On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

Abstract

Given that(𝑀,𝑔)is a smooth compact and symmetric Riemannian 𝑛-manifold, 𝑛≥2, we prove a multiplicity result for antisymmetric sign changing solutions of the problem −𝜀²Δ_𝑔𝑢+𝑢=|𝑢|^𝑝−2𝑢 in 𝑀. Here 𝑝>2 if 𝑛=2 and 2<𝑝<2^∗=2𝑛/(𝑛−2) if 𝑛≥3.

1. Introduction

Let (𝑀,𝑔) be a smooth compact connected Riemannian manifold without boundary of dimension 𝑛≥2. Let us consider the problem −𝜀²Δ_𝑔𝑢+𝑢=|𝑢|^𝑝−2𝑢in𝑀,𝑢∈H¹_𝑔(𝑀), (1.1) where 𝑝>2 if 𝑛=2,2<𝑝<2𝑛/(𝑛−2) if 𝑛≥3 and 𝜀 is a positive parameter. Here H¹_𝑔(𝑀) is the completion of 𝐶^∞(𝑀) with respect to ‖𝑢‖²_𝑔∶=_𝑀|∇_𝑔𝑢|²𝑑𝜇_𝑔+_𝑀𝑢²𝑑𝜇_𝑔. (1.2) It is well known that any critical point of the energy functional 𝐽_𝜀∶H¹_𝑔(𝑀)→ℝ constrained to the Nehari manifold 𝒩_𝜀 is a solution to (1.1). Here 𝐽_𝜀1(𝑢)∶=𝜀^𝑛_𝑀(12𝜀²|∇_𝑔𝑢|²+12𝑢²−1𝑝|𝑢|^𝑝)𝑑𝜇_𝑔,𝒩_𝜀∶={𝑢∈H¹_𝑔(𝑀)⧵{0}∶𝐽^′_𝜀(𝑢)[𝑢]=0}. (1.3)(1.4) In [1] the authors show that the least energy solution of (1.1), that is, the minimum of 𝐽_𝜀 on 𝒩_𝜀 is a positive solution with a spike layer, whose peak converges to the maximum point of the scalar curvature 𝒮_𝑔 of (𝑀,𝑔) as 𝜀 goes to zero. Successively, in [2] (see also [3, 4]) the authors point out that the topology of the manifold 𝑀 influences the multiplicity of positive solutions of (1.1), that is, (1.1) has at least cat(𝑀) nontrivial solutions provided that 𝜀 is small enough. Here cat(𝑀) denotes the Lusternik-Schnirelman category of 𝑀. Recently, in [5–7] it has been proved that the existence of positive solutions is strongly related to the geometry of 𝑀, that is stable critical points of the scalar curvature 𝒮_𝑔 generate positive solutions with one or more peaks as 𝜀 goes to zero.

As far as it concerns the existence of sign changing solutions to (1.1), a few results are known. The first result has been obtained in [7] where it has been constructed solutions with one positive peak and one negative peak, which approach, as 𝜀 goes to zero, the minimum point and the maximum point of 𝒮_𝑔, provided the scalar curvature is not constant. In [8] the authors assume the following:

(S)the manifold 𝑀 is a regular submanifold of ℝ^𝑁 invariant with respect to 𝜏, where 𝜏∶ℝ^𝑁→ℝ^𝑁 is an orthogonal linear transformation such that 𝜏≠𝐼 and 𝜏²=𝐼,𝐼 being the identity of ℝ^𝑁.

They prove problem (1.1) has at least 𝐺_𝜏−cat(𝑀−𝑀_𝜏) pairs of sign changing solutions which change sign exactly once. Here 𝐺_𝜏−cat(𝑀−𝑀_𝜏) denotes the 𝐺_𝜏-equivariant Lusternik-Schnirelman category for the group 𝐺_𝜏∶={𝐼,𝜏} and 𝑀_𝜏∶={𝑥∈𝑀∶𝜏𝑥=𝑥}.

In this paper we assume 𝑀 satisfies (𝑆) in the particular case 𝜏=−𝐼. We look for solutions of the problem −𝜀²Δ_𝑔𝑢+𝑢=|𝑢|^𝑝−2𝑢in𝑀,𝑢∈H¹_𝑔(𝑀),𝑢(−𝑥)=−𝑢(𝑥). (1.5) We evaluate the number of solutions of problem (1.5) using Morse theory. Our main result reads as following.

Theorem 1.1. Assume that for 𝜀 small enough all the solutions to problem (1.5) with energy close to 2𝑚_∞ are nondegenerate. Then there are at least 𝑃₁(𝑀/𝐺) pairs (𝑢,−𝑢) of nontrivial solutions to (1.5) which change sign exactly once, where 𝑚_∞∶=inf_{∫ℝ𝑛|∇𝑢|2+𝑢2=∫ℝ𝑛|𝑢|𝑝ℝ𝑛}(12|∇𝑢|²+12𝑢²−1𝑝|𝑢|^𝑝)𝑑𝑥. (1.6) Here 𝐺={𝐼,−𝐼} and 𝑃₁(𝑀/𝐺) is the Poincaré polynomial 𝑃_𝑡(𝑀/𝑔) when 𝑡=1.

Concerning the assumptions of nondegeneracy of all the critical points with energy close to 2𝑚_∞, we think that it is true “generically” in some sense with respect to (𝜀,𝑔) where 𝜀 is a positive parameter and 𝑔 is a Riemannian metric.

We point out that problem (1.1) has been widely studied when the manifold 𝑀 is replaced by an open bounded and smooth domain in ℝ^𝑁 with Dirichlet or Neumann boundary condition. In particular, it has been studied the effect of the domain topology or the domain geometry on the number of solutions. See, for example, [9–19] for the Dirichlet problem and [20–32] for the Neumann problem,

The paper is organized as follows. In Section 2 we set the problem and we recall some known results; in Section 3 we give the proof of Theorem 1.1; in Section 4 we prove the technical Lemma 4.5, which is crucial for the proof of Theorem 1.1.

2. Setting of the Problem

First of all, we will recall some topological notions which are used in the paper.

Definition 2.1 (Poincaré polynomial). If (𝑋,𝑌) is a couple of the topological spaces, the Poincaré polynomial 𝑃_𝑡(𝑋,𝑌) is defined as the following power series in 𝑡: 𝑃_𝑡(𝑋,𝑌)∶=_𝑘dim𝐻_𝑘(𝑋,𝑌)𝑡^𝑘, (2.1) where 𝐻_𝑘(𝑋,𝑌) is the 𝑘th homology group with coefficients in some fields. Moreover, we set 𝑃_𝑡(𝑋)∶=𝑃_𝑡(𝑋,∅)=_𝑘dim𝐻_𝑘(𝑋)𝑡^𝑘. (2.2) If 𝑋 is a compact manifold, we have that dim𝐻_𝑘(𝑋)<+∞ and in this case 𝑃_𝑡(𝑋) is a polynomial and not a formal series.

Definition 2.2 (Morse index). Let 𝐽 be a 𝐶2-functional on a Banach space 𝑋 and 𝑢∈𝑋 an isolated critical point of 𝐽 with 𝐽(𝑢)=𝑐. If 𝐽^𝑐∶={𝑣∈𝑋∶𝐽(𝑣)≤𝑐} then the (polynomial) Morse index 𝑖_𝑡(𝑢) of 𝑢 is the following series: 𝑖_𝑡(𝑢)∶=_𝑘dim𝐻_𝑘(𝐽^𝑐,𝐽^𝑐⧵{𝑢})𝑡^𝑘, (2.3) where 𝐻_𝑘(𝐽^𝑐,𝐽^𝑐⧵{𝑢}) is the 𝑘th homology group of the couple (𝐽^𝑐,𝐽^𝑐⧵{𝑢}). If 𝑢 is a nondegenerate critical point of 𝐽 then 𝑖_𝑡(𝑢)=𝑡^𝜇(𝑢), where 𝜇(𝑢) is the (numerical) Morse index of 𝑢 and it is given by the dimension of the maximal subspace on which the bilinear form 𝐽^′′(𝑢)[⋅,⋅] is negatively definite.

It is useful to recall the following result (see [33]).

Remark 2.3. Let 𝑋 and 𝑌 be topological spaces. If 𝑓∶𝑋→𝑌 and 𝑔∶𝑌→𝑋 are continuous maps such that 𝑔∘𝑓 is homotopic to the identity map on 𝑋 then 𝑃_𝑡(𝑌)=𝑃_𝑡(𝑋)+𝑍(𝑡), where 𝑍(𝑡) is a polynomial with non negative coefficients.

Now, let us point out that the transformation 𝜏=−𝐼∶𝑀→𝑀 induces a transformation on H¹_𝑔(𝑀). We define the linear operator 𝜏^∗ as follows: 𝜏^∗∶H¹_𝑔(𝑀)→H¹_𝑔(𝑀),𝜏^∗(𝑢(𝑥))∶=−𝑢(−𝑥). (2.4) The operator 𝜏^∗ is selfadjoint with respect to the following scalar product on H¹_𝑔(𝑀), which is equivalent to the usual one: ⟨𝑢,𝑣⟩_𝜀1∶=𝜀^𝑛_𝑀(𝜀²∇_𝑔𝑢∇_𝑔𝑣+𝑢𝑣)𝑑𝜇_𝑔, (2.5) which induces the norm ‖𝑢‖²_𝜀1∶=𝜀^𝑛_𝑀(𝜀²|∇_𝑔𝑢|²+𝑢²)𝑑𝜇_𝑔. (2.6) In particular, we have ‖𝜏^∗𝑢‖_𝜀,𝑝=‖𝑢‖_𝜀,𝑝,‖𝜏^∗𝑢‖_𝜀=‖𝑢‖_𝜀,𝐽_𝜀(𝜏^∗𝑢)=𝐽_𝜀(𝑢). (2.7) Here ‖𝑢‖^𝑝_𝜀,𝑝1∶=𝜀^𝑛_𝑀|𝑢|^𝑝𝑑𝜇_𝑔 (2.8) denotes the norm in L^𝑝(𝑀), which is equivalent to the usual one. Therefore, in virtue of the Palais Principle, the nontrivial solutions of (1.5) are the critical points of the restriction of 𝐽_𝜀 to the 𝜏-invariant Nehari manifold 𝒩^𝜏_𝜀∶={𝑢∈𝒩_𝜀∶𝑢(−𝑥)=−𝑢(𝑥)}=𝒩_𝜀∩𝐻^𝜏_𝑔, (2.9) where 𝐻^𝜏_𝑔∶={𝑢∈H¹_𝑔(𝑀)∶𝑢(−𝑥)=−𝑢(𝑥)}.

In fact, since 𝐽−𝜀(𝜏^∗𝑢)=𝐽_𝜀(𝑢) and 𝜏^∗ is a selfadjoint operator, we have ⟨∇𝐽_𝜀(𝜏^∗𝑢),𝜏^∗𝜑⟩_𝜀=⟨∇𝐽_𝜀(𝑢),𝜑⟩_𝜀∀𝜑∈H¹_𝑔(𝑀) (2.10) and so ∇𝐽_𝜀(𝑢)=𝜏^∗∇𝐽_𝜀(𝜏^∗𝑢)=𝜏^∗∇𝐽_𝜀(𝑢) if (𝜏^∗𝑢)(𝑥)=𝑢(𝑥)=−𝑢(−𝑥).

Let us set 𝑚_𝜀∶=inf_𝒩𝜀𝐽_𝜀,𝑚^𝜏_𝜀∶=inf_𝒩𝜏𝜀𝐽_𝜀 (2.11) and let 𝑚_∞ be as in (1.6).

It is easy to verify that 𝐽_𝜀 satisfies the Palais-Smale condition on 𝒩^𝜏_𝜀. Then, there exists 𝑣_𝜀 minimizer of 𝑚^𝜏_𝜀 and 𝑣_𝜀 is a critical point of 𝐽_𝜀 on H¹_𝑔(𝑀). Thus 𝑣⁺_𝜀 and 𝑣⁻_𝜀 belong to 𝒩_𝜀, then 𝑚^𝜏_𝜀=𝐽_𝜀(𝑣_𝜀)≥2𝑚_𝜀. We recall that lim_𝜀→0𝑚_𝜀=𝑚_∞ as it has been shown in [2, Remark 5.9].

It is well known that there exists a unique positive spherically symmetric (with respect to the origin) function 𝑈∈H¹(ℝ^𝑛) minimizer of 𝑚_∞. Obviously this fact implies that −Δ𝑈+𝑈=𝑈^𝑝−1 in ℝ^𝑛 and for any 𝜀>0 we can define a family of functions 𝑈_𝜀(𝑥)∶=𝑈(𝑥/𝜀) satisfying the following equation −𝜀²Δ𝑈_𝜀+𝑈_𝜀=𝑈_𝜀^𝑝−1 in ℝ^𝑛.

On the tangent bundle of any compact connected Riemannian manifold 𝑀, it is defined the exponential map exp∶𝑇𝑀→𝑀 which is a 𝐶^∞-map. Then for 𝜌 sufficiently small (smaller than the injectivity radius of 𝑀) the manifold 𝑀 possesses a special set of charts given by exp_𝑥∶𝐵(0,𝜌)→𝐵_𝑔(𝑥,𝜌), where 𝑇_𝑥𝑀 is identified with ℝ^𝑛 for 𝑥∈𝑀. Here 𝐵(0,𝜌) denotes the ball in ℝ^𝑛 centered at 0 with radius 𝜌 and 𝐵_𝑔(𝑥,𝜌) denotes the ball in 𝑀 centered at 𝑥 with radius 𝜌 with the distance given by the metric 𝑔. The system of coordinates corresponding to those charts are called normal coordinates.

3. The Main Ingredient of the Proof

Let us sketch the proof of our main result.

Since lim_𝜀→0𝑚^𝜏_𝜀=2𝑚_∞ (see Lemma 4.3), given 𝛿∈(0,𝑚_∞/4) for 𝜀 small enough, we have 0<2(𝑚_∞−𝛿)<𝑚^𝜏_𝜀<2(𝑚_∞+𝛿). Thus 2(𝑚_∞−𝛿) is not a critical value of 𝐽_𝜀 for any 𝜀. Fixed 𝜀, if the number of critical points of 𝐽_𝜀 is finite in 𝐽^2(𝑚∞_𝜀^+𝛿), we can choose 𝛿 such that 2(𝑚_∞+𝛿) is not a critical value of 𝐽_𝜀.

Let 𝒩^𝜏_𝜀/ℤ₂ be the set obtained by identifying antipodal points of the Nehari manifold 𝒩^𝜏_𝜀. It is easy to check that the set 𝒩^𝜏_𝜀/ℤ₂ is homeomorphic to the projective space 𝑃^∞∶=𝜕Σ₁/ℤ₂, which is obtained by identifying antipodal points in un unit sphere 𝜕Σ₁ in the space 𝐻^𝜏_𝑔.

We are looking for pairs of nontrivial critical points (𝑢,−𝑢) if the functional 𝐽_𝜀∶𝐻^𝜏_𝑔→ℝ, that is we are searching critical points for the functional 𝐽_𝜀̃∶𝐻^𝜏_𝑔⧵{0}/ℤ₂→ℝ defined by𝐽_𝜀̃([𝑢])∶=𝐽_𝜀(𝑢)=𝐽_𝜀(−𝑢). We use the same arguments as in [33]. The following relation can be proved as in [33, 34] (see [33,Lemma 5.2]):

𝑃_𝑡(𝐽̃^2(𝑚∞_𝜀^+𝛿),𝐽̃^2(𝑚∞_𝜀^−𝛿))=𝑡𝑃_𝑡(𝐽̃^2(𝑚∞_𝜀^+𝛿)∩𝑁^𝜏_𝜀/ℤ₂). (3.1) By Lemma 4.5 we deduce that Φ𝑀/𝐺−−−→^𝜀𝐽̃^2(𝑚∞_𝜀^+𝛿)∩𝒩^𝜏_𝜀ℤ₂𝛽𝑀−→^𝑑𝐺, (3.2) where 𝛽∘Φ_𝜀 is homotopic to the identity map and 𝑀_𝑑/𝐺 is homotopically equivalent to 𝑀_𝑔. Therefore by Remark 2.3 we get 𝑃_𝑡(𝐽̃^2(𝑚∞_𝜀^+𝛿)∩𝒩^𝜏_𝜀ℤ₂)=𝑃_𝑡(𝑀𝐺)+𝑍(𝑡), (3.3) where 𝑍(𝑡) is a polynomial with nonnegative integer coefficients.

By our assumption we have that for 𝜀 small enough all the critical points 𝑢 such that 𝐽_𝜀̃(𝑢)<2(𝑚_∞+𝛿) are nondegenerate. Moreover the functional 𝐽_𝜀̃ satisfies the Palais-Smale condition. Then by Morse theory and relations (3.1) and (3.3) we get at least 𝑃₁(𝑀/𝐺) pairs (𝑢,−𝑢) of nontrivial solutions for (1.5). By Remark (4.7) these solutions change sign exactly once. That concludes the proof of Theorem 1.1.

Remark 3.1. By [33, Lemma 5.2] we deduce that 𝑃_𝑡(𝐻^𝜏_𝑔⧵{0}ℤ₂,𝐽̃^2(𝑚∞_𝜀^−𝛿))=𝑡𝑃_𝑡(𝒩^𝜏_𝜀ℤ₂). (3.4) Since 𝑃^∞ is homeomorphic to 𝒩^𝜏_𝜀/ℤ₂ we get 𝑃_𝑡(𝒩^𝜏_𝜀/ℤ₂)=𝑃_𝑡(𝑃^∞). Provided the homology is evaluated with 𝒵₂-coefficients (see, e.g., [35, Theorem 7.4]), we have 𝑃₁(𝑃^∞)=+∞. Then, if all the critical points are nondegenerate, we get infinitely many pairs (𝑢,−𝑢) of nontrivial solutions for (1.5).

4. Technical Results

Let 𝜒_𝑟 be a smooth cut-off function such that 𝜒_𝑟𝑟(𝑧)=1if𝑧∈𝐵(0,2),𝜒_𝑟(𝑧)=0if𝑧∈ℝ^𝑁⧵𝐵(0,𝑟),|∇𝜒_𝑟(𝑧)|≤2∀𝑧∈ℝ^𝑁. (4.1) Fixing a point 𝑞∈𝑀 and 𝜀>0, let us define the function 𝑤_𝜀,𝑞 on 𝑀 as𝑤_𝜀,𝑞(𝑥)∶=𝑈_𝜀(exp_𝑞⁻¹(𝑥))𝜒_𝑟(exp_𝑞⁻¹(𝑥))if𝑥∈𝐵_𝑔(𝑞,𝑟)𝑤_𝜀,𝑞(𝑥)∶=0otherwise. (4.2) We choose 𝑟 smaller than the injectivity radius of 𝑀 and such that 𝐵_𝑔(𝑞,𝑟)∩𝐵_𝑔(−𝑞,𝑟)=∅ for any 𝑞∈𝑀. For any 𝜀>0 we can define a positive number 𝑡(𝑤_𝜀,𝑞) such that Φ_𝜀(𝑞)∶=𝑡(𝑤_𝜀,𝑞)𝑤_𝜀,𝑞∈H¹_𝑔(𝑀)∩𝒩_𝜀forany𝑞∈𝑀. (4.3) Namely, 𝑡(𝑤_𝜀,𝑞) verifies 𝑡(𝑤_𝜀,𝑞∫)=[^𝑀(𝜀²|∇_𝑔𝑤_𝜀,𝑞|²+𝑤²_𝜀,𝑞)𝑑𝜇_𝑔∫_𝑀𝑤²_𝜀,𝑞𝑑𝜇_𝑔]^1/𝑝−2. (4.4) In [2, Proposition 4.2] the following lemma has been proved.

Lemma 4.1. Given 𝜀>0 the map Φ_𝜀∶𝑀→H¹_𝑔(𝑀)∩𝒩_𝜀 is continuous. Moreover, given 𝛿>0 there exists 𝜀₀(𝛿) such that if 𝜀∈(0,𝜀₀(𝛿)) then Φ_𝜀(𝑞)∈𝒩_𝜀∩𝐽^𝑚∞_𝜀^+𝛿.

Now, fixing a point 𝑞∈𝑀 let us define the function Φ^𝜏_𝜀(𝑞)∶=𝑡(𝑤_𝜀,𝑞)𝑤_𝜀,𝑞−𝑡(𝑤_{𝜀,𝜏𝑞})𝑤_{𝜀,𝜏𝑞}. (4.5) It holds that _𝑀|𝑤_𝜀,𝑞|²𝑝=_𝑀|𝑤_{𝜀,𝜏𝑞}|²𝑝,_𝑀|∇_𝑔𝑤_𝜀,𝑞|²𝑑𝜇_𝑔=_𝑀|∇_𝑔𝑤_{𝜀,𝜏𝑞}|²𝑑𝜇_𝑔. (4.6) By (4.4) and (4.6), we deduce 𝑡(𝑤_𝜀,𝑞)=𝑡(𝑤_{𝜀,𝜏𝑞}). (4.7) The proof of the next results follows the same arguments as in [8].

Lemma 4.2. Given 𝜀>0 the map Φ^𝜏_𝜀∶𝑀→H¹_𝑔(𝑀)∩𝒩^𝜏_𝜀 is continuous. Moreover, given 𝛿>0 there exists 𝜀₀(𝛿) such that if 𝜀∈(0,𝜀₀(𝛿)) then Φ^𝜏_𝜀(𝑞)∈𝒩^𝜏_𝜀∩𝐽^2(𝑚∞_𝜀^+𝛿).

Proof. Since 𝑈_𝜀𝜒_𝑟 is a radially symmetric function, we set 𝑈_𝜀(|𝑧|)∶=𝑈_𝜀(𝑧)𝜒_𝑟(𝑧). Moreover, since we have |exp⁻¹_𝜏𝑞(𝜏𝑥)|=𝑑_𝑔(−𝑥,−𝑞)=𝑑_𝑔(𝑥,𝑞)=|exp_𝑞⁻¹(𝑥)|,|exp_𝑞⁻¹(𝜏𝑥)|=𝑑_𝑔(−𝑥,𝑞)=𝑑_𝑔(𝑥,−𝑞)=|exp⁻¹_𝜏𝑞(𝑥)|, (4.8) we get 𝜏^∗Φ^𝜏_𝜀(𝑞)(𝑥)=−𝑡(𝑤_𝜀,𝑞)𝑤_𝜀,𝑞(−𝑥)+𝑡(𝑤_{𝜀,𝜏𝑞})𝑤_{𝜀,𝜏𝑞}(−𝑥)=−𝑡(𝑤_𝜀,𝑞)𝑈_𝜀(|exp_𝑞⁻¹(−𝑥)|)+𝑡(𝑤_{𝜀,𝜏𝑞})𝑈_𝜀(|exp_𝑞⁻¹(−𝑥)|)=𝑡(𝑤_{𝜀,𝜏𝑞})𝑈_𝜀(|exp_𝑞⁻¹(𝑥)|)−𝑡(𝑤_𝜀,𝑞)𝑈_𝜀(|exp_𝑞⁻¹(𝜏𝑥)|)=𝑡(𝑤_𝜀,𝑞)𝑈_𝜀(|exp_𝑞⁻¹(𝑥)|)−𝑡(𝑤_𝜀,𝑞)𝑈_𝜀(|exp_𝑞⁻¹(𝑥)|)=Φ^𝜏_𝜀(𝑞)(𝑥), (4.9)(4.10)(4.11)(4.12) because by (4.7) we have 𝑡(𝑤_𝜀,𝑞)=𝑡(𝑤_{𝜀,𝜏𝑞}). Hence Φ^𝜏_𝜀(𝑞)∈𝒩^𝜏_𝜀.
To get that Φ^𝜏_𝜀(𝑞)∈𝐽^2(𝑚∞_𝜀^+𝛿), it is enough to prove that 𝐽_𝜀(Φ^𝜏_𝜀(𝑞))=2𝐽_𝜀(Φ_𝜀(𝑞)), because by Lemma 4.1 the statement will follow. Since the support of the function Φ^𝜏_𝜀(𝑞) is 𝐵_𝑔(𝑞,𝑟)∪𝐵_𝑔(−𝑞,𝑟) and 𝐵_𝑔(𝑞,𝑟)∩𝐵_𝑔(−𝑞,𝑟)=∅, by (4.6) and the definition of the function Φ^𝜏_𝜀, we get 𝐽_𝜀(Φ^𝜏_𝜀1(𝑞))=(2−1𝑝)1𝜀^𝑛_𝑀|Φ^𝜏_𝜀(𝑞)|^𝑝𝑑𝜇_𝑔1=(2−1𝑝)1𝜀^𝑛(_{𝐵𝑔(𝑞,𝑟)}|Φ_𝜀(𝑞)|^𝑝𝑑𝜇_𝑔+_{𝐵𝑔(−𝑞,𝑟)}|Φ_𝜀(𝜏𝑞)|^𝑝𝑑𝜇_𝑔)=2𝐽_𝜀(Φ_𝜀(𝑞)). (4.13) That concludes the proof.

Lemma 4.3. One has that lim_𝜀→0𝑚^𝜏_𝜀=2𝑚_∞.

Proof. By Lemma 4.2 and (4.12) we have that for any 𝛿>0 there exists 𝜀₀(𝛿) such that for any 𝜀∈(0,𝜀₀(𝛿)) it holds that 2𝑚_𝜀≤𝑚^𝜏_𝜀≤𝐽_𝜀(Φ^𝜏_𝜀(𝑞))=2𝐽_𝜀(Φ_𝜀(𝑞))≤2(𝑚_∞+𝛿). (4.14) Since lim_𝜀→0𝑚_𝜀=2𝑚_∞ (see [2,Remark 5.9]) we get the claim.

For any function 𝑢∈𝒩^𝜏_𝜀 we can define a point 𝛽(𝑢)∈ℝ^𝑁 by ∫𝛽(𝑢)∶=^𝑀𝑥|𝑢⁺(𝑥)|^𝑝𝑑𝜇_𝑔∫_𝑀|𝑢⁺(𝑥)|^𝑝𝑑𝜇_𝑔. (4.15)

Lemma 4.4. There exists 𝛿₀>0 such that for any 𝛿∈(0,𝛿₀), for any 𝜀∈(0,𝜀₀(𝛿)) (as in Lemma 4.2), and for any function 𝑢∈𝒩^𝜏_𝜀∩𝐽^2(𝑚∞_𝜀^+𝛿), it holds that 𝛽(𝑢)∈𝑀_𝑑, where 𝑀_𝑑∶={𝑥∈ℝ^𝑁∶𝑑(𝑥,𝑀)<𝑑}.

Proof. Let 𝑢∈𝒩^𝜏_𝜀∩𝐽^2(𝑚∞_𝜀^+𝛿). Since 𝑢(𝑥)=−𝑢(−𝑥) we set 𝑀⁺∶={𝑥∈𝑀∶𝑢(𝑥)>0} and 𝑀⁻∶={𝑥∈𝑀∶𝑢(𝑥)<0}. It is easy to see that 𝑀⁺={−𝑥∶𝑥∈𝑀⁻}. Then we have 𝐽_𝜀1(𝑢)=(2−1𝑝)1𝜀^𝑛_𝑀|𝑢|^𝑝𝑑𝜇_𝑔1=(2−1𝑝)1𝜀^𝑛(_𝑀+|𝑢⁺|^𝑝𝑑𝜇_𝑔+_𝑀−|𝑢⁻|^𝑝𝑑𝜇_𝑔)=2𝐽_𝜀(𝑢⁺). (4.16) Since 𝐽_𝜀(𝑢)≤2(𝑚_∞+𝛿), we have 𝐽_𝜀(𝑢⁺)≤𝑚_∞+𝛿 and by [2, Proposition 5.10] we get the claim.

It is easy to check that Φ^𝜏_𝜀(−𝑞)=−𝜙^𝜏_𝜀(𝑞) and 𝛽(−𝑢)=−𝛽(𝑢). Moreover, by Lemmas 4.1 and 4.2, we can define a map Φ_𝜀∶𝑀/𝐺→𝐽̃^2(𝑚∞_𝜀^+𝛿)∩𝒩^𝜏_𝜀/ℤ₂ by Φ_𝜀([𝑞])∶=[Φ^𝜏_𝜀(𝑞)]={Φ^𝜏_𝜀(𝑞),Φ^𝜏_𝜀(−𝑞)}. (4.17) By Lemma 4.4 we can define a map 𝛽∶𝐽̃^2(𝑚∞_𝜀^+𝛿)∩𝒩^𝜏_𝜀/ℤ₂→𝑀_𝑑/𝐺 by 𝛽([𝑢])∶=[𝛽(𝑢)]={𝛽(𝑢),𝛽(−𝑢)}. (4.18)

Lemma 4.5. There exists 𝜀₀>0 such that for any 𝜀∈(0,𝜀₀) the map 𝐼_𝜀∶=𝛽∘Φ^𝜏_𝜀∶𝑀𝐺→𝑀_𝑑𝐺 (4.19) is well defined, continuous, and homotopic to the identity map.

Proof. By Lemmas 4.2 and 4.4, 𝐼_𝜀 is well defined. In order to show that 𝐼_𝜀 is homotopic to the identity, we estimate the following difference: |𝛽Φ^𝜏_𝜀∫(𝑞)−𝑞|=^𝑀(𝑥−𝑞)|(Φ^𝜏_𝜀(𝑞))⁺|^𝑝𝑑𝜇_𝑔∫_𝑀|(Φ^𝜏_𝜀(𝑞))⁺|^𝑝𝑑𝜇_𝑔=∫_𝐵(0,𝑟)𝑦|𝑈(𝑦/𝜀)𝜒_𝑟(|𝑦|)|^𝑝|𝑔_𝑞(𝑦)|^1/2𝑑𝑦∫_𝐵(0,𝑟)|𝑈(𝑦/𝜀)𝜒_𝑟(|𝑦|)|^𝑝|𝑔_𝑞(𝑦)|^1/2=𝜀∫𝑑𝑦_{𝐵(0,𝑟/𝜀)}𝑧|𝑈(𝑧)𝜒_𝑟(|𝜀𝑧|)|^𝑝|𝑔_𝑞(𝜀𝑧)|^1/2𝑑𝜇_𝑔∫_{𝐵(0,𝑟/𝜀)}|𝑈(𝑧)𝜒_𝑟(|𝜀𝑧|)|^𝑝|𝑔_𝑞(𝜀𝑧)|^1/2𝑑𝜇_𝑔. (4.20) Hence |𝛽Φ^𝜏_𝜀(𝑞)−𝑞|,|𝛽Φ^𝜏_𝜀(−𝑞)+𝑞|≤𝑐𝜀, because 𝛽Φ^𝜏_𝜀(−𝑞)=−𝛽Φ^𝜏_𝜀(𝑞), for a constant 𝑐 which does not depend on the point 𝑞. Therefore |𝐼_𝜀(𝑞)−𝑞|<𝑐𝜀; that concludes the proof.

Remark 4.6. We have only to prove that any solution 𝑢 of (1.5) such that 𝐽_𝜖(𝑢)<2(𝑚_∞+𝛿) changes sign exactly once. In fact, assume that the set {𝑢∈𝑀∶𝑢(𝑥)>0} has ℎ connected components 𝑀₁,…,𝑀_ℎ. Set 𝑢_𝑖(𝑥)∶=𝑢(𝑥) if 𝑥∈𝑀_𝑖∪(−𝑀_𝑖) and 𝑢_𝑖(𝑥)∶=0 otherwise. We have 𝑢_𝑖∈𝒩^𝜏_𝜀 and 32ℎ𝑚_∞≤𝑚^𝜏_𝜀≤𝐽_𝜀(𝑢)=^ℎ_𝑖=1𝐽_𝜀(𝑢_𝑖)≤2(𝑚_∞+𝛿)<3𝑚_∞. (4.21) Then ℎ=1. This concludes the proof.