Spectral asymptotics of radial solutions and nonradial bifurcation for the H\'enon equation

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u&=|x|^\alpha |u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} in the unit ball ${\bf B} \subset \mathbb{R}^N,N\geq 3$, $p>2$ in the limit $\alpha \to +\infty$. More precisely, for a given positive integer $K$, we derive asymptotic $C^1$-expansions for the negative eigenvalues of the linearization of the unique radial solution $u_\alpha$ of $(H)$ with precisely $K$ nodal domains and $u_\alpha(0)>0$. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch $\alpha \mapsto (\alpha,u_\alpha)$ which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.


Introduction
We consider the Dirichlet problem for the generalized Hénon equation where B ⊂ R N , N ≥ 3 is the unit ball and p > 2, α > 0. This equation originally arose through the study of stellar clusters in [11]. One of the first results on (1.1) is due to Ni [18], who proved the existence of a positive radial solution in the subcritical range of exponents 2 < p < 2 * α , where 2 * α := 2N +2α N −2 . In another seminal paper, Smets, Willem and Su [22] observed that symmetry breaking occurs for fixed p and large α, i.e., there exists α * > 0 depending on p such that ground state solutions of (1.1) are nonradial for α > α * . In the sequel, the existence and shape of radial and nonradial solutions of the Hénon equation has received extensive attention, see e.g. [1-6, 14, 19-21]. In particular, bifurcation of nonradial positive solutions in the parameter p is studied in [1] for fixed α > 0. Moreover, a related critical parameter-dependent equation on R N is considered in [9].
The main motivation for the present paper is the investigation of bifurcation of nonradial nodal (i.e., sign changing) solutions -in the parameter α > 0 -from the set of radial nodal solutions. To explain this in more detail, let us fix K ∈ N, an exponent p > 2 and consider α > α p := max (N − 2)p − 2N 2 , 0 , which amounts to the subcriticality condition p < 2 * α . Under these assumptions, it has been proved by Nagasaki [16] that (1.1) admits a unique classical radial solution u α ∈ C 2 (B) with u α (0) > 0 and with precisely K nodal domains (i.e., K − 1 zeros in the radial variable r = |x| ∈ (0, 1)). In order to decide whether the branch α → u α admits bifurcation of nonradial solutions for large α, we need to analyze its spectral asymptotics as α → ∞. More precisely, we wish to derive asymptotic expansions of the eigenvalues of the linearizations of (1.1) at u α as α → ∞. For this we consider the linearized operators ϕ → L α ϕ := −∆ϕ − (p − 1)|x| α |u α | p−2 ϕ, α > α p , (1.2) which are self-adjoint operators in L 2 (B) with compact resolvent, domain H 2 (B) ∩ H 1 0 (B) and form domain H 1 0 (B). In particular, they are Fredholm operators of index zero. As usual, u α is called nondegenerate if L α : H 2 (B) ∩ H 1 0 (B) → L 2 (B) is an isomorphism, which amounts to the property that the equation L α ϕ = 0 only has the trivial solution ϕ = 0 in H 2 (B) ∩ H 1 0 (B). Otherwise, u α is called degenerate. By a classical observation, only values α such that u α is degenerate can give rise to bifurcation from the branch α → u α . Moreover, properties of the kernel of L α and the change of the Morse index are of key importance to establish bifurcation. Here we recall that the Morse index of u α is defined as the number of negative eigenvalues of the operator L α .
The first step in deriving asymptotic spectral information of the operator family L α , α > α p is to characterize the limit shape of the solutions u α after suitable transformations. Inspired by Byeon and Wang [4], we transform the radial variable and derive a corresponding limit problem. Here, for simplicity, we also regard u α = u α (r) as a function of the radial variable r = |x| ∈ [0, 1]. Our first preliminary result is the following. Proposition 1.1. Let p > 2, K ∈ N. Moreover, for α > α p , let u α denote the unique radial solution of (1.1) with K nodal domains and u α (0) > 0, and define ) is characterized as the unique bounded solution of the limit problem with U ′ (0) > 0 and with precisely K − 1 zeros in (0, ∞).
The asymptotic description derived in Proposition 1.1 implies that the solutions u α blow up everywhere in B as α → ∞, in contrast to the nonradial ground states considered in [22]. It is therefore reasonable to expect that the Morse index of u α tends to infinity as α → ∞. This fact has been proved recently and independently for more general classes of problems in [2,14], extending a result for the case N = 2 given in [15]. To obtain a more precise description of the distribution of eigenvalues of L α as α → ∞, we rely on complementary approaches of [2,14] and implement new tools. We note here that [14] uses the transformation (1.3) in a more general context together with Liouville type theorems for limiting problems on the half line. In the present paper, we build on very useful results obtained recently by Amadori and Gladiali in [2]. In particular, we use the fact that the Morse index of u α equals the number of negative eigenvalues (counted with multiplicity) of the weighted eigenvalue problem see [2,Prop. 5.1]. In various special cases, this observation had already been used before, see e.g. [7,Section 5]. In order to avoid regularity issues related to the singularity of the weight 1 |x| 2 , it is convenient to consider (1.5) in weak sense via the quadratic form q α associated with L α , see Section 3 below. The problem (1.5) is easier to analyze than the standard eigenvalue problem L α ϕ = λϕ without weight. Indeed, every eigenfunction of (1.5) is a sum of functions of the form where ψ ∈ H 1 0,rad (B) and Y ℓ is a spherical harmonic of degree ℓ, see [2,Prop. 4.1]. Here H 1 0,rad (B) denotes the space of radial functions in H 1 0 (B). We recall that the space of spherical harmonics of degree ℓ ∈ N ∪ {0} has dimension d ℓ : and that every such spherical harmonic is an eigenfunction of the Laplace-Beltrami operator on the unit sphere S N −1 corresponding to the eigenvalue λ ℓ := ℓ(ℓ + N − 2). For functions ϕ of the form (1.6), the eigenvalue problem (1.5) reduces to an eigenvalue problem for radial functions given by where µ = λ − λ ℓ . In [2,p.19 and Prop. 3.7], it has been proved that (1.7) admits precisely K negative eigenvalues Combining this fact with the observations summarized above, one may then derive the following facts which we cite here in a slightly modified form from [2].
In order to describe the asymptotic distribution of negative eigenvalues of L α , it is essential to study the asymptotics of the eigenvalues α → µ i (α), i = 1, . . . , K. With regard to this aspect, we mention the estimate which has been derived in [2,Lemma 5.11 and Remark 5.12]. In particular, it follows that µ i (α) → −∞ as α → ∞ for i = 1, . . . , K − 1. In our first main result, we complement this estimate by deriving asymptotics for µ i (α). Theorem 1.3. Let p > 2 and α > α p . Then the negative eigenvalues of (1.7) are given as . . , K satisfying the asymptotic expansions where c * i , i = 1, . . . , K are constants and the values ν * 1 < ν * 2 < · · · < ν * K < 0 are precisely the negative eigenvalues of the eigenvalue problem with U ∞ given in Proposition 1.1. In particular, there exists α * > 0 such that the curves µ i , i = 1, . . . , K are strictly decreasing on [α * , ∞).
Remark 1.4. The strict monotonicity of the curves µ i on [α * , ∞) will be of key importance for the derivation of bifurcation of nonradial solutions via variational bifurcation theory. For this we require the derivative expansion in (1.10), but we do not need additional information on the constants c * i since ν * i < 0 for i = 1, . . . , K. Our proof of (1.10) gives rise to the following characterization of the constants c * i : For fixed i ∈ {1, . . . , K}, we have where U ∞ is given in Proposition 1.1, V is the unique bounded solution of the problem and Ψ is the (up to sign unique) eigenfunction of (1.11) associated with the eigenvalue ν * i with ∞ 0 Ψ 2 dt = 1. The strict monotonicity of the curves µ i for large α asserted in Theorem 1.3 allows us to deduce the following useful properties related to nondegeneracy and a change of the Morse index of the functions u α . Corollary 1.5. Let p > 2. For every i ∈ {1, . . . , K}, there exist ℓ i ∈ N ∪ {0} and sequences of numbers α i,ℓ ∈ (α p , ∞), ε i,ℓ > 0, ℓ ≥ ℓ i with the following properties: (iv) For ε ∈ (0, ε i,ℓ ) the Morse index of u α i,l +ε is strictly larger than the Morse index of u α i,l −ε .
With the help of Corollary 1.5 and an abstract bifurcation result in [13], we will derive our second main result on the bifurcation of nonradial solutions from the branch α → u α . Theorem 1.6. Let 2 < p < 2N N −2 , and let K ∈ N, i ∈ {1, . . . , K} be fixed. Then the points α i,ℓ , ℓ ≥ ℓ i are bifurcation points for nonradial solutions of (1.1).
As mentioned above, Theorem 1.6 will be derived from Corollary 1.5 and variational bifurcation theory. For this we reformulate (1.1) as a bifurcation equation in the Hilbert space H 1 0 (B) and show that, as a consequence of Corollary 1.5, the crossing number of an associated operator family is nonzero at the points α i,ℓ . Thus the main theorem in [13] applies and yields that the points α i,ℓ , ℓ ≥ ℓ i are bifurcation points for solutions of (1.1) along the branch α → u α . To see that bifurcation of nonradial solutions occurs, it suffices to note that the solutions u α are radially nondegenerate for α > 0, i.e., the kernel of L α does not contain radial functions. A proof of the latter fact can be found in [2,Theorem 1.7], and it also follows from results in [23].
Since Corollary 1.5 is a rather direct consequence of Theorem 1.3, the major part of this paper is concerned with the proofs of Proposition 1.1 and Theorem 1.3. It is not difficult to see that, via the transformation given in (1.3), the Hénon equation (1.1) transforms into a family of problems depending on the new parameter γ = N −2 N +α which admits a well-defined limit problem as γ → 0 + given by (1.4). It is then necessary to choose a proper function space which allows to apply the implicit function theorem at γ = 0, and this yields the convergence statement in Proposition 1.1. The idea of the proof of Theorem 1.3 is similar, as we use the same transformation (up to scaling) to rewrite the α-dependent eigenvalue problem (1.7) as a γ-dependent eigenvalue problem on the interval [0, ∞). We shall then see that (1.11) arises as the limit of the transformed eigenvalue problems as γ → 0 + . In order to obtain C 1 -expansions of eigenvalue curves, we wish to apply the implicit function theorem again at the point γ = 0. Here a major difficulty arises in the case where p ∈ (2, 3], as the map U → |U | p−2 fails to be differentiable between standard function spaces. We overcome this problem by restricting this map to the subset of C 1 -functions on [0, ∞) having only a finite number of simple zeros and by considering its differentiability with respect to a weighted uniform L 1 -norm, see Sections 3 and 4. This is certainly the hardest step in the proof of Theorem 1.3.
It seems instructive to compare the transformations used in the present paper with the ones used in [2,15]. Transforming a radial solution u of (1.1) by setting w(τ ) = ( 2 2+α ) 2 p−2 u(τ 2 2+α ) for τ ∈ (0, 1) leads to the problem with M = M (α) = 2(N +α) 2+α . Via this transformation, the associated weighted singular eigenvalue problem (1.7) corresponds to the even more singular eigenvalue equation 13) which is considered in M -dependent function spaces in [2]. In principle, it should be possible to carry out our approach also via these transformations, but we found it easier to find appropriate parameter-independent function spaces in the framework we use here. We stress again that finding parameter-independent function spaces is essential for the application of the implicit function theorem. The paper is organized as follows. In Section 2, we first recall some known results on radial solutions of (1.1) and properties of the associated linearized operators. We then study the asymptotic behavior of the functions u α as α → ∞ and prove Proposition 1.1. Section 3 is devoted to the proofs of Theorem 1.3 and Corollary 1.5. In Section 4 we prove, in particular, the differentiability of the map U → |U | p−2 for p ∈ (2, 3] in a suitable functional setting. In Section 5, we finally prove the bifurcation result stated in Theorem 1.6. This section is devoted to the asymptotics of branches of sign changing radial solutions of (1.1) as α → ∞. In particular, we will prove Proposition 1.1. As before, we let K ∈ N be fixed, and we first recall a result on the existence, uniqueness and radial Morse index of a radial solution u α of (1.1) with K nodal domains.
Theorem 2.1. For every p > 2 and α > α p , equation (1.1) has a unique radial solution u α ∈ C 2 (B) with precisely K nodal domains such that u α (0) > 0. Furthermore, the linearized operator is a Fredholm operator of index zero having the following properties for every α ≥ 0: (i) u α is radially nondegenerate in the sense that the kernel of L α does not contain radial functions.
(ii) u α has radial Morse index K in the sense that L α has precisely K negative eigenvalues corresponding to radial eigenfunctions in Theorem 2.1 is merely a combination of results in [16] and [2]. More precisely, the existence and uniqueness of u α is proved in [16]. Note that the operator L α is a compact perturbation of the isomorphism −∆ : , which implies that it is a Fredholm operator of index zero. A proof of the radial nondegeneracy and radial Morse index can be found in [2, Theorem 1.7]. We remark here that the radial nondegeneracy can also be deduced from results in [23]. (ii) In [16] it is also shown that for p ≥ 2N +2α N −2 , the trivial solution is the only radial solution of equation (1.1).
Next we recall that, in the radial variable, u α solves Inspired by Byeon-Wang [4], we transform equation (2.1), considering By direct computation, we see that U α is a bounded solution of the problem We first note the following facts regarding (2.3).
In the following, it is more convenient to work with the parameter γ = N −2 N +α ∈ (0, N −2 N ) in place of α. Hence, from now on, we will write U γ in place of U α . We also set U 0 : We wish to consider (1.4) and (2.2) in suitable spaces of continuous functions. For δ ≥ 0, we We note the following.
Proof. This is a straightforward consequence of the Arzelà-Ascoli theorem.
For the remainder of this section, we fix δ = 2 N and consider the spaces As note above, F is a Banach space with norm · F = · C δ . Moreover, for every v ∈ E we have Since C 1 δ is a Banach space, it easily follows that E is a Banach space as well. We also note that lim It remains to show that lim t→∞ U (t) = 0. For this we consider the nonincreasing function m(t) := sup s≥t |U (s)|. Using (2.2) and the fact that U ∈ E, we find that and therefore Consequently, We intend to use the implicit function theorem to show that U γ → U 0 in E as γ → 0. This requires uniqueness and nondegeneracy properties as given in the following two lemmas. Lemma 2.6. Let p > 2, γ ∈ (0, N −2 N +αp ) and letŨ ∈ E be a solution of (2.2) with precisely K − 1 zeros in (0, ∞) and lim Proof. Let α > 0 be the unique value such that γ = γ(α) = N −2 N +α , and consider the function SinceŨ ∈ E, the latter limit exists. We then have u ∈ C 2 ((0, 1]) ∩ C([0, 1]), and u solves N+αŨ ′ (t).
also follows that lim r→0 u ′′ (r) exists, and that u also satisfies the boundary conditions in (2.1).
Moreover, we have u(0) > 0 since lim t→∞Ũ (t) > 0 by assumption. The uniqueness result in Theorem 2.1 then yields that u is equal to u α . Transforming back, we conclude that Then the solution U γ of problem (2.2) is nondegenerate in the sense that the equation has no bounded nontrivial solution.
Proof. We consider the auxiliary function w := U ′ γ + γ−1 p−2 U γ , which, by direct computation, solves the linearized equation Sturm comparison with w yields that v can only have finitely many zeros in I.
In the first case we then have v ≡ 0 and the proof is finished. In the other case it also follows that there exists c = 0 such that cw We may now state a continuation result for the map γ → U γ which in particular implies Proposition 1.1.
Proof. We consider the map Since e (γ−1)t ≤ e − 2 N t for γ < N −2 N +αp , G is well-defined and of class C 1 . Moreover, by definition of U γ we have We first show that the linear map is an isomorphism for γ ∈ [0, N −2 N +αp ). For this, we first note that Moreover, if f ∈ F is given and ϕ : I → R is defined by for t ≥ 0 and therefore ϕ ∈ E. We thus infer (2.11).
By a continuation argument based on (2.10), an application of the implicit function theorem in points (γ, U γ ) for γ > 0 and the same continuity considerations as above , we then see that the map is of class C 1 . The proof is thus finished.
Remark 2.9. Using the function g and ε 0 > 0 from Proposition 2.8, it is convenient to define With this definition, it follows from Proposition 2.8 that the map Moreover, implicit differentiation of (2.2) at γ = 0 shows that V = ∂ γ γ=0 U γ is given as the unique bounded solution of the problem (2.14)

Spectral asymptotics
This section is devoted to the proofs of Theorem 1.3 and Corollary 1.5. We fix p > 2, and we start by recalling some results from [2] on the eigenvalue problem (1.5) and its relationship to the Morse index of u α . Recall that we consider (1.5) in weak sense. More precisely, we say that ϕ ∈ H 1 0 (B) is an eigenfunction of (1.5) corresponding to the eigenvalue λ ∈ R if where is the quadratic form associated with the operator L α . Note that the RHS of (3.1) is welldefined for ϕ, ψ ∈ H 1 0 (B) by Hardy's inequality. (ii) Let ϕ ∈ H 1 0 (B) be an eigenfunction of (1.5) corresponding to the eigenvalue λ ∈ R. Then there exists a number ℓ 0 ∈ N ∪ {0}, spherical harmonics Y ℓ of degree ℓ and functions ϕ ℓ ∈ H 1 0,rad (B), ℓ = 1, . . . , ℓ 0 with the property that Moreover, for every ℓ ∈ {1, . . . , ℓ 0 }, we either have ϕ ℓ ≡ 0, or ϕ ℓ is an eigenfunction of (1.7) corresponding to the eigenvalue µ = λ − λ ℓ .
Regarding the reduced weighted eigenvalue problem (1.7), we also recall the following. Let α > α p . Then 0 is not an eigenvalue of (1.7), and the negative eigenvalues of (1.7) are simple and given by Here we point out that Theorem 2.1(i) already implies that zero is not an eigenvalue of (1.7). We also note that Proposition 1.2 now merely follows by combining Lemma 3.1 and Lemma 3.2.
We now turn to the proof of Theorem 1.3. For this we transform the radial eigenvalue problem (1.7). Note that, if we write an eigenfunction ψ ∈ H 1 0,rad (B) as a function of the radial variable r = |x|, it solves We transform this problem by considering again I := (0, ∞) and setting This gives rise to the eigenvalue problem Here, we have added the condition Ψ ∈ L ∞ (I) since we focus on eigenfunctions corresponding to negative eigenvalues, and in this case eigenfunctions ψ ∈ H 1 0,rad (B) of (1.7) are bounded by Lemma 3.2. In the following, we also consider the case γ = 0 in (3.5), which corresponds to the linearization of (2.3) at U 0 : (3.6) We note that for γ ∈ [0, N −2 N +α ) and every solution Ψ of (3.5) there exists a sequence t n → ∞ with Ψ ′ (t n ) → 0, which implies that for t ≥ 0. We also note that problem (3.5) can be rewritten as We need the following estimate in terms of the space C 2 δ (I) defined in Section 2.
By (3.10) and the definition of δ, the function v ε : This implies that v ε cannot attain a negative minimum in the set (t 0 , ∞). Moreover, by definition of v ε we have v ε (t 0 ) ≥ 0 and lim t→∞ v ε (t) = ∞.
. It is clear that w has a zero between any two zeros of U 0 on [0, ∞). Moreover, letting t * > 0 denote the largest zero of U 0 , we find that the numbers have opposite sign, hence w also has a zero in (t * , ∞). Since U 0 has K − 1 zeros in (0, ∞) and U 0 (0) = 0, we infer that w has at least K zeros in (0, ∞). From this, it is standard to deduce that ν K (0) < 0. We thus have proved (3.12). Next we note that eigenfunctions Ψ of (3.5) corresponding to an eigenvalue ν j (γ) < 0 have precisely j − 1 zeros in I. Indeed, this follows from standard Sturm-Liouville theory since any such eigenfunction decays exponentially as t → ∞ together with their first and second derivatives by Lemma 3.3. It also follows that ν j (γ) is simple in this case, i.e., the corresponding eigenspace is one-dimensional.
Next, we wish to derive some information on the derivative ∂ γ ν j (γ) of the negative eigenvalues of (3.5) as γ → 0 + . We intend to derive this information via the implicit function theorem applied to the map G : Here, ε 0 is given in Proposition 2.8, so that (−ε 0 , N −2 N +αp ) → C 1 0 (I), γ → U γ is a well defined C 1 -map by Remark 2.9. Moreover,Ẽ andF are suitable spaces of functions on I chosen in a way that eigenfunctions and eigenvalues of (3.8) and (3.6) correspond to zeros of this map. However, in the case p ∈ (2, 3], the function | · | p−2 is not differentiable at zero and therefore it is not a priori clear howẼ andF need to be chosen to guarantee that G is of class C 1 . In particular, spaces of continuous functions will not work in this case, so we need to introduce different function spaces. For δ > 0 and 1 ≤ r < ∞, we let L r δ (I) denote the space of all functions f ∈ L r loc (I) such that The completeness of L r -spaces readily implies that the spaces L r δ (I) are also Banach spaces. We will need the following observation: and t 0 e µs |f (s)| ds ≤ D δ,µ f 1,δ e (µ−δ)t for µ > δ, t ≥ 0 with D δ,µ := e 2µ−δ e µ−δ − 1 . (3.20) Proof. Let f ∈ L 1 δ (I) and t ≥ 0. If µ < δ, we have and in the case µ > δ we have with C µ,δ and D δ,µ given above.
Next, for δ > 0, we define the function space and endow this space with the norm We first note that Proof. Consider a Cauchy sequence (u n ) n in W 2 δ (I). Then we have by (3.19). From (3.22) we deduce that u ′′ = v ∈ L 1 δ (I) in weak sense. Then it follows from (3.21) that u n → u in W 2 δ (I).
The following simple lemma is essential.
The key observation of this section is the following. Proposition 3.9. Let ε 0 > 0 be given by Proposition 2.8, so that (−ε 0 , γ ⋄ ) → C 1 0 (I), γ → U γ is a well defined C 1 -map by Remark 2.9. Moreover, let the map be defined by (3.18). Then G is of class C 1 with We postpone the somewhat lengthy proof of this proposition to the next section and continue the main argument first. We fix j ∈ {1, . . . , K} and for γ ≥ 0 we let Ψ γ,j denote an eigenfunction of the eigenvalue problem (3.8) corresponding to the eigenvalue ν j (γ). We thus have By (3.26) we have Ψ γ,j ∈ E δ . Moreover, we can assume ∞ 0 Ψ 2 γ,j dt = 1 so that G(γ, Ψ γ,j , ν j (γ)) = 0.
To apply the implicit function theorem to G at the point (γ, Ψ γ,j , ν j (γ)), we need the following property.
Proof. Since, by definition, we may apply Lemma 3.8 with µ = −ν j (γ). Hence the map is a Fredholm operator of index zero. The kernel of this map is one dimensional, since it consists of eigenfunctions corresponding to ν j (γ). Hence the codimension of the image of T is one, and we claim that Ψ γ,j is not contained in the image of T . Otherwise, there exists Multiplying with Ψ γ,j and integrating by parts then yields a contradiction. It follows that We now show that L is an isomorphism. First assume L(ϕ, ρ) = 0 for some (ϕ, ρ) ∈ E δ × R, i.e., Since Ψ γ,j ∈ range T , the first equality yields ρ = 0. But then ϕ itself is an eigenfunction and therefore ϕ = cΨ γ,j for some c ∈ R. The second equality then yields c = 0, and thus (ϕ, ρ) = (0, 0). Hence L is injective. Now let (g, σ) ∈ F δ ×R. By (3.27) there exist g 0 ∈ range T , κ ∈ R such that g = g 0 +κΨ γ,j . Since g 0 ∈ range T , there exists a solution ϕ 0 ∈ E δ of Furthermore, for any η ∈ R, ϕ 0 + ηΨ γ,j ∈ E δ is also a solution. Taking η = σ − ∞ 0 Ψ γ,j ϕ 0 dt yields ∞ 0 Ψ γ,j (ϕ 0 + ηΨ γ,j ) dt = σ.

Consequently, we have
Hence L is surjective.
With the help of Propositions 3.9 and 3.10, we may now apply the implicit function theorem to G at (γ, Ψ γ,j , ν j (γ)). This yields the following result.

Differentiability of the map G
In this section, we give the proof of Proposition 3.9, which we restate here in a slightly more general form. As before, we fix p > 2 and γ ⋄ ∈ [0, N −2 N +αp ).
Here we identify |u| q−2 u with sgn(u) in the case q = 1.
Proof. We only consider the case q ∈ (0, 1). The proof in the case q = 1 is similar but simpler, and the proof in the case q > 1 is standard. We first prove Claim 1: If 1 ≤ r < 1 1−q , then the map σ q : U → L r 0 (I), σ q (u) = |u| q−2 u is well defined and continuous. To see this, we note that, by definition of U , for every u ∈ U we have More generally, if K ⊂ U is a compact subset (with respect to · C 1 0 ), we also have that κ K := sup u∈K κ u < ∞.
As a consequence of (4.1), we have for every u ∈ U and t ≥ 0, since 1 (q−1)r < −1 by assumption. Hence σ q (u) ∈ L r 0 (I) for every u ∈ U , so the map σ q is well defined. To see the continuity of σ q , let (u n ) n ⊂ U be a sequence such that u n → u ∈ U as n → ∞ with respect to the C 1 0 -norm. We then consider the compact set K := {u n , u : n ∈ N}. For given ε > 0, we fix c ∈ (0, 1) sufficiently small such that c (q−1)r+1 < ε 2 r κ K 2 1+(q−1)r 1+(q−1)r .
We conclude that for i = 1, . . . , K − 1 and each direction w ∈ S N −1 the function (r i − δ, r i + δ) → R, t → u n (tw) has precisely one zero, which we denote by r i,n (w). In particular, the nodal domains of u n are given by Ω 1 := x ∈ B : |x| < r 1,n x |x| and Ω i := x ∈ B : r i−1,n x |x| < |x| < r i,n x |x| for i = 2, . . . K. Consequently, 0 ∈ Ω 1 , Ω 1 is homeomorphic to a ball, and Ω 2 , . . . , Ω K are homeomorphic to annuli. Finally, we note that u n = v n + u αn is nonradial, since v n ≡ 0 and u αn is the unique radial solution of (1.1) with α = α n and with K nodal domains.