_{n,q} = 2nq(n2s)>0\).
Proof
Up to translation, let us assume that \(X_0=0\). Thus
By Definition 2.1, we need to integrate by parts only the last term in (14). More precisely, since \(\nabla _x F_{\lambda _+,\lambda _}(u) = q(\partial ^a_y u) \nabla _x u\) in \(\partial ^0 B^+_1\), we get
Summing together the previous equalities, we finally get the claimed result. We remark that the previous computations are also valid in the case \(q = 1\), but require some justification. More precisely, as observed in [33, Proposition 2.7], the GaussGreen formula holds for all vector fields \(Y \in C(\overline{B_r},\mathbb {R}^{n+1})\) with \(\mathrm {div}Y \in L^1(B_r)\). In particular in (19), the GaussGreen formula is applied to the vector field
where
The previous quantity is absolutely integrable in \(\partial ^0 B^+_r\) as a direct consequence of the characterization of the class of trace of \(H^{1,a}(B^+_r)\) with \(r\in (0,1)\) (see [22, Theorem 2.11]). \(\square \)
Combining the previous estimate, we finally get a lower bound for the derivative of the Almgrentype frequency formula.
Corollary 3.2
Let \(X_0 \in \Gamma (u)\) and \(r_1,r_2 \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\) such that \(H(X_0,u,r) \ne 0\) for a.e. \(r\in (r_1,r_2)\). Then
for a.e. \(r \in (r_1,r_2)\).
Proof
The proof follows essentially the ideas of the similar results in literature and it is based on a straightforward combination of (18), Proposition 3.1 and the validity of the CauchySchwarz inequality on \(\partial ^+ B^+_r(X_0)\). \(\square \)
Now, we are ready to show the validity of the classical weak unique continuation principle for solution of the sublinear nonlocal equation.
Theorem 3.3
Let \(q \in [1,2), \lambda _+>0, \lambda _\ge 0\) and \(u \in H^{1,a}(B^+_1)\) be a weak solution of (3) which vanishes in a neighbourhood in \(\mathbb {R}^n\times \{0\}\) of a point on \(\Gamma (u)\). Then \(u \equiv 0\) in \(\partial ^0 B^+_1\).
Proof
Let us define the vanishing set on \(\mathbb {R}^n\times \{0\}\) as
Since \(u\not \equiv 0\) and \(\partial ^0 B^+_1\) is connected, we already know that U is open, nonempty and \(\partial U \cap \partial ^0 B^+_1 \ne \emptyset \), where \(\partial U\) is the topological boundary of U as a subset of \(\mathbb {R}^n\). Let \(X^* \in U\) and \(r>0\) be such that \(B_r(X^*) \subset U\), then by (3) the function u satisfies
By [28, Proposition 5.9], it follows that necessary \(u\equiv 0\) in \(B^+_r(X^*)\) and then, by the weak unique continuation principle for the \(L_a\)operator (see [16, Theorem 1.4]) we get that \(u\equiv 0\) in \(B_1^+\). Since we already know that u is Hölder continuous, we get \(u\equiv 0\) in \(\partial ^0 B^+_1\). \(\square \)
In order to prove the strong unique continuation principle, we now introduce a 2parameter family of Weisstype monotonicity formulas, that will be the fundamental tool of our analysis. Indeed, inspired by [31], for \(X_0 \in \Gamma (u)\) and \(r \in (0,\text{ dist }(X_0,\partial ^+ B^+_1))\) we consider the functional
and similarly we introduce the twoparameters families of functionals
Notice that for \(t=q\), we recover the functionals in (16) and their associated Almgrentype formula.
Proceeding exactly as in Proposition 3.1 and Corollary 3.2, we get
where
and \(C^s_{n,t}= 2nt(n2s)\).
Proposition 3.4
Let \(X_0 \in \Gamma (u)\) and \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\). Then we have
In particular, for \(t=2\) and \(k\ge k_q\) the function \(r\mapsto W_{k,2}(X_0,u,r)\) is monotone nondecreasing.
Proof
Up to translation, let us suppose \(X_0=0\) and \(r \in (0,1)\). A direct computation gives
where in the second inequality we used the estimate (18). By the GaussGreen formula in (21) we get
and by taking care of the estimate (23), we finally obtain
Now, for \(t=2\) and \(k\ge k_q\) (see (8) for the definition of the critical exponent \(k_q\)) the monotonicity follows straightforwardly by the previous computations. Indeed, we have
where \(C_{n,2}^s  2k(2q)\le 0\) if and only if \(k\ge k_q\). \(\square \)
Thus, as simple corollaries of the monotonicity result, we deduce the following results for \(k\ge k_q\).
Corollary 3.5
Let \(X_0\in \Gamma (u)\) and \(k\ge k_q\). Then, there exists the limit
Moreover, the map \(r\mapsto W_{k_q,2}(X_0,u,r)\) is constant if and only if u is \(k_q\)homogeneous in \(\overline{\mathbb {R}^{n+1}_+}\) with respect to \(X_0\).
Corollary 3.6
For \(X_0 \in \Gamma (u)\), there exists \(k\ge k_q\) such that
Moreover, if \(W_{k_1,2}(X_0,u,0^+)<0\) then \(W_{k_2,2}(X_0,u,0^+)=\infty \) for every \(k_2>k_1\).
Proof
Up to translation, let us consider \(X_0=0\) and \(r \in (0,1)\). By Theorem 3.3, since \(u\not \equiv 0\) there exists \(r_1 \in (0,1)\) such that \(H(u,r_1)\ne 0\). Now, there exists \(k\ge k_q\) sufficiently large, such that
and by the monotonicity result in Proposition 3.4 we obtain \(W_{k,2}(u,0^+) \le W_{k,2}(u,r_1) <0\), for k sufficiently large.
Now, fixed \(k_1>0\) such that \(W_{k_1,2}(u,0^+)<0\), let us consider \(k_2>k_1\). Thus, for \(r \in (0,1)\)
which implies the claimed conclusion. \(\square \)
Finally, by Corollary 3.6 we are able to prove the existence of a transition exponent \({\overline{k}}\) for the frequency \(W_{k,2}(X_0,u,0^+)\) which characterize the possible behaviours of the Weiss frequency for every \(k\ge 0\).
Corollary 3.7
For every \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0) \ge k_q\), there exists finite
Moreover, the limit \(W_{k,2}(X_0,u,0^+)\) exists for every \(k\ge 0\) and it satisfies
Proof
The existence of \({\overline{k}} \ge 0\) follows by Corollary 3.6. Now, let us consider separately the cases \(k<k_q\) and \(k\ge k_q\). In the first one, since \({\mathcal {O}}(u,X_0)\ge k_q\), by (6) there exists \(\varepsilon >0\) such that
and two constant \(C>0,r_0>0\), depending on \(\varepsilon \), such that
for every \(r\in (0,r_0)\). By definition of the Weisstype formula, we get
with \(\alpha = k_q \varepsilon \). Finally, since \(q\in [1,2)\), we get
which leads to the claimed result ad \(r\rightarrow 0^+\). In particular, this estimate suggests that \({\overline{k}}\ge k_q\).
Instead, in the case \(k> k_q\) the existence of a nonnegative limit for \(k< {\overline{k}}\) follows by the monotonicity result in Proposition 3.4 and by Corollary 3.6. \(\square \)
The previous result emphasizes an hidden relation between the notion of \(H^{1,a}\)vanishing order and the transition exponent \({\overline{k}}\) defined in Corollary 3.7, which will be deeply examined in Sect. 4. Finally, we can prove the main result of the Section.
Proof of Theorem 1.4
By contradiction, suppose that \(u\not \equiv 0\) on \(\partial ^0 B^+_1\) and \({\mathcal {O}}(u,X_0)=+\infty \), i.e.
In particular, given \({\overline{k}}>0\) as in Corollary 3.7, let us fix \(k>{\overline{k}}\) and \(\beta =2k/q\). Thus, there exists \(r_0>0\) and \(C>0\) such that
On one side, since \(2k/q>k\) for \(q \in [1,2)\), by the previous inequality we easily have
while, by an integration by parts, fixed \(\Lambda = \max \{\lambda _+,\lambda _\}\) we get
where in the second inequality we use Lemma 2.2 and in the last one (28). Finally, collecting the previous estimate, for every \(r \in (0,r_0)\) we have
and in particular \(W_{k,2}(X_0,u,0^+)>\infty \), in contradiction with the fact that, being \(k >{\overline{k}}\), by Corollary 3.7 we must have \(W_{k,2}(X_0,u,0^+)=\infty \) for any \(k > {\overline{k}}\). \(\square \)
4 The transition exponent for the Weisstype formula
In this Section we develop a finer analysis of the transition exponent \({\overline{k}}\) for the Weisstype monotonicity formula \(W_{k,2}\) in the case \({\mathcal {O}}(u, X_0) \ge k_q\). The main result of the Section is a characterization of \({\overline{k}}\) in terms of the critical exponent \(k_q\) and the \(H^{1,a}\)vanishing order, which allows to prove an upper bound for the admissible vanishing orders \({\mathcal {O}}(u,X_0)\) of u at \(X_0\) (see Proposition 4.7).
First, we start by proving the following partial characterization of the transition exponent \({\overline{k}}\) of \(W_{k,2}\) (see (3.7)) in terms of \(k_q\).
Proposition 4.1
For every \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0) \ge k_q\), we have
Moreover, combining the previous estimate with Corollary 3.7 we deduce that \(W_{k,2}(X_0,u,0^+)\) exists for every \(k\ge 0\) and
Following the strategy presented in [31], this result will be a consequence of the following Lemmata in which we assume that \({\overline{k}}> k_q\).
Remark 4.2
Since we never use the assumption \({\mathcal {O}}(u,X_0)\ge k_q\), we highlight that the following results are still true near nodal point with \({\mathcal {O}}(u,X_0)< k_q\).
Lemma 4.3
Let \(X_0 \in \Gamma (u)\) and assume that \({\overline{k}}\ge k_q\). Then
for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\). Moreover, if \(k>{\overline{k}}\), we deduce
Proof
Fixed \(k \in (k_q,{\overline{k}}]\), we already know by Proposition 3.4 that \(r \mapsto W_{k,2}(X_0,u,r)\) is monotone nondecreasing and, by Corollary 3.7, that \(W_{k,2}(X_0,u,0^+)\ge 0\). Hence, for every \(r \in (0,\mathrm {dist}(X_0,\partial B^+))\) we get
Moreover, since \(W_{k,q}(X_0,u,r) \ge W_{k,2}(X_0,u,r) \ge 0\) for every \(r \in (0,\mathrm {dist}(X_0,\partial B^+))\), by (18) we deduce
Finally, if \(H(X_0,u,r_1)=0\) for some \(r_1 >0\), by the monotonicity of (30), we deduce that \(u\equiv 0\) in \(B^+_{r_1}(X_0)\), in contradiction with Theorem 3.3.
Hence, collecting the previous inequality, we get \(N_2(X_0,u,r)\ge 0\) and in particular, since \(k> {\overline{k}}\) we get
Also, since \(H(X_0,u,r)/r^{2k}\ge 0\), we finally deduce
which implies the desired claim. \(\square \)
As a consequence, for every \(t \in (0,2)\) the associated Almgrentype formula \(N_t(X_0,u,r)\) is nonnegative for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B+))\).
Since in this Section we are proceeding by assuming by contradiction that \({\overline{k}}>k_q\), consider t the medium point between \(k_q\) and \({\overline{k}}\).
Lemma 4.4
Let \(X_0 \in \Gamma (u)\) and assume that \({\overline{k}}> k_q\). Given
then the map \(r \mapsto W_{k,{\tilde{t}}}(X_0,u,r)\) is monotone nondecreasing in \((0,\mathrm {dist}(X_0,\partial ^+ B^+))\), for every \(k\ge {\widetilde{k}}\).
Proof
The proof is a direct corollary of Proposition 3.4. More precisely, since \(q \in [1,2)\) and \(k\ge {\widetilde{k}}> k_q\) we get that
which implies, by (27), the claimed result. \(\square \)
Therefore, under the absurd assumption \({\overline{k}}>k_q\), we can prove that the transition exponent \({\overline{k}}\) associated to \(W_{k,2}(X_0,u,0^+)\) coincides with the transition exponent associated to the frequency \(W_{k,{\tilde{t}}}(X_0,u,0^+)\).
Lemma 4.5
Assume that \({\overline{k}}>k_q\), then
In particular, for every \(k > {\overline{k}}\) we get
Proof
Following the reasoning in Corollary 3.6, we can immediately deduce the existence of \(k \ge {\tilde{k}}\) such that \(W_{k,{\tilde{t}}}(X_0,u,0^+) <0\). Hence, we can reasonably define the quantity
for which
Since \({\tilde{t}}<2\), we fist have \(W_{k,{\tilde{t}}}(X_0,u,r) \ge W_{k,2}(X_0,u,r)\) for every \(0<r<R\) and \(k>0\). Now, on one side \(W_{k,{\tilde{t}}}(X_0,u,0^+)=\infty \) implies \(W_{k,2}(X_0,u,0^+)=\infty \) and hence \(\overline{{\overline{k}}}\ge {\overline{k}}\). So, let us suppose by contradiction that \(\overline{{\overline{k}}}>{\overline{k}}\), hence there exists \(k \in ({\overline{k}},\overline{{\overline{k}}})\) such that \(W_{k,{\tilde{t}}}(X_0,u,0^+)\ge 0\).
By the monotonicity result in Lemma 4.4 we get \(W_{k,{\tilde{t}}}(X_0,u,r) \ge 0\) for \(r>0\) and, since \({\tilde{t}}\in (q,2)\), we deduce
for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+))\). Finally, recalling the relation in (30), by (32) it follows that \(r \mapsto r^{2k}H(X_0,u,r)\) is monotone nondecreasing and in particular there exists finite
which contradicts Lemma 4.3. \(\square \)
Lemma 4.6
Let \(X_0 \in \Gamma (u)\) and \({\overline{k}}\ge k_q\). There exists a sequence \((r_n)_n\) such that \(r_i \rightarrow 0^+\) and
Proof
Let \(k \in [k_q,{\overline{k}})\), by Corollary 3.4 and Corollary 3.7 we have \(W_{k,2}(X_0,u,r)\ge 0\) for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+))\). Since for any fixed radius \(r>0\) the function \(k \mapsto W_{k,2}(X_0,u,r)\) is continuous, we infer as \(k \rightarrow {\overline{k}}^\) that \(W_{{\overline{k}},2}(X_0,u,r)\ge 0\), which implies by continuity that \(W_{{\overline{k}},2}(X_0,u,0^+)\ge 0\).
Thus, for any \({\overline{r}}\in (0,\mathrm {dist}(X_0,\partial ^+ B^+))\) we get
On the other side, by (27) we deduce
which implies, combined with the nonintegrability of \(s\mapsto s^{1}\) in 0, that if
then (33) would not be true. Thus, this implication suggests that the previous liminf has to be null. \(\square \)
Proof of Proposition 4.1
The proof is based on a blowup argument: given \(X_0 \in \Gamma (u)\) assume that \({\overline{k}}> k_q\) and let \((r_n)_n\) be the sequence introduced in Lemma 4.6. Therefore, consider the blowup sequence
where \(R= \mathrm {dist}(X_0,\partial ^+ B^+)\). Thanks to Lemma 4.3, we have \(H(X_0,u,r_n)>0\) and \(E_2(X_0,u,r_n)\ge 0\), which lead to
On the other hand by Lemma 4.3 we deduce
which implies, since \({\tilde{t}}<2\) that
As a consequence of the previous estimates and Lemma 4.5, we get
Since the sequence \((u_n)_n\) is uniformly bounded in \(H^{1,a}(B^+_1)\), the compactness of the Sobolev embedding implies that \((u_n)_n\) converges weakly in \(H^{1,a}(B_+^1)\) and strongly in \(L^{2,a}(\partial ^+ B_1^+)\) to a function \({\overline{u}}\in H^{1,a}(B^1_+)\).
Moreover, since by [22, Theorem 2.11] the space of the trace of functions in \(H^{1,a}(B^+)\) on the set \(\partial ^0B^+\) coincides with the SobolevSlobodeckij space \(H^s(\partial ^0B^+)\), by the Riesz–Frechet–Kolmogorov Theorem, the trace operator
is well defined an compact for every \(p\in [1,2]\) (see Lemma 2.3). Hence, since \(q \in [1,2)\), we get
Since the first equality implies that \({\overline{u}}\not \equiv 0\) on \(\partial ^+ B^+_1\), we deduce by the trace embedding that \({\overline{u}}\not \equiv 0\) on the whole \(B^+_1\). On the other side, we get
By direct computation, since we are assuming \({\overline{k}}> k_q\), we have \(2(2{\overline{k}}+a1)/q > 2 {\overline{k}}\) and, for n sufficiently large, it implies
where, by Lemma 4.3 and Lemma 4.6, the right hand side goes to 0 as \(n\mapsto +\infty \). By (34) we infer that
On the other hand, since \((u_n)_n\) is uniformly bounded in \(H^{1,a}(B_1^+)\) and \(u_n \rightharpoonup {\overline{u}}\) weakly in \(H^{1,a}\), from
we deduce that the limit function \({\overline{u}} \in H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{R^{n+1}_+})\) is a weak solution of
such that \({\overline{u}}\not \equiv 0\) on \(\mathbb {R}^{n+1}_+\). The contradiction follows immediately by the unique continuation principle for the traces of \(L_a\)harmonic functions (see [28, Proposition 5.9]). \(\square \)
The following result completes the previous characterization in the case \({\mathcal {O}}(u,X_0)\ge k_q\) by relating the critical exponent \(k_q\) and the transition exponent \({\overline{k}}\) to the \(H^{1,a}\)vanishing order of u at \(X_0\). More precisely, it implies that the solutions of (3) can vanish with order less or equal than \(k_q\).
Proposition 4.7
Let u be a solution of (3) and \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0)\ge k_q\). Then, the vanishing order \({\mathcal {O}}(u,X_0)\) is characterized by
Furthermore, we get
Proof
The proof of this result follows the one of its local counterpart in [31]. For the sake of simplicity, let us denote with \(\left\ {\cdot } \right\ _{H^{1,a}(B^+_r(X_0))} = \left\ {\cdot } \right\ _{X_0,r}\). Now, fixed \(X_0 \in \Gamma (u)\), let us prove that
where \({\overline{k}}=k_q\). After that, the result will follow by Proposition 4.1 and (29). By contradiction, let us suppose there exists a sequence \(r_n\rightarrow 0^+\) such that
Then, consider the blowup sequence associated to the \(H^{1,a}\)norm, defined as
As we deduce in the proof of Proposition 4.1, since the blowup sequence \((u_n)_n\) is uniformly bounded in \(H^{1,a}(B^+_1)\), the compactness of the Sobolev embedding implies that \((u_n)_n\) converges weakly in \(H^{1,a}(B_1^+)\) and strongly in \(L^{2,a}(\partial ^+ B_1^+)\) to a function \({\overline{u}}\in H^{1,a}(B_1^+)\). Similarly, the traces on \(\partial ^0 B^+_1\) converge strongly in \(L^q(\partial ^0 B^+_1)\) to the trace of \({\overline{u}}\), for every \(q\in [1,2)\). In particular,
Thus, since the limit \(W_{{\overline{k}},2}(X_0,u,0^+)\) exists, by the monotonicity result in Proposition 3.4, we get that \(W_{{\overline{k}},2}(X_0,u,r)\ge 0\) and \(W_{{\overline{k}},q}(X_0,u,r)\ge 0\) for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\) and \(q< 2\).
First, by Lemma 4.3, we know that \(E_2(X_0,u,r)\ge 0\) and \(H(X_0,u,r)>0\) for every \(r \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\) and, for every \(k > {\overline{k}}\) we get
Now, let us compute the same limit in the case \({\overline{k}}\). Since the function \(r \mapsto H(X_0,u,r)/r^{2{\overline{k}}}\) is monotone nondecreasing, there exists the limit as \(r\rightarrow 0^+\) and, by (37), we get
which implies
In order to reach a contradiction, we need to prove that the blowup limit satisfies \({\overline{u}}\equiv 0\), in contradiction with the normalization (38) (see the conclusion of the Section). \(\square \)
Lemma 4.8
Fixed \(X_0 \in \Gamma (u)\) and \({\overline{k}}=k_q\) let us suppose that (40) holds true. Then, we get
Proof
Let us consider first the limit associated to the case \(t=q\) and, by contradiction, assume that \(\varepsilon >0\) and \(r_0 \in (0,\mathrm {dist}(X_0,\partial ^+ B^+_1))\) such that
By (18), we deduce that
and integrating by parts the previous inequality between \(r \in (0,r_0)\) and \(r_0\) we get
In particular
in contradiction with (39) with \(k = {\overline{k}}+\varepsilon \).
Now, for \(t=2\) and \({\overline{k}}=k_q\) we already know by Proposition 3.4 that
In the remaining part of the proof, for the sake of simplicity we omit the dependence with respect to u and \(X_0\). Hence, combining the previous derivative with (30) we get
and since \(0\le W_{{\overline{k}},2}(r) \le W_{{\overline{k}},q}(r)\) we infer that
which is nonnegative by the CauchySchwarz inequality. Since \(H(r)>0\) and \(0\le W_{{\overline{k}},2}(r) \le W_{{\overline{k}},q}(r)\), the previous part of the proof yields that the second limit in (41) exists and is equal to zero. \(\square \)
Conclusion of the proof of Proposition 4.7
Since \(\left\ {u} \right\ _{X_0,r}^2 \ge H(X_0,u,r)\), by Lemma 4.8, there exists a sequence \(r_m\rightarrow 0^+\) such that
Now, let \(u_m\) be the blowup subsequence of (38) associated to the sequence \((r_m)_m\) which converges to a limit function \({\overline{u}}\). First, by (41) we infer
which implies, combined with the strong convergence in \(L^q(\partial ^0 B^+_1)\) and (40), that \({\overline{u}} \equiv 0\) on \(\partial ^0 B^+_1\). On the other side, by (42) we deduce that
as \(m\rightarrow +\infty \). Therefore, collecting the previous results we get
which implies that \(\left\ {u_m} \right\ _{0,1}^2 \rightarrow ({\overline{k}}+1)\left\ {{\overline{u}}} \right\ _{L^{2,a}(\partial ^+ B^+_1)}^2\). Since by (38) the normalization implies \(\left\ {u_m} \right\ _{0,1}=1\) for every m, we immediately deduce that \({\overline{u}} \not \equiv 0\) in \(B^+_1\). Therefore, the conclusion follows as in the proof of Proposition 4.1. \(\square \)
5 Blowup analysis for \({\mathcal {O}}(u,X_0)< k_q\)
In this Section we initiate the blowup analysis of the nodal set starting from those points with vanishing order smaller than the critical value \(k_q=2s/(2q)\). The main strategy is to develop a blowup argument based on the validity of Almgrentype and Weisstype monotonicity formulas, which provide a Taylor expansion of the solutions near the nodal set in terms of \(L_a\)harmonic polynomials symmetric with respect to \(\{y=0\}\).
We initiate the analysis by introducing an Almgrentype monotonicity formula. More precisely, by using the upper bound on the \(H^{1,a}\)vanishing order of u, we prove the validity of a monotonicity result for functional \(N(X_0,u,r)=N_q(X_0,u,r)\) introduced in (17).
Proposition 5.1
Let \(K \subset \subset \partial ^0 B^+_1\) and suppose there exists \(\delta >0\) such that
Then there exists \(r_0>0\) such that for every \(X_0 \in \Gamma (u)\cap K\)
is monotone nondecreasing for \(r \in (0,\min (r_0,\mathrm {dist}(K,\partial ^0 B^+)))\), for some constant \(\alpha = \alpha (\delta ,n,s,q)\) and \({{\tilde{C}}}={{\tilde{C}}}(\delta ,n,s,q)\). Moreover, for every \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0)< k_q\) there exits the limit
and the map \(X_0\mapsto N(X_0,u,0^+)\) is upper semicontinuous on \(\Gamma (u)\).
Proof
Let \(K\subset \subset \partial ^0 B^+_1\) and \(\alpha >0\) to be made precise later. Let \(X_0 \in K\) and, for the sake of simplicity, we omit the dependence of the functionals with respect to u and \(X_0\). By Corollary 3.2, we easily get
with \(C^s_{n,q} = 2n  q(n2s)\). On the other hand, by Lemma 2.2
Now, we want to show that there exists \(\alpha , r_0, C_2>0\) such that
for every \(r\in (0,r_0)\). Then, combining the previous inequality with (44) and (45), we will get
as we claimed. First, by (43), let us choose \(\alpha = \delta /2\) and consider
for every \(X_0 \in \Gamma (u)\cap K\). Indeed, by the definition of \(H^{1,a}\)vanishing order, there exists \(r_2>0\) and \(C_2>0\) such that, for every \(r\in (0,r_2)\)
Since \(\delta =\delta (K)\), the constant \(C_2,\alpha \) and \(r_2\) depend only on the choice of the compact K. Finally, the upper semicontinuity follows by a standard argument. \(\square \)
Using this monotonicity result we can prove the equivalence between the notion of \(H^{1,a}\)vanishing order \({\mathcal {O}}(u,X_0)\) and the one introduced in Definition 1.3.
Corollary 5.2
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {O}}(u,X_0)<k_q\). Then
Proof
Suppose by contradiction that \({\mathcal {O}}(u,X_0) < {\mathcal {V}}(u,X_0)\) and consider \(k \in ({\mathcal {O}}(u,X_0),{\mathcal {V}}(u,X_0))\). Let us write
for some \(\alpha >0\). Now, let \(r \in (0,\mathrm {dist}(X_0,\partial ^0 B^+_1))\), by (45) we get
which implies
As in (47), in the proof of Proposition 5.1, there exists \(r_0>0\) and \(C_0>0\) such that
for every \(r\in (0,r_0)\). With a slight abuse of notations, it is not restrictive to assume that \(r_0\) corresponds to the radius introduced in Proposition 5.1.
Finally, by the monotonicity result, fixed \(R=\min \{r_0,\mathrm {dist}(X_0,\partial ^0 B^+)\}\) we deduce, for every \(r \in (0,R)\), that
where \(C>0\) depends only on \(C_0\). Thus, by Definition 1.3 we get that \({\mathcal {O}}(u,X_0)\ge {\mathcal {V}}(u,X_0)\) that, in combination with the opposite inequality, implies the desired result. \(\square \)
Similarly, we show that in the case \({\mathcal {O}}(u,X_0)={\mathcal {V}}(u,X_0)< k_q\), the possible vanishing orders correspond to the possible limits of the Almgrentype frequency formula. For the sake of completeness, we report the proof of this result which is deeply based on the validity of the Almgrentype monotonicity result.
Corollary 5.3
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {V}}(u,X_0)<k_q\). Then \({\mathcal {V}}(u,X_0) = N(X_0,u,0^+).\)
Proof
By (16) and Definition 1.3, we claim that
It is not restrictive to assume that \(X_0=0\) and \(r \in (0,R)\), for some \(R>0\) that will be choose later. By definition of \(r\mapsto H(0,u,r)=H(u,r)\) we immediately get for every \(r \in (0,R)\) that
and in particular for every \(k>0\), by Proposition 5.1, there exists \(\alpha , {\tilde{C}}>0\) such that
with
Suppose first \({\mathcal {V}}(u,0)< N(u,0^+)\), so there exists \(\varepsilon >0\) such that \(k := N(u,0^+)  \varepsilon >{\mathcal {V}}(u,0)\). Let \(R>0\) be such that
where \({\tilde{C}},\alpha >0\) are introduced in Proposition 5.1. Thus, we get \({\underline{N}}k >\varepsilon /2 \) and consequently by (51)
for some constant \(C_2>0\) depending only on \(R>0\). The absurd follows immediately since \(k>{\mathcal {V}}(u,0)\), namely
Similarly, if \({\mathcal {V}}(u,0)> N(u,0^+)\) consider \(k=N(u,0^+)+\varepsilon \), with \(\varepsilon >0\) sufficiency small so that \({\mathcal {V}}(u,0) > k\). By the monotonicity result Proposition 5.1, let \(R>0\) be such that
Hence, since \({\overline{N}}k < \varepsilon /2\), we get by (51)
for some constant \(C_2>0\) depending only on \(R>0\). The contradiction follows by Definition 1.3. \(\square \)
In particular, from the previous equivalences, for those points satisfying \({\mathcal {O}}(u,X_0)<k_q\), it holds
Moreover, for every \(k_1<N(X_0,u,0^+)<k_2\) there exist \(C_1,C_2>0\) such that
for \(r \in (0,R)\), for some \(R>0\) sufficiently small.
Finally, we can introduce the following notion of stratum of the nodal set.
Definition 5.4
Let \(k< k_q\) we define
While in the local case, in [31, 33] the authors proved the existence of a generalized Taylor expansion of the solution near the nodal set by applying an iteration argument based on the results of [6], we apply a blowup analysis in order to understand how the solutions behave near the nodal set \(\Gamma (u)\).
Hence, given \(X_0 \in \Gamma (u)\), for any \(r_k \downarrow 0^+\), we define as normalized blowup sequence
such that
Let us introduce the notation
Since we are assuming \({\mathcal {O}}(u,X_0)< k_q\), the sequence \((\alpha _k)_k\) is bounded and converges to 0 as \(k \rightarrow \infty \).
Theorem 5.5
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {O}}(u,X_0)< k_q\) and \(u_k\) be a normalized blowup sequence centered in \(X_0\) and associated with some \(r_k \downarrow 0^+\). Then, there exists \(p \in H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) such that, up to a subsequence, \(u_k\rightarrow p\) in \(C^{0,\alpha }_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) for every \(\alpha \in (0,1)\) and strongly in \(H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\). In particular, the blowup limit satisfy
The proof will be presented in a series of lemmata.
Lemma 5.6
Let \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0)<k_q\). For any given \(R>0\), we have
where \(C>0\) is a constant independent on \(k>0\). Moreover \(u_k \rightarrow p\) strongly in \(H^{1,a}(B^+_R)\) for every \(R>0\), for some \(p\in H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) such that \(\left\ {p} \right\ _{L^{2,a}(\partial ^+ B^+)}=1\).
Proof
Set \(\rho ^2_k = H(X_0,u,r_k)\), then by the definition of \(u_k\), (50) and Proposition 5.1 we obtain
which gives us \(\left\ {u_k} \right\ _{L^{2,a}(\partial ^+ B^+_R)}^2\le C(R) R^{n+a}\). Instead, inspired by Corollary 5.2, let
for some \(\alpha >0\), then
and by Lemma 2.2 and Proposition 5.1, we infer
which finally implies the uniform bound.
Thus, up to a subsequence, we have proved the existence of a non trivial function \(p \in H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) such that \(\left\ {p} \right\ _{L^{2,a}(\partial B^+_1)}=1\) and \(u_k \rightharpoonup p\) in \(H^{1,a}(B^+_R)\) for every \(R>0\). Moreover, since \(u_k\) is uniformly bounded in \(H^{1,a}(B^+_R)\), we get that, up to a subsequence, \(u_k \rightarrow p\) strongly in \(L^2(B^+_R)\) and in \(L^p (\partial ^0 B^+_R)\), for every \(p \in [1,2^*)\).
On the other hand, we omit the details of the strong convergence in \(H^{1,a}(B^+_R)\) since it simply follows by testing (53) with \((u_k p)\eta \), where \(\eta \in C^\infty _c(B_R)\) is an arbitrary cutoff function, and passing then to the limit. \(\square \)
So far we have proved the existence of a nontrivial solution \(p\in H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\cap L^\infty _{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) of (54) such that, up to a subsequence, we have \(u_k \rightarrow p\) strongly in \(H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\). With the following result we complete the compactness result by showing the uniform convergence in \(C^{0,\alpha }_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) for \(\alpha \in (0,1)\).
Lemma 5.7
For every \(R > 0\) there exists \(C > 0\), independent of k, such that
for every \(\alpha \in (0,1)\).
Proof
The proof follows essentially the ideas of the similar results in [28, 34, 35]: the critical exponent \(\alpha =1\) is related to a Liouville type theorem for \(L_a\)harmonic function in \(\mathbb {R}^{n+1}\) symmetric with respect to the characteristic manifold \(\{y=0\}\), as given in [29]. \(\square \)
As a first Corollary we deduce that the possible vanishing orders of u in the case \({\mathcal {O}}(u,X_0)<k_q\) are completely classified as the possible vanishing orders of \(L_a\)harmonic function even with respect to \(\{y=0\}\) (see [28] for a complete analysis of the nodal set of \(L_a\)harmonic functions).
Corollary 5.8
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\). Then \(k\in 1+\mathbb {N}\) and every blowup limit centered at \(X_0\) is a khomogeneous solution of (54).
Proof
Let \(k= {\mathcal {O}}(u,X_0)<k_q\). By Theorem 5.5 we already know that given \((u_j)_j\) a normalized blowup sequence centered in \(X_0\) and associated to some \(r_j\rightarrow 0^+\), it converges strongly in \(H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) and uniformly on every compact set of \(\overline{\mathbb {R}^{n+1}_+}\) to some \(p \in H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) such that
On the other hand, by Corollary 5.2 and Corollary 5.3 we get \(N(X_0,u,0^+)=k\). Moreover, by the strong convergence of \((u_j)_j\), we have
where
Since p is a global \(L_a\)harmonic function even with respect to \(\{y=0\}\), by [28, Lemma 4.7] we deduce that p is khomogeneous in \(\mathbb {R}^{n+1}_+\) with \(k=1+\mathbb {N}\). \(\square \)
Inspired by the notations introduced by [28], we denote with \(\mathfrak {sB}^a_k(\mathbb {R}^{n+1})\) the class of \(L_a\)harmonic polynomial symmetric with respect to \(\{y=0\}\) and homogeneous of order k.
In order to conclude the local analysis near the points of the nodal set such that \({\mathcal {O}}(u,X_0)<k_q\) we need to better understand the Taylor expansion of the function u near nodal points. Inspired by quite standard techniques (see [17, 18] for similar results in the context of obstacle type problems with weights) we start by introducing the following Weisstype monotonicity formula.
Proposition 5.9
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\). Given \(\delta = 2s(2q)k >0\), there exist \(R_1>0\) and \(C_2>0\) such that
is monotone nondecreasing, for every \(r \in (0,\min \{R_1,\mathrm {dist}(X_0,\partial ^+ B^+_1)\})\) and \(\varepsilon <\delta \). In particular, we get
Proof
For \(k>0\), by Proposition 3.4 and Lemma 2.2, we get
where \(C=C(n,q,s,\Lambda )\). By definition of \(H^{1,a}\)vanishing order, for every \(k_1< {\mathcal {O}}(u,X_0)\) there exists \(R_1,C_1>0\) such that
for every \(r<R_1\). Since \(k< k_q\), there exist \(\delta >0\) such that
Thus, for every \(\varepsilon <\delta \), if we take \(k_1=k\varepsilon /q\) we get that \(r\mapsto W_k(X_0,u,r) + C_2 r^{\delta \varepsilon }\) is monotone nondecreasing, where \(C_2\) does not depend on \(\varepsilon >0\).
Finally, since by Corollary 5.2 and Corollary 5.3 we have \(k={\mathcal {O}}(u,X_0)=N(X_0,u,0^+)\), we get
\(\square \)
The next monotonicity formulas is a Monneautype formula, which will allow to prove uniqueness of the blowup near nodal points satisfying \({\mathcal {O}}(u,X_0)< k_q\).
Proposition 5.10
Let \(X_0 \in \Gamma (u)\) such that \(k={\mathcal {O}}(u,X_0)<k_q\). Given \(\delta = 2s(2q)k >0\), there exist \(R_1>0\) and \(C_2>0\) such that, for every homogenous \(L_a\)harmonic polynomial \(p\in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\), the map
satisfies
for every \(r \in (0,\min \{R_1,\mathrm {dist}(X_0,\partial ^+ B^+_1)\}\) and \(\varepsilon <\delta \), with \(p_{X_0}(X)=p(XX_0)\).
Proof
The strategy is inspired by known result for the thinobstacle problem (see [17]) and for the study of the nodal set of \(L_a\)harmonic functions (see [28]). First, since \(k={\mathcal {O}}(u,X_0)\) we already know \(W_k(X_0,u,0^+)=0\). Now, let \(w = u  p_{X_0}\), then on one hand we have
On the other hand, looking at the expression of the kWeiss functional, we have
where \(C=C(\lambda _+,\lambda _)\) and in the second equality we used the khomogeneity of \(p_{X_0}\in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\). Hence we finally infer
On one side by Proposition 5.9 we have
with \(\delta = 2s(2q)k >0\) and \(\varepsilon <\delta \). On the other, under the notations of Proposition 5.9, for every \(\varepsilon \in (0,\delta )\) let us introduce
Then, by (52) we infer the existence of \(R>0\) sufficiently small such that
for \(r \in (0,R)\). Hence, there exist \(R_1>0\) and \(C=C(n,s,q,\Lambda ,k)\) such that
is monotone nondecreasing \(r \in (0,\min \{R_1,\mathrm {dist}(X_0,\partial ^+ B^+_1)\})\) and \(\varepsilon <\delta \). \(\square \)
For the sake of simplicity, we will use through the paper the following notation for the previous monotonicity formula
Starting from these results, we will improve our knowledge of the blowup convergence by proving the existence of a unique non trivial blowup limit at every point of the nodal set \(\Gamma (u)\), which will be called the tangent map \(\varphi ^{X_0}\) of u at \(X_0\).
Lemma 5.11
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\). Then, there exists \(r_0>0\) and \(C>0\) such that
Proof
Let \(k={\mathcal {O}}(u,X_0)\) and \(\delta = k_qk\). By (50) and Proposition 5.1, there exist \(r_0>0, \alpha = \alpha (\delta ,n,s,q)\) and \({{\tilde{C}}}={{\tilde{C}}}(\delta ,n,s,q)\) such that
for every \(\rho \in (0,r_0)\). Thus, given \(r<r_0\) and integrating between r and \(r_0\) we get
as we claimed. \(\square \)
Lemma 5.12
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\). Then, there exists \(C>0\) such that
where \(R=1\mathrm {dist}(X_0,\partial ^0 B_1)\).
Proof
Since the proof is an adaptation of a similar result for the thinobstacle problem (see [17, Lemma 2.8.1]), we briefly sketch the main ideas.
Fix \(X_0\in \Gamma (u)\) and suppose by contradiction that given a decreasing sequence \(r_j\downarrow 0\) we have
Thus, for \(r_j\le R=\min (r_0,\mathrm {dist}(X_0,\partial ^0 B^+))\), consider the blowup sequence
centered in \(X_0 \in \Gamma (u)\). By Theorem 5.5 and Corollary 5.8 the sequence \((u_j)_j\) converges, up to a subsequence, strongly in \(H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) and uniformly on every compact set of \(\mathbb {R}^{n+1}_+\) to some \(L_a\)harmonic homogenous polynomial p of degree k symmetric with respect to \(\{y=0\}\) such that \(H(0,p,1)=1\).
Therefore, under the notations in Proposition 5.10, by taking \(p_{X_0}\) as above we get
where in the third equality we used the assumption on the growth of u. By the monotonicity result of Proposition 5.10, we obtain
and similarly
On the other hand, rescaling the previous inequality and using the notion of blowup sequence \(u_k\) defined as above, we obtain
Since \({\mathcal {V}}(u,X_0)= {\mathcal {O}}(u,X_0)=k\), by Definition 1.3 we get
and consequently, passing to the limit as \(j\rightarrow \infty \) in the previous inequality, we obtain
in contradiction with \(p \not \equiv 0\). \(\square \)
Theorem 5.13
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\). Then there exists a unique nonzero \(\varphi ^{X_0} \in \mathfrak {sB}_k^a(\mathbb {R}^{n+1})\) blowup limit such that
Moreover, we define as tangent map of u at \(X_0\) the unique \(\varphi ^{X_0} \in \mathfrak {sB}_k^a(\mathbb {R}^{n+1})\) that satisfies (57).
Proof
The proof is a straightforward consequence of the Weiss and Monneau monotonicity formulas. Indeed, up to a subsequence \(r_j\rightarrow 0^+\), we have that \(u_{X_0,r_j}\rightarrow p\) in \({\mathcal {C}}^{0,\alpha }_{{\tiny {{\text{ loc }}}}}\) for every \(\alpha \in (0,1)\). The existence of such limit follows directly from the growth estimate of Lemma 5.11 and, by Lemma 5.12, we have p is not identically zero. Now, by Proposition 5.9, for any \(r>0\) we have
where
In particular, by [28, Proposition 5.2] it implies that the p is khomogeneous \(L_a\)harmonic function even with respect to \(\{y=0\}\), that is \(p \in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\). Now, by Proposition 5.10, the limit of the Monneautype formula exists and can be computed by
Now, suppose by contradiction that for any other subsequence \(r_i\rightarrow 0^+\) we have that \((u_{X_0,r_i})_i\) converges to another blowup limit \(q\in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\) with \(q\not \equiv p\), then
which contradicts \(M(X_0,u,p_{X_0},0^+)=0\). \(\square \)
Thanks to the uniqueness and the nondegeneracy of the blowup limit, we can also construct the generalized Taylor expansion of the solution on the nodal set.
Theorem 5.14
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\) and \(\varphi ^{X_0}\) be the tangent map of u at \(X_0\). Then
Moreover, the map \(X_0 \mapsto \varphi ^{X_0}\) from \(\Gamma _k(u)\) to \(\mathfrak {sB}_k^a(\mathbb {R}^{n+1})\) is continuous.
Proof
The proof is inspired by a similar result for the thinobstacle problem (see [17, Lemma 2.8.1] for more details) based on the validity of the Monneau monotonicity formula. Indeed, fixed \(X_0\in \Gamma (u)\) let \(\varphi ^{X_0}\) be the unique tangent map of u at \(X_0\) defined by Theorem 5.13. Therefore, given \(\varepsilon >0\), Proposition 5.10 implies the existence of \(r_\varepsilon =r_\varepsilon (X_0), \delta _\varepsilon = \delta _\varepsilon (X_0)\) such that
and for \(X_1\in \Gamma _k(u)\cap \Sigma \cap B_{\delta _\varepsilon }(X_0)\). Thus, by rescaling the monotonicity formula in \(X_1\), we can conclude
\(\square \)
Finally, we improve the convergence rate \(o(\left{XX_0} \right^k)\) of the previous generalized Taylor’s expansion into a quantitative bound of the form \(O(XX_0^{k+\delta })\) for some \(\delta >0\).
This result is obtained by proving the validity of an Almgrentype monotonicity result for the difference between the solution u and its tangent map \(\varphi ^{X_0}\) at \(X_0\). Since \({\mathcal {O}}(u,x_0)<k_q\), this analysis resembles the starting point of the iteration argument already used in the case of \(L_a\)harmonic function in [28] to obtain higher order Taylor expansion near nodal points. Notice that this methodology has been used in the last years to study the stratification of obstacle type problems (see [14] for a finer analysis of higher order iterations in the context of thinobstacle problems).
Theorem 5.15
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {O}}(u,X_0)< k_q\) and set
where \(\varphi ^{X_0}\) is the tangent map of u at \(X_0\). Then, there exist \(r_0\) and an absolutely continuous map \(\Psi (r)\) satisfying
and some \(\alpha >0\), such that
is monotone nondecreasing for \(r \in (0,r_0)\). Consequently, there exists the limit
Proof
For the sake of simplicity, it is not restrictive to assume that \(X_0=0\). Set \(k={\mathcal {O}}(u,0)<k_q\), then by Lemma 5.11, Lemma 5.12 and Theorem 5.13 there exists \(C_1,C_2>0\) such that
Now, given \(\varphi \in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\) the unique tangent map of u at \(0\in \Gamma (u)\), let us consider the difference \(w=u\varphi \in H^{1,a}(B^+_r)\) which solves
Thus, following the same computation of the last Section, we easily deduce by an integration by parts (see the proof of Proposition 3.1) that
where E, H are defined by (16) and
Therefore, by the CauchySchwarz inequality on \(\partial ^+ B^+_r\), the associated Almgrentype formula (17) satisfies
On one hand, we have
where \(f(t)=\lambda _+t_+^{q1}\lambda _t_^{q1}\). On the other hand, by Lemma 2.2 and (56) we get
In order to estimate the last remainder R(w, r) we need the introduce the auxiliary function
for \(h\in (0,1)\) to be chosen later. A direct computation yields the identity
which implies, by Lemma 2.2, that
Finally, we get
and consequently
for some \(h\in (0,1)\). By (59) and (46), there exists \(\alpha >0\) such that
Hence, let
Then, by Lemma 2.2 we first deduce \(0\le \psi (r)\le Cr^{1+kqh}\) and then
for r sufficiently small. Therefore, we deduce that the function
is absolutely continuous and increasing for \(r\in (r_1, r_2)\), for some \(0<r_1<r_2\). Following a standard argument, the modified Almgrentype formula (63) can be defined for all \(r \in (0,r_2)\), and it can be extended for \(r = 0\) by taking its limit for \(r \rightarrow 0^+\). \(\square \)
Remark 5.16
Notice that, under the notations of Theorem 6.1, all the computations up to (62) still hold in the critical case \({\mathcal {O}}(u,0)=k_q\) with \(k_q\in \mathbb {N}\) (that is \(\mu =0\) with \(\mu \) defined by (68)). Indeed, in Sect. 6 we will prove that if \(k_q\in \mathbb {N}\) the blowup limit p is an homogeneous \(L_a\)harmonic function symmetric with respect to \(\{y=0\}\), and the function \(w=up\) still satisfies (60). However, in that context the computations will lead to
By the dichotomy (67), even if \(\mu =0\) yields to \(\alpha _k\rightarrow 0^+\), this is not enough to ensure the integrability of the right hand side of the previous inequality. As remarked in [31], is possible that a sophisticated Fourier expansion finally lead to uniqueness: indeed it will imply that \(r_k\mapsto \alpha _k(r_k)\) is Dinicontinuous, which will be enough to ensure the validity of an Almgrentype monotonicity result.
As a simple corollary of the monotonicity result in Proposition 5.10 for the Monneautype formula, we easily deduce a lower bound for the Almgrentype formula evaluated on w.
Corollary 5.17
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {O}}(u,X_0)< k_q\). Then \(N(X_0,u\varphi ^{X_0},0^+)\ge {\mathcal {O}}(u,X_0)\).
In order to improve the growth order of the remainder in (58), we start by proving a blowup argument based on the validity of the previous Almgrentype monotonicity formula. Hence, given \(X_0 \in \Gamma (u)\) and \(w\in H^{1,a}(B^+_r)\) as in Theorem 5.15, we consider the normalized blowup sequence \((w_k)_k\) centered in \(X_0\) associated to some \(r_k \downarrow 0^+\) (see (5) for the definition of normalized blowup sequence), such that
with
Lemma 5.18
Under the previous notations, let \(0<k_1\le k_2\) be such that \({\mathcal {O}}(u,X_0)\le k_1<k_2\). Then, if \(k_1\le N(X_0,u\varphi ^{X_0},0^+)\le k_2\) we infer
for some \(C>0\) and k sufficiently large.
Proof
First, by Proposition 5.10 we already know that \(\beta _k\rightarrow 0^+\). Now, let \(k_1,k_2>0\) be such that \({\mathcal {O}}(u,X_0)\le k_1<k_2\) and \(k_1\le N(X_0,u\varphi ^{X_0},0^+)\le k_2\). By (18) and Theorem 5.15 we have that if \(k_1\le N(X_0,u\varphi ^{X_0},0^+)\le k_2\) then there exits \(C_1,C_2,{\overline{r}}>0\) such that
for every \(r \in (0,{\overline{r}})\). Thus
for k sufficiently large such that \(r_k\le {\overline{r}}\). Finally, by Corollary 5.17, if \(k_1={\mathcal {O}}(u,X_0)<k_q\) we get
as we claimed. \(\square \)
In the following Proposition we finally compute the vanishing order of \(u\varphi ^{X_0}\) in terms of \({\mathcal {O}}(u,X_0)\).
Proposition 5.19
Let \(X_0 \in \Gamma (u)\) be such that \({\mathcal {O}}(u,X_0)< k_q\). Then
Proof
Let \(w=u\varphi ^{X_0}\) and \((w_k)_k\) be the normalized blowup sequence centered at \(X_0\) and associated to some \(r_k\rightarrow 0^+\). As we did in Lemma 5.6, exploiting the normalization with respect to the \(L^{2,a}(\partial ^+ B^+_1)\)norm and the validity of the Almgrentype monotonicity formula, it is easy to see that \((w_k)_k\) is uniformly bounded in \(H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) and it converges, up to subsequence, to some \(p\in H^{1,a}_{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\cap L^\infty _{{\tiny {{\text{ loc }}}}}(\overline{\mathbb {R}^{n+1}_+})\) such that \(\left\ {p} \right\ _{L^{2,a}(\partial ^+ B^+_1)}=1\).
On the other hand, since \({\mathcal {O}}(u,X_0)<k_q\), by Lemma 5.18 we get that both \((\alpha _k)_k\) and \((\beta _k)_k\) approach zero, as k goes to infinity. Therefore, following the same contradiction argument of Lemma 5.7, the sequence \((w_k)_k\) is uniformly bounded in \(C^{0,\alpha }_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) for every \(\alpha \in (0,1)\) and it converges uniformly on every compact set to some \(p \in \mathfrak {sB}^a_k(\mathbb {R}^{n+1})\). Indeed, by the strong convergence of \((w_k)_k\), we get
where
Therefore, p is an homogeneous \(L_a\)harmonic function even with respect to \(\{y=0\}\) of order N(0, p, 1). By [28, Lemma 4.7] we first get that \(N(0,p,1)\in \mathbb {N}\) while by Theorem 5.13 we deduce that \(N(0,p,1) > {\mathcal {O}}(u,X_0)\). Since \(N(0,p,1)=N(X_0,w,0^+)\) we finally get the claimed result. \(\square \)
Thanks to this classification, we can improve the growth order of the remainder in (58).
Corollary 5.20
Let \(X_0 \in \Gamma (u)\) be such that \(k={\mathcal {O}}(u,X_0)< k_q\) and \(\varphi ^{X_0}\) be the tangent map of u at \(X_0\). Then
Moreover, the map \(X_0 \mapsto \varphi ^{X_0}\) from \(\Gamma _k(u)\) to \(\mathfrak {sB}_k^a(\mathbb {R}^{n+1})\) is Hölder continuous.
Having established Theorem 5.14 and Proposition 5.19, we can finally show the validity of the first part of Theorem 1.7 and Theorem 1.9.
Proof of Theorem 1.7
Let us consider the case \({\mathcal {V}}(u,X_0)<k_q\). By Corollary 5.2 and Corollary 5.3 we already know that
Therefore the results of this Section hold true also for the case \({\mathcal {V}}(u,X_0)< k_q\). In particular, by Corollary 5.8, we know that \({\mathcal {V}}(u,X_0)\) must be a positive integer and, by Theorem 5.14 and Proposition 5.19, it follows the validity of expansion (10). \(\square \)
Finally, by applying a variant of the classical Federer’s dimension reduction principle [7, Theorem 8.5] (for the classical result we refer to [27, Appendix A]), and the Whitney’s extension theorem (we refer to [13] and the reference therein) we can easily estimate the Hausdorff dimension of the singular strata.
Proof of Theorem 1.9
First, since \(\Gamma (u) = {\mathcal {T}}(u)\) for those values of \(s\in (0,1)\) and \(q \in [1,2)\) such that \(k_q\le 1\), let us concentrate on the opposite case. Seeing that on \({\mathcal {R}}(u)\cup {\mathcal {S}}(u)\) all the notions of vanishing order coincide, that is
we can easily adapt the general approach of [28] by using the validity of the Almgrentype monotonicity formula. More precisely, by a straightforward application of Corollary 5.20 and the implicit function theorem, we already deduce that
which is relatively open in \(\Gamma (u)\) and it is a \((n1)\)dimensional regular set of class \(C^{1,\alpha }\). Moreover, by the upper semicontinuity of \(X_0\mapsto N(X_0,u,0^+)\), the proof of the Hausdorff estimate
follows the one of [28, Theorem 6.3]).
On the other hand, it is possible to apply step by step the proof of [28, Theorem 7.7] and [28, Theorem 7.8] (using Corollary 5.20 instead of [28, Theorem 5.12] and the generalized formulation of the Whitney’s extension theorem in [13] for \(C^{m,\omega }\)functions), obtaining the desired result for the stratification of the singular set. The crucial idea is that the Whitney’s extension allows to study the structure of the nodal set just by using the generalized Taylor expansion (10) without the highorder differentiability of the function itself. \(\square \)
6 Blowup analysis for \({\mathcal {O}}(u,X_0)=k_q\)
The previous analysis terminates the study of the nodal set in those points where the local behaviour of the solutions resemble the one of the sharmonic functions. In this Section we will complete our study by considering the threshold case \({\mathcal {O}}(u,X_0)=k_q\). The following result is the second part of Theorem 1.7.
Theorem 6.1
Let \(q \in [1,2), \lambda _+, \lambda _ > 0\) and \(u \in H^{1,a}(B_1), u\ne 0\) be a solution of (3).
If \(X_0 \in \Gamma (u)\) satisfies \({\mathcal {O}}(u, X_0) = k_q\), then for every sequence \(r_k \rightarrow 0^+\) we have, up to a subsequence, that
for every \(\alpha \in (0,\min (1,2s))\), where \({\overline{u}}\) is a \(k_q\)homogeneous nontrivial solution to
for some \(\mu \ge 0\). Moreover, the case \(\mu = 0\) is possible if and only if \(k_q\in \mathbb {N}\).
The proof will be presented in a series of lemmata. Given \(X_0 \in \Gamma (u)\) such that \({\mathcal {O}}(u,X_0)=k_q\) and \(r_k \rightarrow 0^+\), we consider normalized blowup sequence
for \(0<r_k<R<\mathrm {dist}(X_0,\partial B_1)\). Thus \(\left\ {u_k} \right\ _{0,1} = 1\) and
By Proposition 4.7 (in particular by (36)), there exists \(C>0\) such that
for every \(r_k<R\). As we pointed out in the previous Sections, the \(H^{1,a}\)normalization seems to be more suitable for the critical case \({\mathcal {O}}(u,X_0)=k_q\) and it overcomes the lack of monotonicity of the Almgrentype formula. The following is a compactness result for the blowup sequence.
Lemma 6.2
For every \(R>0\), there exists \(k_R>0\) such that, for every \(k>k_R\), the sequence \((u_k)_k\) is uniformly bounded in \(H^{1,a}(B_{R}^+)\) and, up to a subsequence, it converges strongly in \(L^{2,a}(B_{R}^+)\) and \(H^{1,a}(B_{R}^+)\).
Proof
The convergence of the sequence \((u_k)_k\) with respect to the strong topology in \(H^{1,a}(B_R^+)\) is a straightforward consequence of the uniform bound in \(H^{1,a}(B_R^+)\). Indeed, suppose there exists \(k_R>0\) such that, for every \(k>k_R\) the sequence is uniformly bounded in \(H^{1,a}(B_R^+)\), then it implies that up to a subsequence \((u_k)_k\) weakly converges in \(H^{1,a}(B_{R}^+)\) and strongly in \(L^{2,a}(B_{R}^+)\). Moreover, by trace embedding, the traces of \((u_k)_k\) on \(\Sigma \) strongly converge in \(L^p(\partial ^0 B^+_R)\), for every \(p \in [1,2^\star )\).
Finally, by testing the equation against \((u_ku)\eta \), where \(\eta \in C^\infty _c(B_{R})\), we easily deduce the validity of the strong convergence by passing to the limit as \(k\rightarrow +\infty \). More precisely,
Since \((u_k)_k\) is uniformly bounded in \(H^{1,a}(B_R)\) and it converges strongly in \(L^{2,a}(B^+_R)\), the first term in the right hand side tends to 0 as \(k \rightarrow \infty \). Similarly, since \(\alpha _k\) is bounded and \(u_k \rightarrow u\) strongly in \(L^p(\partial ^0 B^+_R)\) for \(p \in [1,2^\star )\), the second term vanishes too. Finally, regarding the left hand side, by the weak convergence we get
as k goes to \(+\infty \), which leads to the claimed result.
Hence, it remains to prove the validity of a uniform bounds in \(H^{1,a}\). By definition of \((u_k)_k\), since
the first part of the result follows if there exists \(k_R, C_R>0\) such that
Thus, suppose by contradiction that, up to a subsequence, for \(r_k\searrow 0\) it results
We claim, in such case, that
as \(k\rightarrow \infty \). If not, by (36), we would have that
against the absurd hypothesis. Thus, by Lemma 2.2, we get for every \(r>0\) that
where \(\Lambda = \max \{\lambda _+,\lambda _\}\), which implies
as \(k\rightarrow \infty \). On the other hand, by combining
with the monotonicity of \(r \mapsto W_{k_q,2}(X_0,u,r)\), we deduce that
for k sufficiently large. Together with (66), it implies
as k sufficiently large. Therefore, if we consider the new sequence
since it is uniformly bounded in \(H^{1,a}(B_1)\) and it satisfies
we deduce from the first part of the proof that, up to a subsequence, it converges strongly in \(L^{2,a}(B_1), L^{2,a}(\partial B_1)\) and in \(H^{1,a}(B_1)\) to a function \({\overline{v}} \in H^{1,a}(B_1)\). Moreover, by (66), it solves
Now, the strong convergence in \(L^{2,a}(\partial B_1)\) implies
that is \({\overline{v}} \not \equiv 0\) on \(\partial ^0 B^+_1\). On the other hand, by the absurd assumption, we have
which implies that \({\overline{v}}\equiv 0\) on \(\partial ^0 B^+_{1/R}\). The contradiction follows by the unique continuation property for \(L_a\)harmonic function even with respect to \(\{y=0\}\) (see [11] for the classic unique continuation theorem of \(L_a\)harmonic functions). \(\square \)
Lemma 6.3
Under the previous notations, the sequence \((u_k)_k\) is uniformly bounded in \(C^{0,\alpha }_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) for every \(\alpha \in (0,\min (1,k_q))\). Moreover, up to a subsequence, it converges uniformly on every compact set of \(\mathbb {R}^{n+1}_+\).
Proof
The proof follows essentially the ideas of the similar results in [28, 34, 35] and the result of Proposition 2.4. \(\square \)
So far we have proved the strong convergence of the blowup sequence \((u_k)_k\) in \(H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) and uniformly on every compact set, to a function \({\overline{u}} \in H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\cap L^\infty _{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\). The next step is to prove the homogeneity of the blowup limit and the complete characterization of the possible limits.
Conclusion of the proof of Theorem 6.1
Since by Proposition 4.7 there exists \(C>0\) such that \(\alpha _{k}\in (0,C)\), up to a subsequence, we have either
First, suppose that the limit l is finite. By Lemma 6.2, together with a diagonal argument, we get that \(u_k \rightarrow {\overline{u}}\) strongly in \(H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) and uniformly on every compact set. It is also clear that the limit \({\overline{u}}\) solves (64) with
and \({\overline{u}} \not \equiv 0\) since, by strong \(H^{1,a}(B_1^+)\)convergence, we have \(\left\ {{\overline{u}}} \right\ _{0,1}=1\). Now, since it remains to prove that \({\overline{u}}\) is homogeneous, we start by considering the Weiss type formula \(W_{k_q,2}(0,u_k,R)\), that is
Indeed, passing to the limit as \(k \rightarrow \infty \), we deduce by the uniform convergence that
for any \(R>0\), namely the map \(R\mapsto W_{k_q,2}(0,{\overline{u}},R)\) is constant. Therefore, by Corollary 3.5, it follows that \({\overline{u}}\) is \(k_q\)homogeneous.
Let us deal with the second case in (67). By following the same arguments of the case \(l\in (0,+\infty )\) up to the validity of a Weisstype monotonicity result, we already know that, up to a subsequence, \((u_k)_k\) converges uniformly on every compact set, to a function \({\overline{u}}\in H^{1,a}_{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\cap L^\infty _{\tiny {{\text{ loc }}}}(\overline{\mathbb {R}^{n+1}_+})\) which satisfies
Now, even if (69) still holds true, we can not conclude that \({\overline{u}}\) is \(k_q\)homogeneous as before. Instead, by (69) and the monotonicity of \(R \mapsto W_{k_q,2}(X_0,u,R)\), we get
with \(R_0 \in (0, \mathrm {dist}(X_0, \partial B_1))\) arbitrarily chosen and k sufficiently large. By the previous estimate, we have
where the terms in the right hand side go to zero since \(\alpha _k \rightarrow 0^+\) and \(\left\ {u} \right\ _{X_0,r_k}/r_k^{k_q}\rightarrow +\infty \). Finally, passing to the limit as \(k \rightarrow \infty \), we get
for every \(R>0\). On the other hand, since \({\mathcal {O}}(u, X_0) = k_q\), we get
By Lemma 6.2 and (71), for every \(\alpha >0\) we have
which yields that \({\mathcal {O}}({\overline{u}},0) \ge k_q\). Since we already know that \({\overline{u}}\) is a weak solution of (70), by [28, Lemma 4.7] we get that
which implies with (71) that \(k_q \in 1+\mathbb {N}\) and that \({\overline{u}}\) is \(k_q\)homogeneous in \(\mathbb {R}^{n+1}_+\). \(\square \)
By having established the compactness of the blowup sequence for those point satisfying \({\mathcal {O}}(u,X_0)=k_q\), we can finally prove the equivalence between the two notion of vanishing order.
Corollary 6.4
For every \(X_0 \in \Gamma (u)\), we have \({\mathcal {O}}(u,X_0)={\mathcal {V}}(u,X_0)\).
Proof
Since we already proved in Corollary 5.2 the previous equivalence for the case \({\mathcal {O}}(u,X_0)<k_q\), let us focus on the case \({\mathcal {O}}(u,X_0)=k_q\) and let us prove that
Since the upper estimate follows by the definition of the norm in \(H^{1,a}(B_r(X_0))\), suppose by contradiction that there exists \(r_k \rightarrow 0^+\) such that
Since \({\mathcal {O}}(u,X_0)=k_q\), the normalized blowup sequence
converges, up to a subsequence, to an homogenous nontrivial solution \({\overline{u}}\) of (64) in \(\mathbb {R}^{n+1}\). On the other hand, by (72) we get
By homogeneity, it implies that \({\overline{u}}\equiv 0\) on \(\mathbb {R}^{n+1}\), a contradiction. \(\square \)
Up to the previous Corollary, we knew that Theorem 1.5 was valid for the \(H^{1,a}\)vanishing order. Finally, we can complete the proof in terms of the classic vanishing order \({\mathcal {V}}(u,X_0)\).
Proof of Theorem 1.5
By Proposition 4.7 we already know that the maximum admissible \(H^{1,a}\)vanishing order is equal to \(k_q=k_q\). If \({\mathcal {O}}(u,X_0)<k_q\), by Corollary 5.2 and Corollary 5.3 we already know that
Therefore by Corollary 5.8 we know that \({\mathcal {V}}(u,X_0)\) must be a positive integer.
If instead \({\mathcal {O}}(u,X_0)=k_q\), by Corollary 6.4 we finally deduce that \({\mathcal {V}}(u,X_0)=k_q\), as we claimed. \(\square \)
7 Onedimensional \(k_q\)homogeneous solution
By Theorem 1.5 we already know that for those values of \(s\in (0,1), q \in [1,2)\) such that \(k_q \le 1\) it holds \( \Gamma (u)= {\mathcal {T}}(u)\) with
In this Section, we prove the existence of \(k_q\)homogeneous solutions of (64) whose traces on \(\mathbb {R}^n\times \{0\}\) are onedimensional, for those values of the parameters s and q such that \(k_q<1\).
Thanks to the Federer’s reduction principle, this result allows to control the Hausdorff dimension of \({\mathcal {T}}(u)\) and to prove that the nodal set is a collection of point with vanishing order \(k_q\) and Hausdorff dimension less or equal than \((n1)\), in contrast with the case \(s=1\).
Remark 7.1
The classification of \(k_q\)homogeneous solutions depending only on twovariables \((x_1,y)\) is the starting point for a complete understanding of the regularity of the sublinear set \({\mathcal {T}}(u)\). Indeed, we claim that a possible improvement of flatness approach, via a viscosity formulation of the sublinear set \({\mathcal {T}}(u)\), could give a complete picture of the biggest stratum of \({\mathcal {T}}(u)\). Moreover, we think that this strategy can be easily extended to the case \(k_q>1, k_q \not \in \mathbb {N}\) by taking care of the classification of \(L_a\)harmonic polynomial in [28].
Theorem 7.2
For every \(s \in (0,1), q \in [1,2)\) and \(\lambda _+,\lambda _>0\) such that \(k_q<1\), there exists a \(k_q\)homogeneous function u such that \(u(0,0)=0\) and
In particular, by exploiting the homogeneity of u, the previous problem is equivalent to consider
with \(\mu =k_q\left( k_q+12s\right) \) and \(u(X)=\left{X} \right^k\varphi (X\left{X} \right^{1})\).
In order to simplify the proofs, we will first address the case \(\lambda _+=\lambda _\) by proving existence of solutions of (73) whose traces on \(\mathbb {R}\times \{0\}\) are either of the form
for some positive constants \(A_1,A_2\) depending only on s, q and \(\lambda _+\). Indeed, since these prototypes of solution are either symmetric of antisymmetric with respect to x, the construction of Theorem 7.4 will imply the existence of solution of (73) for every \(\lambda _+,\lambda _>0\).
In the following Lemma we prove a sufficient condition for the existence of nontrivial solutions to (74) in (0, T) for some \(T \in (0,\pi )\), in the case \(\lambda _+=\lambda _\).
Lemma 7.3
Given \(T\in (0,\pi )\) and
let us consider the mixed DirichletNeumann eigenvalue associated to (0, T)
Then, if \(\mu =k_q\left( k_q+12s\right) < \lambda _M(T)\) there exists an unique positive function \(\varphi \in X\) such that
Proof
Under the previous notations, let us consider the minimization problem \(\min _{\varphi \in X} J(\varphi ) \) with
Since \(q\in [1,2)\), for every \(u\in X\) there exists \({\overline{t}}>0\) small enough such that \(J(tu)<0\) for every \(t \in (0,{\overline{t}})\).
Notice that critical point of J in X are solution of (75), i.e. for every \(\phi \in X\) we get
By the Sobolev embedding, for every \(n>2s, q \in [1,2)\) it holds
with
Thus, for \(n=1\) and \(\mu =k_q\left( k_q+12s\right) \) we get
with
Moreover, since by the Poincaré inequality in X we have
for some positive constant \(C_p\), we get
Finally, since
it follows that \(C_p\mu <1\), which implies that J is bounded from below and coercive. Since X is weakly closed, the direct method of the calculus of variations implies the existence of a minimizer u which solves (75). Moreover, we can prove that u is positive: indeed, since if u is a minimizer the same holds also for \(\left{u} \right\), we can already suppose that \(u\ge 0\). Now the strong maximum principle implies that either \(u>0\) or \(u \equiv 0\), but the latter options can be easily ruled out observing that \(J(u)<0\).
Finally, if we suppose there exists two different solutions \(\varphi _1,\varphi _2\) of (75), it is straightforward to see that there exists a linear combination \(w=\varphi _1C\varphi _2\), with \(C>0\) such that \(\varphi _1^{q1}(0)=C\varphi _2^{q1}(0)\) and
Moreover
Necessary w must vanishes identically in (0, T): indeed, if not either the function is strictly positive in (0, T) or it changes sign in (0, T), both in contradiction with the assumption \(\mu < \lambda _M(T)\). Hence, \(\varphi _1\equiv C\varphi _2\) in [0, T], which contradicts the definition of C. \(\square \)
Theorem 7.4
Let \(k_q<1\), then for every \(\lambda _+>0\) there exist only two \(k_q\)homogeneous solutions \(u_1,u_2 \in H^{1,a}(\mathbb {R}^{2}_+)\) of
such that
for some positive constants \(A_1,A_2\) depending only on s, q and \(\lambda _+\).
Proof
Notice first that the condition \(k_q<1\) immediately implies \(s \in (0,1/2)\). Since we plan to prove the existence of a \(k_q\)homogeneous function, it is obvious that its trace must be of the form (77). Moreover, if we suppose by contradiction that there exist two solutions u and v with the same type of traces (either like \(u_1(\cdot ,0)\) or \(u_2(\cdot ,0)\)) then, it must exist a constant \(C>0\) such that \(u^{q1}_\pm (x,0)=Cv^{q1}_\pm (x,0)\) in \(\mathbb {R}\). Consequently, the function \(w=uCv\) is a \(k_q\)homogeneous solution of
By the classification of [28, Lemma 4.7] we already know that either \(k_q\in 1 + \mathbb {N}\) or \(w\equiv 0\). Since \(k_q<1\), necessary \(w\equiv 0\), in contradiction with the choice of \(C>0\).
In order to construct two functions with these features, let us consider the symmetric and antisymmetric solution of the eigenvalue problem associated the traces on \(S^1\) of u.
Hence, for the antisymmetric case, fixed \(T=\pi /2\), by Lemma 7.3 there exists \(\varphi \in H^{1,a}(0,\pi /2)\) such that \(\varphi (\pi /2)=0\) and
Hence, we define
an antisymmetric solution of (74) with \(\lambda _+=\lambda _\). On the other hand, let us consider the symmetric eigenfunction \(\phi \) defined as
for \(T \in (0,\pi /2)\), where \(\lambda _1(T)\) is the fist eigenvalue associated to \((T,\pi T)\). By monotonicity of the eigenvalue with respect to the set inclusion, we already know that \(T \mapsto \lambda _1(T)\) is increasing and it satisfies
Thus, since \(s<1/2\), there exists \(T^* \in (0,\arctan ( \sqrt{2(1s)}))\) such that \(\lambda _1(T^*)=k_q\). Furthermore, by applying Lemma 7.3 with \(T=T^*\), there exists a function \(\psi \in H^{1,a}(0,T^*)\) such that \(\psi (T^*)=0\) and
Finally, let \(C>0\) be such that \(C\phi '(T^*)=\psi '(T^*)\), then if we define
we get a symmetric solution of (74) with Thus, the solutions \(u_i\) are defined as the homogeneous extension of \(\varphi _i\) in \(\overline{\mathbb {R}^{n+1}_+}\)
which gives the claimed result. \(\square \)
Finally, by applying the Federer’s reduction principle in the form of [7, Theorem 8.5], we can conclude the proof of Theorem 1.9 as a byproduct of the results of this Section.
Proof (Conclusion of the proof of Theorem 1.9)
Let us consider the class of functions \({\mathcal {F}}\) defined as
endowed with the topology associated to the uniform convergence and
We already know that \({\mathcal {F}}\) is close under rescaling, translation and normalization. Moreover, by Theorem 6.1 the hypothesis of the existence of a blowup limit in \({\mathcal {F}}\) is satisfied, as well as the singular set assumption. Thus, the Federer’s reduction principle [7, Theorem 8.5] is applicable and it implies the existence of an integer \(d \in [0,n]\) such that
for every function \(u\in {\mathcal {F}}\). Suppose by contradiction that \(d = n\), this would implies the existence of \(\varphi \in {\mathcal {F}}\) such that \(\overline{{\mathcal {S}}}(\varphi ) = \mathbb {R}^n\), that is \(\varphi \equiv 0\) on \(\mathbb {R}^n\). Thus \(\varphi \equiv 0\) on the whole \(\mathbb {R}^{n+1}\), which contradicts the fact the \(0 \not \in {\mathcal {F}}\). Actually, since Theorem 7.2 ensures the existence of a \((n1)\)linear subspace \(E\subset \mathbb {R}^n\) and a \(k_q\)homogeneous function \(\varphi \in {\mathcal {F}}\) such that \(\overline{{\mathcal {S}}}(\varphi )=E\), we get \(d=n1\). \(\square \)
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Work partially supported by the ERC Advanced Grant 2013 n. 339958 COMPAT and by the GNAMPA project “Esistenza e proprietà qualitative per soluzioni di EDP non lineari ellittiche e paraboliche”. We would like to thank Chiara Ferrante for some fruitful discussions and suggestions.
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Tortone, G. The nodal set of solutions to some nonlocal sublinear problems. Calc. Var. 61, 82 (2022). https://doi.org/10.1007/s00526022021975
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DOI: https://doi.org/10.1007/s00526022021975