Show Summary Details
More options …

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …
Volume 8, Issue 1

# Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four

Jérôme Vétois
• Corresponding author
• Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
• Email
• Other articles by this author:
/ Shaodong Wang
• Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
• Email
• Other articles by this author:
Published Online: 2017-08-18 | DOI: https://doi.org/10.1515/anona-2017-0085

## Abstract

We extend Chen, Wei and Yan’s constructions of families of solutions with unbounded energies [5] to the case of cubic nonlinear Schrödinger equations in the optimal dimension four.

MSC 2010: 35J61

## 1 Introduction and main results

In this note, we consider the cubic nonlinear Schrödinger equation

(1.1)

where $\left(M,g\right)$ is a Riemannian manifold of dimension 4, ${\mathrm{\Delta }}_{g}:=-{\mathrm{div}}_{g}\nabla$ is the Laplace–Beltrami operator and $f\in {C}^{0,\alpha }\left(M\right)$, $\alpha \in \left(0,1\right)$.

For $\left(M,g\right)=\left({𝕊}^{4},{g}_{0}\right)$, where ${g}_{0}$ is the standard metric on the sphere ${𝕊}^{4}$, we obtain the following result.

#### Theorem 1.1.

Assume that $\mathrm{\left(}M\mathrm{,}g\mathrm{\right)}\mathrm{=}\mathrm{\left(}{\mathrm{S}}^{\mathrm{4}}\mathrm{,}{g}_{\mathrm{0}}\mathrm{\right)}$ and $f\mathrm{>}\mathrm{2}$ is constant. Then there exists a family of positive solutions ${\mathrm{\left(}{u}_{\epsilon }\mathrm{\right)}}_{\epsilon \mathrm{>}\mathrm{0}}$ to (1.1) such that ${\mathrm{\parallel }\mathrm{\nabla }\mathit{}{u}_{\epsilon }\mathrm{\parallel }}_{{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{S}}^{\mathrm{4}}\mathrm{\right)}}\mathrm{\to }\mathrm{\infty }$ as $\epsilon \mathrm{\to }\mathrm{0}$.

Theorem 1.1 extends a result obtained by Chen, Wei and Yan [5], in dimensions $n\ge 5$, for positive solutions of the equation

(1.2)

where ${2}^{*}:=\frac{2n}{n-2}$. The dimension four is optimal for this result since Li and Zhu [11] obtained the existence of a priori bounds on the energy of positive solutions to (1.2) in dimension three.

It is also interesting to mention that in the case where $n\notin \left\{3,6\right\}$ and $f>\frac{n\left(n-2\right)}{4}$ on ${𝕊}^{n}$ (or, more generally, $f>\frac{n-2}{4\left(n-1\right)}{\mathrm{Scal}}_{g}$ on a general closed manifold, where ${\mathrm{Scal}}_{g}$ is the scalar curvature), Druet [6] obtained a compactness result for families of positive solutions ${\left({u}_{\epsilon }\right)}_{\epsilon >0}$ of (1.2) with bounded energies, i.e., such that ${\parallel \nabla {u}_{\epsilon }\parallel }_{{L}^{2}\left(M\right)} for some constant C independent of ε. Theorem 1.1, together with the result of Chen, Wei and Yan [5] in dimensions $n\ge 5$, shows that the energy assumption in Druet’s result is necessary at least in the case of the standard sphere.

In the case where $f\equiv \frac{n\left(n-2\right)}{4}$ and $\left(M,g\right)=\left({𝕊}^{n},{g}_{0}\right)$, the positive solutions of (1.2) have been classified by Obata [12] (see also [4]). In this case, the solutions are not bounded in ${L}^{\mathrm{\infty }}\left({𝕊}^{n}\right)$ but they all have the same energy. We refer to [2, 3, 10] and the references therein for results on the set of solutions of (1.2) for $f\equiv \frac{n-2}{4\left(n-1\right)}{\mathrm{Scal}}_{g}$ and $\left(M,g\right)\ne \left({𝕊}^{n},{g}_{0}\right)$. On the other hand, for $f<\frac{n-2}{4\left(n-1\right)}{\mathrm{Scal}}_{g}$ on a general closed manifold, Druet [7] obtained pointwise a priori bounds on the set of positive solutions of (1.2). Note that if, moreover, $0 is constant, then $u\equiv {f}^{\left(n-2\right)/4}$ is the unique positive solution of (1.2), see [1]. We refer to the books of Druet, Hebey and Robert [8], and Hebey [9] for more results on equations of type (1.1) on a closed manifold.

As in [5], we obtain Theorem 1.1 by proving a more general result for the case $\left(M,g\right)=\left({ℝ}^{4},{\delta }_{0}\right)$, where ${\delta }_{0}$ is the Euclidean metric on ${ℝ}^{4}$. We let ${D}^{1,2}\left({ℝ}^{4}\right)$ be the completion of the set of smooth functions with compact support in ${ℝ}^{4}$ with respect to the norm ${\parallel u\parallel }_{{D}^{1,2}\left({ℝ}^{4}\right)}={\parallel \nabla u\parallel }_{{L}^{2}\left({ℝ}^{4}\right)}$. For simplicity, we will use the notation $\mathrm{\Delta }:={\mathrm{\Delta }}_{{\delta }_{0}}$, $〈\cdot ,\cdot 〉:={〈\cdot ,\cdot 〉}_{{\delta }_{0}}$ and $|\cdot |:=|\cdot {|}_{{\delta }_{0}}$. We say that the operator $\mathrm{\Delta }+f$ is coercive in ${D}^{1,2}\left({ℝ}^{4}\right)$ if

for some constant $C>0$. We have the following result.

#### Theorem 1.2.

Assume that $\mathrm{\left(}M\mathrm{,}g\mathrm{\right)}\mathrm{=}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{,}{\delta }_{\mathrm{0}}\mathrm{\right)}$ and that $f\mathrm{\in }{C}^{\mathrm{0}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$ is radially symmetric about the point 0. Assume, moreover, that the operator $\mathrm{\Delta }\mathrm{+}f$ is coercive in ${D}^{\mathrm{1}\mathrm{,}\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$ and the function $r\mathrm{↦}{r}^{\mathrm{2}}\mathit{}f\mathit{}\mathrm{\left(}r\mathrm{\right)}$ has a strict local maximum point ${r}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that $f\mathit{}\mathrm{\left(}{r}_{\mathrm{0}}\mathrm{\right)}\mathrm{>}\mathrm{0}$. Then there exists a family of positive solutions ${\mathrm{\left(}{u}_{\epsilon }\mathrm{\right)}}_{\epsilon \mathrm{>}\mathrm{0}}$ in ${C}^{\mathrm{2}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}\mathrm{\cap }{D}^{\mathrm{1}\mathrm{,}\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$ of (1.1) such that ${\mathrm{\parallel }\mathrm{\nabla }\mathit{}{u}_{\epsilon }\mathrm{\parallel }}_{{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}}\mathrm{\to }\mathrm{\infty }$ as $\epsilon \mathrm{\to }\mathrm{0}$.

The proof of Theorem 1.2 relies on a Lyapunov–Schmidt-type method, as in [5]. This method for constructing solutions with infinitely many peaks was invented and successfully used in previous works by Wang, Wei and Yan [13, 14] and Wei and Yan [15, 16, 17, 18]. A specificity in our case is that the number of peaks in the construction behaves as a logarithm of the peak’s height, while it behaves as a power of the peak’s height in the higher dimensional case (see [5]). Due to this logarithm behavior, we need to introduce some suitable changes of variables in order to find the critical points of the reduced energy in this case (see the proof of Theorem 1.2 at the end of Section 2).

## 2 Proof of Theorems 1.1 and 1.2

This section is devoted to the proof of Theorems 1.1 and 1.2. For any integer $k\ge 1$, we let ${H}_{k}$ be the set of all functions $u\in {D}^{1,2}\left({ℝ}^{4}\right)$ such that u is even in ${x}_{2},{x}_{3},{x}_{4}$ and

$u\left(r\mathrm{cos}\left(\theta \right),r\mathrm{sin}\left(\theta \right),{x}_{3},{x}_{4}\right)=u\left(r\mathrm{cos}\left(\theta +2\pi /k\right),r\mathrm{sin}\left(\theta +2\pi /k\right),{x}_{3},{x}_{4}\right)$

for all $r>0$ and $\theta ,{x}_{3},{x}_{4}\in ℝ$. Assuming that the operator $\mathrm{\Delta }+f$ is coercive in ${D}^{1,2}\left({ℝ}^{4}\right)$, we can equip ${H}_{k}$ with the inner product

and the norm

For any $k\ge 1$ and $r,\mu >0$, we define

with

${x}_{i,k,r}:=\left(r\mathrm{cos}\left(2\left(i-1\right)\pi /k\right),r\mathrm{sin}\left(2\left(i-1\right)\pi /k\right),0,0\right).$

Moreover, we define

where

${Z}_{i,1,k,r,\mu }:=\frac{1}{\mu }\frac{d}{dr}\left[{U}_{i,k,r,\mu }\right]\mathit{ }\text{and}\mathit{ }{Z}_{i,2,k,r,\mu }:=\mu \frac{d}{d\mu }\left[{U}_{i,k,r,\mu }\right].$

First, in Proposition 2.1 below, we solve the equation

${Q}_{k,r,\mu }\left({W}_{k,r,\mu }+\varphi -{\left(\mathrm{\Delta }+f\right)}^{-1}\left({\left({W}_{k,r,\mu }+\varphi \right)}_{+}^{3}\right)\right)=0,$(2.1)

where $\varphi \in {P}_{k,r,\mu }$ is the unknown function, ${Q}_{k,r,\mu }$ is the orthogonal projection of ${H}_{k}$ onto ${P}_{k,r,\mu }$ and, as usual, ${u}_{+}:=\mathrm{max}\left(u,0\right)$ for all $u:{ℝ}^{4}\to ℝ$.

We will prove the following result in Section 3.

#### Proposition 2.1.

Let $f\mathrm{\in }{C}^{\mathrm{0}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$ be a radially symmetric function about the point 0 and such that the operator $\mathrm{\Delta }\mathrm{+}f$ is coercive in ${D}^{\mathrm{1}\mathrm{,}\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$. Then, for any $a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{>}\mathrm{0}$, with $a\mathrm{<}b$ and $c\mathrm{<}d$, there exist constants ${k}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ and ${C}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that for any $k\mathrm{\ge }{k}_{\mathrm{0}}$, $r\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}$ and $\mu \mathrm{\in }\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$, there exists a unique solution ${\varphi }_{k\mathrm{,}r\mathrm{,}\mu }\mathrm{\in }{P}_{k\mathrm{,}r\mathrm{,}\mu }$ of (2.1) satisfying

${\parallel {\varphi }_{k,r,\mu }\parallel }_{{H}_{k}}\le {C}_{0}k/\mu .$(2.2)

Moreover, the map $\mathrm{\left(}r\mathrm{,}\mu \mathrm{\right)}\mathrm{↦}{\varphi }_{k\mathrm{,}r\mathrm{,}\mu }$ is continuously differentiable and if there exists a critical point $\mathrm{\left(}{r}_{k}\mathrm{,}{\mu }_{k}\mathrm{\right)}\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}\mathrm{×}\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$ of the function

$\left(r,\mu \right)↦{\mathcal{ℐ}}_{k}\left(r,\mu \right):=I\left({W}_{k,r,\mu }+{\varphi }_{k,r,\mu }\right),$

where

$I\left(u\right):=\frac{1}{2}{\int }_{{ℝ}^{4}}\left(|\nabla u|+f{u}^{2}\right)𝑑x-\frac{1}{4}{\int }_{{ℝ}^{4}}{u}_{+}^{4}𝑑x,$

then the function ${W}_{k\mathrm{,}{r}_{k}\mathrm{,}{\mu }_{k}}\mathrm{+}{\varphi }_{k\mathrm{,}{r}_{k}\mathrm{,}{\mu }_{k}}$ is a positive solution in ${C}^{\mathrm{2}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}\mathrm{\cap }{H}_{k}$ of the equation

(2.3)

Then we will prove the following result in Section 4.

#### Proposition 2.2.

Let $f\mathrm{\in }{C}^{\mathrm{0}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$ be a radially symmetric function about the point 0 and such that the operator $\mathrm{\Delta }\mathrm{+}f$ is coercive in ${D}^{\mathrm{1}\mathrm{,}\mathrm{2}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{4}}\mathrm{\right)}$. Then there exist constants ${c}_{\mathrm{0}}\mathrm{,}{c}_{\mathrm{1}}\mathrm{,}{c}_{\mathrm{2}}\mathrm{>}\mathrm{0}$ such that for any $a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{>}\mathrm{0}$, with $a\mathrm{<}b$ and $c\mathrm{<}d$, we have

(2.4)

uniformly for $r\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}$ and $\mu \mathrm{\in }\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$, where ${\varphi }_{k\mathrm{,}r\mathrm{,}\mu }$ is as in Proposition 2.1.

Now, we prove Theorem 1.2, by using Propositions 2.1 and 2.2.

#### Proof of Theorem 1.2.

Since $f\left({r}_{0}\right)>0$ and ${r}_{0}$ is a strict local maximum point of the function $r↦{r}^{2}f\left(r\right)$, we obtain that there exists ${\delta }_{0}>0$ such that

(2.5)

For any $k\ge 1$ and $s>0$, we define ${\mu }_{k}\left(s\right):={e}^{s{k}^{2}}$. By applying Proposition 2.2, we obtain

(2.6)

uniformly for $\left(r,s\right)$ in compact subsets of ${\left(0,\mathrm{\infty }\right)}^{2}$. Note that the function

$s↦{e}^{-2s{k}^{2}}\left({c}_{1}f\left(r\right)s-\frac{{c}_{2}}{{r}^{2}}\right)$

attains its maximal value at the point

${s}_{k}\left(r\right):=\frac{{c}_{2}}{{c}_{1}f\left(r\right){r}^{2}}+\frac{1}{2{k}^{2}}$

for all $k\ge 1$ and $r\in \left[{r}_{0}-{\delta }_{0},{r}_{0}+{\delta }_{0}\right]$. We define

${\mathcal{𝒥}}_{k}\left(r,t\right):={\mathcal{ℐ}}_{k}\left(r,{\mu }_{k}\left({s}_{k}\left(r\right)+t\right)\right).$

By using (2.5), we obtain that there exists ${t}_{0}>0$ such that

${t}_{0}<\mathrm{min}\left(\frac{{s}_{k}\left({r}_{0}\right)}{2},\frac{2}{3}\left({s}_{k}\left({r}_{0}+{\delta }_{0}\right)-{s}_{k}\left({r}_{0}\right)\right),\frac{2}{3}\left({s}_{k}\left({r}_{0}-{\delta }_{0}\right)-{s}_{k}\left({r}_{0}\right)\right)\right)$(2.7)

for all $k\ge 1$. Since ${t}_{0}<{s}_{k}\left({r}_{0}\right)/2$, from (2.6) it follows that

(2.8)

uniformly for $\left(r,t\right)\in \left[{r}_{0}-{\delta }_{0},{r}_{0}+{\delta }_{0}\right]×\left[-{t}_{0},{t}_{0}\right]$. Since ${s}_{k}\left(r\right)>{s}_{k}\left({r}_{0}\right)$ and $f\left(r\right)>0$, from (2.8) it follows that

${\mathcal{𝒥}}_{k}\left(r,{t}_{0}\right)<{\mathcal{𝒥}}_{k}\left({r}_{0},{t}_{0}/2\right)$(2.9)

and

${\mathcal{𝒥}}_{k}\left(r,-{t}_{0}\right)<{\mathcal{𝒥}}_{k}\left({r}_{0},{t}_{0}/2\right)$(2.10)

as $k\to \mathrm{\infty }$, uniformly for $r\in \left[{r}_{0}-{\delta }_{0},{r}_{0}+{\delta }_{0}\right]$. Moreover, by using (2.7) and (2.8), we obtain

${\mathcal{𝒥}}_{k}\left({r}_{0}±{\delta }_{0},t\right)<{\mathcal{𝒥}}_{k}\left({r}_{0},{t}_{0}/2\right)$(2.11)

as $k\to \mathrm{\infty }$, uniformly for $t\in \left[-{t}_{0},{t}_{0}\right]$. From (2.9)–(2.11) it follows that the function ${\mathcal{𝒥}}_{k}$ has a local maximum point $\left({r}_{k},{t}_{k}\right)\in \left[{r}_{0}-{\delta }_{0},{r}_{0}+{\delta }_{0}\right]×\left[-{t}_{0},{t}_{0}\right]$ for large k. We then obtain $\nabla {\mathcal{ℐ}}_{k}\left({r}_{k},{\mu }_{k}\left({s}_{k}\left({r}_{k}\right)+{t}_{k}\right)\right)=0$, and so, by applying the second part of Proposition 2.1, we obtain that the function ${W}_{k,{r}_{k},{\mu }_{k}\left({s}_{k}\left({r}_{k}\right)+{t}_{k}\right)}+{\varphi }_{k,{r}_{k},{\mu }_{k}\left({s}_{k}\left({r}_{k}\right)+{t}_{k}\right)}$ is a positive solution of equation (2.3). Moreover, by using (2.2) together with the definition of ${W}_{k,{r}_{k},{\mu }_{k}\left({s}_{k}\left({r}_{k}\right)+{t}_{k}\right)}$, we easily obtain

This ends the proof of Theorem 1.2. ∎

Finally, we prove Theorem 1.1, by using Theorem 1.2.

#### Proof of Theorem 1.1.

By using a stereographic projection, we can see that equation (1.1) on $\left(M,g\right)=\left({𝕊}^{4},{g}_{0}\right)$ is equivalent to the problem

(2.12)

It is easy to check that if $f>2$ is a constant, then the potential function in (2.12) satisfies the assumptions of Theorem 1.2. With this remark, Theorem 1.1 becomes a direct corollary of Theorem 1.2. ∎

## 3 Proof of Proposition 2.1

In this section we prove Proposition 2.1. Throughout this section, we assume that $f\in {C}^{0,\alpha }\left({ℝ}^{4}\right)\cap {L}^{2}\left({ℝ}^{4}\right)$ is radially symmetric about the point 0, and that the operator $\mathrm{\Delta }+f$ is coercive in ${D}^{1,2}\left({ℝ}^{4}\right)$.

We rewrite (2.1) as

${L}_{k,r,\mu }\left(\varphi \right)={Q}_{k,r,\mu }\left({N}_{k,r,\mu }\left(\varphi \right)+{R}_{k,r,\mu }\right),$

where

${L}_{k,r,\mu }\left(\varphi \right):={Q}_{k,r,\mu }\left(\varphi -{\left(\mathrm{\Delta }+f\right)}^{-1}\left(3{W}_{k,r,\mu }^{2}\varphi \right)\right),$${N}_{k,r,\mu }\left(\varphi \right):={\left(\mathrm{\Delta }+f\right)}^{-1}\left({\left({W}_{k,r,\mu }+\varphi \right)}_{+}^{3}-{W}_{k,r,\mu }^{3}-3{W}_{k,r,\mu }^{2}\varphi \right),$${R}_{k,r,\mu }:={\left(\mathrm{\Delta }+f\right)}^{-1}\left({W}_{k,r,\mu }^{3}\right)-{W}_{k,r,\mu }.$

First, we obtain the following result.

#### Lemma 3.1.

For any $a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{>}\mathrm{0}$, with $a\mathrm{<}b$ and $c\mathrm{<}d$, there exist constants ${k}_{\mathrm{1}}\mathrm{>}\mathrm{0}$ and ${C}_{\mathrm{1}}\mathrm{>}\mathrm{0}$ such that for any $k\mathrm{\ge }{k}_{\mathrm{1}}$, $r\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}$ and $\mu \mathrm{\in }\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$, ${L}_{k\mathrm{,}r\mathrm{,}\mu }$ is an isomorphism from ${P}_{k\mathrm{,}r\mathrm{,}\mu }$ to itself and

#### Proof.

The proof of this result follows the same lines as in [5]. ∎

We then estimate the error term ${R}_{k,r,\mu }$. We obtain the following result.

#### Lemma 3.2.

For any $a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{>}\mathrm{0}$, with $a\mathrm{<}b$ and $c\mathrm{<}d$, there exist constants ${k}_{\mathrm{2}}\mathrm{>}\mathrm{0}$ and ${C}_{\mathrm{2}}\mathrm{>}\mathrm{0}$ such that

${\parallel {R}_{k,r,\mu }\parallel }_{{H}_{k}}\le {C}_{2}k/\mu$(3.1)

for all $k\mathrm{\ge }{k}_{\mathrm{2}}$, $r\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}$, and $\mu \mathrm{\in }\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$.

#### Proof.

For any $\varphi \in {H}_{k}$, by integrating by parts, we obtain

${〈{R}_{k,r,\mu },\varphi 〉}_{{H}_{k}}={\int }_{{ℝ}^{4}}\left({W}_{k,r,\mu }^{3}-\mathrm{\Delta }{W}_{k,r,\mu }-f{W}_{k,r,\mu }\right)\varphi 𝑑x$$={\int }_{{ℝ}^{4}}\left({W}_{k,r,\mu }^{3}-\sum _{i=1}^{k}{U}_{i,k,r,\mu }^{3}-f{W}_{k,r,\mu }\right)\varphi 𝑑x$$=\sum _{i=1}^{k}{\int }_{{ℝ}^{4}}\left(3\sum _{j\ne i}{U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }-f{U}_{i,k,r,\mu }\right)\varphi 𝑑x.$(3.2)

By using Hölder’s inequality and Sobolev’s inequality, from (3.2) it follows that

$\parallel {R}_{k,r,\mu }{\parallel }_{{H}_{k}}=\sum _{i=1}^{k}\mathrm{O}\left(\sum {}_{j\ne i}\parallel {U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }{\parallel }_{{L}^{4/3}}+\parallel f{U}_{i,k,r,\mu }{\parallel }_{{L}^{4/3}}\right).$(3.3)

We start with estimating the first term in (3.3). For any $\alpha \in \left\{1,\mathrm{\dots },k\right\}$, we define

${\mathrm{\Omega }}_{\alpha ,k,r}:=\left\{\left({y}_{1},{y}_{2},{y}_{3},{y}_{4}\right)\in {ℝ}^{4}:〈\left({y}_{1},{y}_{2},0,0\right),{x}_{\alpha ,k,r}〉\ge \mathrm{cos}\left(\pi /k\right)\right\}.$

We then write

${\int }_{{ℝ}^{4}}{U}_{i,k,r,\mu }^{8/3}{U}_{j,k,r,\mu }^{4/3}𝑑x=\sum _{\alpha =1}^{k}{\int }_{{\mathrm{\Omega }}_{\alpha ,k,r}}{U}_{i,k,r,\mu }^{8/3}{U}_{j,k,r,\mu }^{4/3}𝑑x.$(3.4)

We observe that if $\alpha \ne j$, then

$|x-{x}_{j,k,r}|\ge |x-{x}_{\alpha ,k,r}|\mathit{ }\text{and}\mathit{ }|x-{x}_{j,k,r}|\ge \frac{1}{2}|{x}_{\alpha ,k,r}-{x}_{j,k,r}|$(3.5)

for all $x\in {\mathrm{\Omega }}_{\alpha ,k,r}$. For any $i,j,\alpha \in \left\{1,\mathrm{\dots },k\right\}$, with $i\ne j$, by using (3.4), we obtain

(3.6)

for all $x\in {\mathrm{\Omega }}_{\alpha ,k,r}\setminus \left\{{x}_{\alpha ,k,r}\right\}$. By using (3.4), (3.6) and straightforward estimates, we obtain

${\int }_{{ℝ}^{4}}{U}_{i,k,r,\mu }^{8/3}{U}_{j,k,r,\mu }^{4/3}𝑑x=\mathrm{O}\left(\frac{{\mu }^{-8/3}}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{8/3}}+\sum _{\alpha \ne i}\frac{{\mu }^{-8/3}}{{|{x}_{i,k,r}-{x}_{\alpha ,k,r}|}^{8/3}}\right)$$=\mathrm{O}\left(\frac{{\mu }^{-8/3}}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{8/3}}+\frac{{k}^{8/3}}{{\mu }^{8/3}}\right).$(3.7)

From (3.7) it follows that

$\sum _{j\ne i}\parallel {U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }{\parallel }_{{L}^{4/3}}=\mathrm{O}\left(k{\left(k/\mu \right)}^{2}\right).$(3.8)

Now, we estimate the second term in (3.4). Since $f\in {L}^{\mathrm{\infty }}\left({ℝ}^{4}\right)\cap {L}^{2}\left({ℝ}^{4}\right)$, by applying Hölder’s inequality and straightforward estimates, we obtain

${\int }_{{ℝ}^{4}\setminus B\left({x}_{i,k,r},1\right)}|f{U}_{i,k,r,\mu }{|}^{4/3}dx=\mathrm{O}\left({\left({\int }_{{ℝ}^{4}\setminus B\left({x}_{i,k,r},1\right)}|{U}_{i,k,r,\mu }{|}^{4}dx\right)}^{1/3}\right)=\mathrm{O}\left({\mu }^{-4/3}\right)$(3.9)

and

${\int }_{B\left({x}_{i,k,r},1\right)}|f{U}_{i,k,r,\mu }{|}^{4/3}dx=\mathrm{O}\left({\int }_{B\left({x}_{i,k,r},1\right)}|{U}_{i,k,r,\mu }{|}^{4/3}dx\right)=\mathrm{O}\left({\mu }^{-4/3}\right).$(3.10)

From (3.9) and (3.10), it follows that

${\parallel f{U}_{i,k,r,\mu }\parallel }_{{L}^{4/3}}=\mathrm{O}\left(1/\mu \right).$(3.11)

Finally, (3.1) follows from (3.8) and (3.11). ∎

We can now prove Proposition 2.1 by using Lemmas 3.1 and 3.2.

#### Proof of Proposition 2.1.

We define

and

${V}_{k,r,\mu }:=\left\{\varphi \in {P}_{k,r,\mu }:{\parallel \varphi \parallel }_{{H}_{k}}\le {C}_{0}k/\mu \right\},$

where ${C}_{0}>0$ is a constant to be fixed later on. From Lemmas 3.1 and 3.2, it follows that

${\parallel {T}_{k,r,\mu }\left(\varphi \right)\parallel }_{{H}_{k}}\le {C}_{1}\left({\parallel {N}_{k,r,\mu }\left(\varphi \right)\parallel }_{{H}_{k}}+{C}_{2}k/\mu \right)$(3.12)

for all $k\ge {k}_{2}$, $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$. By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and straightforward estimates, we obtain

${〈{N}_{k,r,\mu }\left(\varphi \right),\psi 〉}_{{H}_{k}}={\int }_{{ℝ}^{4}}\left({\left({W}_{k,r,\mu }+\varphi \right)}_{+}^{3}-{W}_{k,r,\mu }^{3}-3{W}_{k,r,\mu }^{2}\varphi \right)\psi 𝑑x$$=\mathrm{O}\left(\left({\parallel {W}_{k,r,\mu }\parallel }_{{L}^{4}}{\parallel \varphi \parallel }_{{H}_{k}}^{2}+{\parallel \varphi \parallel }_{{H}_{k}}^{3}\right){\parallel \psi \parallel }_{{H}_{k}}\right)$(3.13)

for all $\psi \in {H}_{k}$. Proceeding as in (3.4)–(3.8), we obtain

${\int }_{{ℝ}^{4}}{W}_{k,r,\mu }^{4}dx=\mathrm{O}\left(\sum _{i=1}^{k}{\int }_{{ℝ}^{4}}\left({U}_{i,k,r,\mu }^{4}+\sum _{j\ne i}{U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }^{2}\right)dx\right)$$=\mathrm{O}\left(k+k{\left(k/\mu \right)}^{4}\mathrm{ln}\mu \right).$(3.14)

From (3.13) and (3.14), it follows that

${\parallel {N}_{k,r,\mu }\left(\varphi \right)\parallel }_{{H}_{k}}=\mathrm{O}\left({k}^{1/4}{\parallel \varphi \parallel }_{{H}_{k}}^{2}+{\parallel \varphi \parallel }_{{H}_{k}}^{3}\right).$(3.15)

Letting ${C}_{0}$ be large enough so that ${C}_{0}>{C}_{1}{C}_{2}$, from (3.12) and (3.15), it follows that there exists a constant ${k}_{3}>0$ such that

${T}_{k,r,\mu }\left({V}_{k,r,\mu }\right)\subset {V}_{k,r,\mu }$(3.16)

for all $k\ge {k}_{3}$, $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$. Now, we prove that if k is large enough, then ${T}_{k,r,\mu }$ is a contraction map from ${V}_{k,r,\mu }$ to itself, i.e.,

(3.17)

for some constant $C\in \left(0,1\right)$. From Lemma 3.1 it follows that

${\parallel {T}_{k,r,\mu }\left({\varphi }_{1}\right)-{T}_{k,r,\mu }\left({\varphi }_{2}\right)\parallel }_{{H}_{k}}\le {C}_{1}{\parallel {N}_{k,r,\mu }\left({\varphi }_{1}\right)-{N}_{k,r,\mu }\left({\varphi }_{2}\right)\parallel }_{{H}_{k}}.$(3.18)

By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and (3.14), we obtain

${〈{N}_{k,r,\mu }\left({\varphi }_{1}\right)-{N}_{k,r,\mu }\left({\varphi }_{2}\right),\psi 〉}_{{H}_{k}}={\int }_{{ℝ}^{4}}\left({\left({W}_{k,r,\mu }+{\varphi }_{1}\right)}_{+}^{3}-{\left({W}_{k,r,\mu }+{\varphi }_{2}\right)}_{+}^{3}-3{W}_{k,r,\mu }^{2}\left({\varphi }_{1}-{\varphi }_{2}\right)\right)\psi 𝑑x$$=\mathrm{O}\left(\left({\parallel {W}_{k,r,\mu }\parallel }_{{L}^{4}}+{\parallel {\varphi }_{1}\parallel }_{{H}_{k}}+{\parallel {\varphi }_{2}\parallel }_{{H}_{k}}\right)\left({\parallel {\varphi }_{1}\parallel }_{{H}_{k}}+{\parallel {\varphi }_{2}\parallel }_{{H}_{k}}\right){\parallel {\varphi }_{1}-{\varphi }_{2}\parallel }_{{H}_{k}}{\parallel \psi \parallel }_{{H}_{k}}\right)$$=\mathrm{O}\left(\left({k}^{1/4}+{\parallel {\varphi }_{1}\parallel }_{{H}_{k}}+{\parallel {\varphi }_{2}\parallel }_{{H}_{k}}\right)\left({\parallel {\varphi }_{1}\parallel }_{{H}_{k}}+{\parallel {\varphi }_{2}\parallel }_{{H}_{k}}\right){\parallel {\varphi }_{1}-{\varphi }_{2}\parallel }_{{H}_{k}}{\parallel \psi \parallel }_{{H}_{k}}\right).$(3.19)

From (3.19) it follows that

(3.20)

uniformly for $r\in \left[a,b\right]$, $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$ and ${\varphi }_{1},{\varphi }_{2}\in {V}_{k,r,\mu }$. We then obtain (3.17) by putting together (3.18) and (3.20). From (3.16) and (3.17), it follows that there exists a constant ${k}_{4}\ge {k}_{3}$ such that for any $k\ge {k}_{4}$, $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$, there exists a unique solution ${\varphi }_{k,r,\mu }\in {V}_{k,r,\mu }$ of (2.1). The continuous differentiability of $\left(r,\mu \right)↦{\varphi }_{k,r,\mu }$ is standard.

Now, we prove the last part of Proposition 2.1. We let $\left({r}_{k},{\mu }_{k}\right)\in \left[a,b\right]×\left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$ be a critical point of ${\mathcal{ℐ}}_{k}$. Since ${\varphi }_{k,r,\mu }$ is a solution of (2.1), there exist ${c}_{1,k}$ and ${c}_{2,k}$ such that

$DI\left({W}_{k,{r}_{k},{\mu }_{k}}+{\varphi }_{k,{r}_{k},{\mu }_{k}}\right)=\sum _{j=1}^{2}{c}_{j,k}\sum _{i=1}^{k}{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\cdot 〉}_{{H}_{k}}.$(3.21)

From (3.21) it follows that

$0=\frac{\partial {\mathcal{ℐ}}_{k}}{\partial r}\left({r}_{k},{\mu }_{k}\right)$$=\sum _{j=1}^{2}{c}_{j,k}\sum _{i=1}^{k}{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{dr}{\left[{W}_{k,r,{\mu }_{k}}+{\varphi }_{k,r,{\mu }_{k}}\right]}_{r={r}_{k}}〉}_{{H}_{k}}$$=\sum _{j=1}^{2}{c}_{j,k}\sum _{i=1}^{k}\left({\mu }_{k}\sum _{\alpha =1}^{k}{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},{Z}_{\alpha ,1,k,{r}_{k},{\mu }_{k}}〉}_{{H}_{k}}+{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{dr}{\left[{\varphi }_{k,r,{\mu }_{k}}\right]}_{r={r}_{k}}〉}_{{H}_{k}}\right)$(3.22)

and

$0=\frac{\partial {\mathcal{ℐ}}_{k}}{\partial \mu }\left({r}_{k},{\mu }_{k}\right)$$=\sum _{j=1}^{2}{c}_{j,k}\sum _{i=1}^{k}{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{d\mu }{\left[{W}_{k,{r}_{k},\mu }+{\varphi }_{k,{r}_{k},\mu }\right]}_{\mu ={\mu }_{k}}〉}_{{H}_{k}}$$=\sum _{j=1}^{2}{c}_{j,k}\sum _{i=1}^{k}\left(\frac{1}{{\mu }_{k}}\sum _{\alpha =1}^{k}{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},{Z}_{\alpha ,2,k,{r}_{k},{\mu }_{k}}〉}_{{H}_{k}}+{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{d\mu }{\left[{\varphi }_{k,{r}_{k},\mu }\right]}_{\mu ={\mu }_{k}}〉}_{{H}_{k}}\right).$(3.23)

For any $i,\alpha \in \left\{1,\mathrm{\dots },k\right\}$ and $j,\beta \in \left\{1,2\right\}$, direct calculations yield

(3.24)

where ${\mathrm{\Lambda }}_{j}>0$ is a constant and ${\delta }_{i\alpha }:=1$ if $\alpha =i$, and ${\delta }_{i\alpha }:=0$ if $\alpha \ne i$. Moreover, since ${\varphi }_{k,r,\mu }\in {P}_{k,r,\mu }$, we obtain

${〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{dr}{\left[{\varphi }_{k,r,{\mu }_{k}}\right]}_{r={r}_{k}}〉}_{{H}_{k}}=-{〈\frac{d}{dr}{\left[{Z}_{i,j,k,r,{\mu }_{k}}\right]}_{r={r}_{k}},{\varphi }_{k,{r}_{k},{\mu }_{k}}〉}_{{H}_{k}},$

and therefore, by using the Cauchy–Schwarz inequality, Sobolev’s inequality, (2.2) and similar estimates as in (3.14), we obtain

$|{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{dr}{\left[{\varphi }_{k,r,{\mu }_{k}}\right]}_{r={r}_{k}}〉}_{{H}_{k}}|\le \parallel \frac{d}{dr}{\left[{Z}_{i,j,k,r,{\mu }_{k}}\right]}_{r={r}_{k}}{\parallel }_{{H}_{k}}\parallel {\varphi }_{k,{r}_{k},{\mu }_{k}}{\parallel }_{{H}_{k}}=\mathrm{O}\left({k}^{3/2}\right).$(3.25)

Similarly, we obtain

$|{〈{Z}_{i,j,k,{r}_{k},{\mu }_{k}},\frac{d}{d\mu }{\left[{\varphi }_{k,{r}_{k},\mu }\right]}_{\mu ={\mu }_{k}}〉}_{{H}_{k}}|\le {\parallel \frac{d}{d\mu }{\left[{Z}_{i,j,k,{r}_{k},\mu }\right]}_{\mu ={\mu }_{k}}\parallel }_{{H}_{k}}{\parallel {\varphi }_{k,{r}_{k},{\mu }_{k}}\parallel }_{{H}_{k}}=\mathrm{O}\left({k}^{3/2}{\mu }_{k}^{-2}\right).$(3.26)

From (3.22)–(3.26) it follows that if k is large enough, then ${c}_{1,k}={c}_{2,k}=0$, i.e., the function ${W}_{k,{r}_{k},{\mu }_{k}}+{\varphi }_{k,{r}_{k},{\mu }_{k}}$ is a weak solution of the equation

By using the coercivity of the operator $\mathrm{\Delta }+f$ in ${D}^{1,2}\left({ℝ}^{4}\right)$, we obtain that $u\ge 0$ a.e. in ${ℝ}^{4}$. Then, from standard elliptic regularity theory and the strong maximum principle, it follows that ${W}_{k,{r}_{k},{\mu }_{k}}+{\varphi }_{k,{r}_{k},{\mu }_{k}}$ is a strong positive solution in ${C}^{2,\alpha }\left({ℝ}^{4}\right)$ of (2.3). ∎

## 4 Proof of Proposition 2.2

In this section we prove Proposition 2.2. Throughout this section we assume that $f\in {C}^{0,\alpha }\left({ℝ}^{4}\right)\cap {L}^{2}\left({ℝ}^{4}\right)$ is radially symmetric about the point 0, and that the operator $\mathrm{\Delta }+f$ is coercive in ${D}^{1,2}\left({ℝ}^{4}\right)$. First, we obtain the following result.

#### Lemma 4.1.

There exist constants ${c}_{\mathrm{0}}\mathrm{,}{c}_{\mathrm{1}}\mathrm{,}{c}_{\mathrm{2}}\mathrm{>}\mathrm{0}$ such that for any $a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{>}\mathrm{0}$, with $a\mathrm{<}b$ and $c\mathrm{<}d$, we have

(4.1)

uniformly for $r\mathrm{\in }\mathrm{\left[}a\mathrm{,}b\mathrm{\right]}$ and $\mu \mathrm{\in }\mathrm{\left[}{e}^{c\mathit{}{k}^{\mathrm{2}}}\mathrm{,}{e}^{d\mathit{}{k}^{\mathrm{2}}}\mathrm{\right]}$.

#### Proof.

By integrating by parts, we obtain

$I\left({W}_{k,r,\mu }\right)=\frac{1}{2}{\int }_{{ℝ}^{4}}\left(\mathrm{\Delta }{W}_{k,r,\mu }+f{W}_{k,r,\mu }\right){W}_{k,r,\mu }𝑑x-\frac{1}{4}{\int }_{{ℝ}^{4}}{W}_{k,r,\mu }^{4}𝑑x$$=\frac{1}{2}{\int }_{{ℝ}^{4}}\left(\sum _{i,j=1}^{k}{U}_{i,k,r,\mu }^{3}{U}_{j,k,r,\mu }+f{W}_{k,r,\mu }^{2}-\frac{1}{2}{W}_{k,r,\mu }^{4}\right)dx$$=\frac{1}{2}\sum _{i=1}^{k}{\int }_{{ℝ}^{4}}\left(f{U}_{i,k,r,\mu }^{2}+\frac{1}{2}{U}_{i,k,r,\mu }^{4}-\sum _{j\ne i}{U}_{i,k,r,\mu }^{3}{U}_{j,k,r,\mu }-3\sum _{j\ne i}{U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }^{2}+f\sum _{j\ne i}{U}_{i,k,r,\mu }{U}_{j,k,r,\mu }\right)𝑑x.$(4.2)

Direct calculations yield

${\int }_{{ℝ}^{4}}{U}_{i,k,r,\mu }^{4}𝑑x={\left(2\sqrt{2}\right)}^{4}{\int }_{{ℝ}^{4}}\frac{dx}{{\left(1+{|x|}^{2}\right)}^{4}}$(4.3)

and

(4.4)

uniformly for $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$. Proceeding as in (3.4)–(3.8), we obtain

$\sum _{j\ne i}{\int }_{{ℝ}^{4}}{U}_{i,k,r,\mu }^{3}{U}_{j,k,r,\mu }dx=\sum _{j\ne i}{\int }_{{\mathrm{\Omega }}_{i,k,r}}\frac{64{\mu }^{2}}{{\left(1+{\mu }^{2}{|x-{x}_{i,k,r}|}^{2}\right)}^{3}}\left(\frac{1+\mathrm{O}\left({𝟏}_{{\mathrm{\Omega }}_{i,k,r}\setminus B\left({x}_{i},|{x}_{1,k,r}-{x}_{2,k,r}|/2\right)}\right)}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{2}}$$+\mathrm{O}\left(\frac{{\mu }^{-2}+|x-{x}_{i,k,r}||{x}_{i,k,r}-{x}_{j,k,r}|}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{4}}{𝟏}_{B\left({x}_{i},|{x}_{1,k,r}-{x}_{2,k,r}|/2\right)}\right)\right)dx$$+\mathrm{O}\left(\sum _{\alpha \ne i}\frac{k\mu }{{|{x}_{i,k,r}-{x}_{\alpha ,k,r}|}^{3}}{\int }_{{\mathrm{\Omega }}_{\alpha ,k,r}}\frac{dx}{{\left(1+{\mu }^{2}{|x-{x}_{\alpha ,k,r}|}^{2}\right)}^{5/2}}\right)$$=\sum _{j\ne i}\left(\frac{64{\mu }^{-2}}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{2}}{\int }_{{ℝ}^{4}}\frac{dx}{{\left(1+{|x|}^{2}\right)}^{3}}+\mathrm{O}\left(\frac{k{\mu }^{-3}}{{|{x}_{i,k,r}-{x}_{j,k,r}|}^{3}}\right)\right)$$=\frac{32{k}^{2}}{{\pi }^{2}{r}^{2}{\mu }^{2}}{\int }_{{ℝ}^{4}}\frac{dx}{{\left(1+{|x|}^{2}\right)}^{3}}\sum _{j=1}^{\mathrm{\infty }}\frac{1}{{j}^{2}}+\mathrm{o}\left(\frac{{k}^{2}}{{\mu }^{2}}\right)$(4.5)

uniformly for $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$. Moreover, straightforward estimates give

$\sum _{j\ne i}{\int }_{\begin{array}{c}{ℝ}^{4}\setminus \left(B\left({x}_{i,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\\ \cup B\left({x}_{j,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\right)\end{array}}{U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }^{2}dx=\mathrm{O}\left({\mu }^{-4}\sum _{j\ne i}{\int }_{\begin{array}{c}{ℝ}^{4}\setminus \left(B\left({x}_{i,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\\ \cup B\left({x}_{j,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\right)\end{array}}|x-{x}_{i,k,r,\mu }{|}^{-4}|x-{x}_{j,k,r,\mu }{|}^{-4}dx\right)$$=\mathrm{O}\left(\sum _{j\ne i}\frac{{\mu }^{-4}}{{|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|}^{4}}\right)$$=\mathrm{O}\left({\left(k/\mu \right)}^{4}\right),$(4.6)$\sum _{j\ne i}{\int }_{\begin{array}{c}B\left({x}_{i,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\\ \cup B\left({x}_{j,k,r,\mu },|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)\end{array}}{U}_{i,k,r,\mu }^{2}{U}_{j,k,r,\mu }^{2}𝑑x=\mathrm{O}\left(\sum _{j\ne i}\frac{{\mu }^{-4}}{{|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|}^{4}}{\int }_{B\left(0,\mu |{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|/2\right)}\frac{dx}{{\left(1+{|x|}^{2}\right)}^{2}}\right)$$=\mathrm{O}\left(\sum _{j\ne i}\frac{{\mu }^{-4}\mathrm{ln}\mu }{{|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|}^{4}}\right)$$=\mathrm{O}\left({\left(k/\mu \right)}^{4}\mathrm{ln}\mu \right),$(4.7)${\sum _{j\ne i}{\int }_{{ℝ}^{4}\setminus \left(B\left({x}_{i,k,r,\mu },1\right)\cup B\left({x}_{j,k,r,\mu },1\right)\right)}f{U}_{i,k,r,\mu }{U}_{j,k,r,\mu }dx=\mathrm{O}\left(\sum _{j\ne i}\left({\int }_{{ℝ}^{4}\setminus B\left({x}_{j,k,r,\mu },1\right)}{U}_{j,k,r,\mu }^{4}dx\right)}^{1/2}\right)$$=\mathrm{O}\left(k{\left({\int }_{{ℝ}^{4}\setminus B\left(0,\mu \right)}\frac{dx}{{\left(1+{|x|}^{2}\right)}^{4}}\right)}^{1/2}\right)$$=\mathrm{O}\left(k/{\mu }^{2}\right),$(4.8)$\sum _{j\ne i}{\int }_{B\left({x}_{i,k,r,\mu },1\right)\cup B\left({x}_{j,k,r,\mu },1\right)}f{U}_{i,k,r,\mu }{U}_{j,k,r,\mu }dx=\mathrm{O}\left(\sum _{j\ne i}{\int }_{B\left({x}_{i,k,r,\mu },1\right)}{U}_{i,k,r,\mu }{U}_{j,k,r,\mu }dx\right)$$=\mathrm{O}\left(\sum _{j\ne i}{\int }_{B\left({x}_{i,k,r,\mu },1\right)}\frac{{\mu }^{-2}dx}{{|x-{x}_{i,k,r,\mu }|}^{2}{|x-{x}_{j,k,r,\mu }|}^{2}}\right)$$=\mathrm{O}\left({\mu }^{-2}\sum _{j\ne i}\mathrm{ln}\frac{1}{|{x}_{i,k,r,\mu }-{x}_{j,k,r,\mu }|}\right)$$=\mathrm{O}\left(\frac{k\mathrm{ln}k}{{\mu }^{2}}\right)$(4.9)

as $k\to \mathrm{\infty }$, uniformly for $r\in \left[a,b\right]$ and $\mu \in \left[{e}^{c{k}^{2}},{e}^{d{k}^{2}}\right]$. Finally, (4.1) follows from (4.2)–(4.9). ∎

We can now prove Proposition 2.2, by using Lemma 4.1.

#### Proof of Proposition 2.2.

By integrating by parts, we obtain

$I\left({W}_{k,r,\mu }+{\varphi }_{k,r,\mu }\right)=I\left({W}_{k,r,\mu }\right)-{〈{R}_{k,r,\mu },{\varphi }_{k,r,\mu }〉}_{{H}_{k}}+\frac{1}{2}{\parallel {\varphi }_{k,r,\mu }\parallel }_{{H}_{k}}^{2}$$-\frac{1}{4}{\int }_{{ℝ}^{4}}\left({\left({W}_{k,r,\mu }+{\varphi }_{k,r,\mu }\right)}_{+}^{4}-{W}_{k,r,\mu }^{4}-4{W}_{k,r,\mu }^{3}{\varphi }_{k,r,\mu }\right)𝑑x.$(4.10)

By using the Cauchy–Schwarz inequality, Lemma 3.1 and Proposition 2.1, we obtain

$-{〈{R}_{k,r,\mu },{\varphi }_{k,r,\mu }〉}_{{H}_{k}}+\frac{1}{2}{\parallel {\varphi }_{k,r,\mu }\parallel }_{{H}_{k}}^{2}=\mathrm{O}\left({\left(k/\mu \right)}^{2}\right).$(4.11)

Moreover, by using Hölder’s inequality, Sobolev’s inequality, (3.14) and Lemma 3.1, we obtain

${\int }_{{ℝ}^{4}}\left({\left({W}_{k,r,\mu }+{\varphi }_{k,r,\mu }\right)}_{+}^{4}-{W}_{k,r,\mu }^{4}-4{W}_{k,r,\mu }^{3}{\varphi }_{k,r,\mu }\right)𝑑x=\mathrm{O}\left({\int }_{{ℝ}^{4}}\left({W}_{k,r,\mu }^{2}+{\varphi }_{k,r,\mu }^{2}\right){\varphi }_{k,r,\mu }^{2}𝑑x\right)$$=\mathrm{O}\left({\parallel {W}_{k,r,\mu }\parallel }_{{L}^{4}}^{2}{\parallel {\varphi }_{k,r,\mu }\parallel }_{{H}_{k}}^{2}+{\parallel {\varphi }_{k,r,\mu }\parallel }_{{H}_{k}}^{4}\right)$$=\mathrm{O}\left(\sqrt{k}{\left(k/\mu \right)}^{2}+{\left(k/\mu \right)}^{4}\right).$(4.12)

Finally, (2.4) follows from (4.10)–(4.12). ∎

## Acknowledgements

This paper was written during the second author’s Ph.D. He wishes to express his gratitude to his supervisors Pengfei Guan and Jérôme Vétois for their support and guidance.

## References

• [1]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539.

• [2]

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008), no. 4, 951–979.  Google Scholar

• [3]

S. Brendle and F. C. Marques, Blow-up phenomena for the Yamabe equation. II, J. Differential Geom. 81 (2009), no. 2, 225–250.

• [4]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.

• [5]

W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in ${ℝ}^{N}$ with critical growth, J. Differential Equations 252 (2012), no. 3, 2425–2447.  Google Scholar

• [6]

O. Druet, From one bubble to several bubbles: The low-dimensional case, J. Differential Geom. 63 (2003), no. 3, 399–473.

• [7]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. IMRN (2004), no. 23, 1143–1191.  Google Scholar

• [8]

O. Druet, E. Hebey and F. Robert, Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Math. Notes 45, Princeton University Press, Princeton, 2004.  Google Scholar

• [9]

E. Hebey, Compactness and stability for nonlinear elliptic equations, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2014.  Google Scholar

• [10]

M. A. Khuri, F. C. Marques and R. M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196.

• [11]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50.

• [12]

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971/72), 247–258.

• [13]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin–Ni’s conjecture, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4581–4615.

• [14]

L. Wang, J. Wei and S. Yan, On Lin–Ni’s conjecture in convex domains, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1099–1126.

• [15]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in ${ℝ}^{N}$, Calc. Var. Partial Differential Equations 37 (2010), no. 3–4, 423–439.  Google Scholar

• [16]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on ${𝕊}^{N}$, J. Funct. Anal. 258 (2010), no. 9, 3048–3081.  Google Scholar

• [17]

J. Wei and S. Yan, On a stronger Lazer–McKenna conjecture for Ambrosetti–Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 2, 423–457.  Google Scholar

• [18]

J. Wei and S. Yan, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9) 96 (2011), no. 4, 307–333.

Accepted: 2017-07-09

Published Online: 2017-08-18

The first author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. The second author was supported by the China Scholarship Council.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 715–724, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

Export Citation