Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms

Abstract

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\mu \frac{u}{|x|^2} = |u|^{2^{*}-2-\varepsilon }u &{} \text {in } \Omega , \\ u = 0&{} \text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$

in a bounded domain $\Omega \subset {\mathbb {R}}^N (N\ge 7)$ with $0\in \Omega$, as $\mu ,\epsilon \rightarrow 0^+$. In [6], we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $\mu =O(\epsilon ^\alpha )$ with $\alpha >\frac{N-4}{N-2}$, $\epsilon \rightarrow 0^+$. Here we prove the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $\mu =O(\epsilon )$, $\epsilon \rightarrow 0^+$.

Appendices

Appendix A: Some lemmas from [6]

In this part we collect some lemmas from [6]. We define for $\eta \in (0,1)$:

$$\begin{aligned} \begin{aligned} {{\mathcal {T}}}_\eta&:=\big \{(\lambda ,\xi )\in {\mathbb {R}}_+^{k+1}\times \Omega ^{k}:\lambda _i\in (\eta ,\eta ^{-1}),\overline{\lambda }\in (\eta ,\eta ^{-1}),\ \mathrm {dist}(\xi _i,\partial \Omega )>\eta ,\\&\quad |\xi _i|>\eta ,\ |\xi _{i_1}-\xi _{i_2}|>\eta ,\ i,i_1,i_2=1,2,\dots ,k,\ i_1\ne i_2\big \}. \end{aligned} \end{aligned}$$

Lemma A.1

(i) For $i=1,2,\dots ,k,$ and $j=0,1,\dots ,N$, there holds

$$\begin{aligned} \Vert P\Psi _i^j-\Psi _i^j\Vert _{2N/(N-2)}={\left\{ \begin{array}{ll} O\left( \delta _i^{\frac{N-2}{2}}\right) \quad &{} \text {if} \; j=1,2,\dots ,N,\\ O\left( \delta _i^{\frac{N-4}{2}}\right) \quad &{} \text {if} \; j=0\end{array}\right. } \end{aligned}$$

as $\delta _i\rightarrow 0$ uniformly for $\xi _i$ in a compact subset of $\Omega$.

(ii) There holds

$$\begin{aligned} \Vert P\overline{\Psi }-\overline{\Psi }\Vert _{2N/(N-2)}=O\left( \sigma ^{\frac{N-4}{2}}\right) \end{aligned}$$

as $\sigma \rightarrow 0$, uniformly for $0<\mu <{\overline{\mu }}$.

Lemma A.2

For $i=1,2,\cdots ,k,$ the following estimates hold uniformly for $(\lambda ,\xi )\in {{\mathcal {T}}}_\eta$:

(i):

For $\mu ,\sigma \rightarrow 0$:

$$\begin{aligned} \int _{\Omega }|\nabla P V_\sigma |^2-\mu \frac{|P V_\sigma |^2}{|x|^2}= & {} S_\mu ^{\frac{N}{2}}-C_0C_\mu ^{2^*-1} H(0,0)\sigma ^{N-2}\\&\times \int _{{\mathbb {R}}^N} \frac{1}{({|z|^{\beta _1}}+|z|^{\beta _2})^{\frac{N+2}{2}}}+O(\mu \sigma ^{N-2})+O(\sigma ^N). \end{aligned}$$

(ii):

For $\delta _i,\sigma \rightarrow 0$:

$$\begin{aligned} \int _{\Omega }|\nabla P U_{\delta _i,\xi _i}|^2=S_0^{\frac{N}{2}}-C_0^{2^*}H(\xi _i,\xi _i)\delta _i^{N-2}\int _{{\mathbb {R}}^N} \frac{1}{(1+|z|^2)^{\frac{N+2}{2}}}+o(\delta _i^{N-2}). \end{aligned}$$

Lemma A.3

For $\mu ,\sigma \rightarrow 0$ there holds

$$\begin{aligned} \int _{\Omega }|P V_\sigma |^{2^*} =S_\mu ^{\frac{N}{2}} -2^*C_0C_\mu ^{2^*-1}H(0,0) \sigma ^{N-2} \int _{{\mathbb {R}}^N} \frac{1}{({|z|^{\beta _1}}+|z|^{\beta _2})^{\frac{N+2}{2}}}+O(\mu \sigma ^{N-2})+O(\sigma ^N). \end{aligned}$$

Lemma A.4

For $\mu \rightarrow 0^+$ there holds

$$\begin{aligned} \int _{{\mathbb {R}}^N}V_{1}^p=\int _{{\mathbb {R}}^N}U_{1,0}^p+o(1)\quad \text {and}\quad \int _{{\mathbb {R}}^N} V_{1}^p\ln V_{1}=\int _{{\mathbb {R}}^N} U_{1,0}^p\ln U_{1,0}+o(1) \end{aligned}$$

for $p>1$ as well as

$$\begin{aligned} C_\mu =C_0-\frac{C_0}{N-2}\mu +O(\mu ^2) \quad \text {and}\quad S_\mu =S_0-\overline{S}\mu +O(\mu ^2), \end{aligned}$$

for some positive constant $\overline{S}$ independent of $\mu$.

Appendix B: Proof of the lemmas from Sect. 3

Lemma B.1

For $i,l=1,2,\dots ,k,$ and $j,h=0,1,\dots ,N$, with $i\ne l$ or $j\ne h$, there are constants $\widetilde{c}_0>0, \widetilde{c}_{i,j}>0$ such that the following estimates hold uniformly for $0<\mu <\overline{\mu }$:

$$\begin{aligned} \begin{aligned} (P\overline{\Psi },P\overline{\Psi })_\mu&=\widetilde{c}_0\frac{1}{\sigma ^2}+o(\sigma ^{-2})\ \text {as}\ \sigma \rightarrow 0,\\ (P\overline{\Psi },P\Psi _i^j)_\mu&=o(\sigma ^{-2})o(\delta _i^{-2})\ \text {as}\ \sigma \rightarrow 0, \ \delta _i\rightarrow 0,\ \text {uniformly for}\ \xi _i \ \text {in a compact subset of}\ \Omega ,\\ (P\Psi _i^j,P\Psi _i^j)_\mu&=\widetilde{c}_{i,j}\frac{1}{\delta _i^2}+o(\delta _i^{-2})\ \text {as}\ \delta _i\rightarrow 0,\ {uniformly for}\ \xi _i \ \text {in a compact subset of}\ \Omega ,\\ (P\Psi _i^j,P\Psi _l^h)_\mu&=o(\delta _i^{-2})\ \text {as}\ \delta _i\rightarrow 0,\ \text {uniformly for}\ \xi _i, \xi _l\ \text {in a compact subset of}\ \Omega . \end{aligned} \end{aligned}$$

Proof

We omit the proof since it is similar to [6, Lemma A.1]. $\square$

Lemma B.2

(i) For $i,l=1,2,\cdots ,k$ there holds

$$\begin{aligned} \left\| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right\| _{2N/(N+2)}=o\left( \delta _l^{-\frac{2N}{N+2}}\right) \end{aligned}$$

as $\sigma , \delta _i, \delta _l \rightarrow 0$ uniformly for $0<\mu <\overline{\mu }$ and $\xi _i$ in a compact subset of $\Omega$.

(ii) There holds

$$\begin{aligned} \left\| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(V_\sigma )\right) \overline{\Psi }\right\| _{2N/(N+2)} = o\left( \sigma ^{-\frac{2N}{N+2}}\right) \end{aligned}$$

as $\sigma , \delta _i\rightarrow 0$ uniformly for $0<\mu <\overline{\mu }$ and $\xi _i$ in a compact subset of $\Omega$.

Proof

We only prove (i) for $h\ne 0$.

$$\begin{aligned} \begin{aligned}&\int _\Omega \left| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{2N/(N+2)}\\&\quad =\bigcup \limits _{i=1}^{k+1}\int _{A_i} \left| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{2N/(N+2)}\\&\qquad +\int _{\Omega \backslash B(0,\rho )} \left| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{2N/(N+2)}. \end{aligned} \end{aligned}$$

As in [26, Lemma A.3], by (2.3) and (2.4) we have

$$\begin{aligned} \begin{aligned}&\int _{A_l} \left| \left( f'_0\left( \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right) -f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{2N/(N+2)}\\&\quad \le C\int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}\varphi _{\delta _l,\xi _l}\Psi _l^h|^{2N/(N+2)}+C\sum \limits _{i\ne l}\int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}U_{\delta _i,\xi _i}\Psi _l^h|^{2N/(N+2)}\\&\qquad +C\int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}V_\sigma \Psi _l^h|^{2N/(N+2)}\\&\quad \le o\left( \delta _l^{-\frac{2N}{N+2}}\right) , \end{aligned} \end{aligned}$$

(B.1)

where we use

$$\begin{aligned} \int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}\varphi _{\delta _l,\xi _l}\Psi _l^h|^{2N/(N+2)} \le C\int _{A_l} \left| \frac{\delta _l^{\frac{N+2}{2}}(x^h-\xi _l^h)}{(\delta _l^2+|x-\xi _l|^2)^3}\right| ^{2N/(N+2)}=O\left( \delta _l^{\frac{2N(N-3)}{N+2}}\right) , \end{aligned}$$

for $i\ne l$,

$$\begin{aligned} \begin{aligned}&\int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}U_{\delta _i,\xi _i}\Psi _l^h|^{2N/(N+2)} =C\int _{A_l} \left| \frac{\delta _l^2(x^h-\xi _l^h)}{(\delta _l^2+|x-\xi _l|^2)^3}\frac{\delta _i^{\frac{N-2}{2}}}{(\delta _i^2+|x-\xi _i|^2)^{\frac{N-2}{2}}}\right| ^{2N/(N+2)}\\&\quad \le C\left( \int _{A_l} \left| \frac{\delta _l^2(x^h-\xi _l^h)}{(\delta _l^2+|x-\xi _l|^2)^3}\right| ^{\frac{N}{2}}\right) ^{\frac{4}{N+2}} \left( \int _{A_l}\left| \frac{\delta _i^{\frac{N-2}{2}}}{(\delta _i^2+|x-\xi _i|^2)^{\frac{N-2}{2}}}\right| ^{2N/(N-2)}\right) ^{\frac{N-2}{N+2}} =o\left( \delta _l^{-\frac{2N}{N+2}}\right) , \end{aligned} \end{aligned}$$

and similarly,

$$\begin{aligned} \int _{A_l} |U_{\delta _l,\xi _l}^{2^*-3}V_\sigma \Psi _l^h|^{2N/(N+2)}=o\left( \delta _l^{-\frac{2N}{N+2}}\right) . \end{aligned}$$

The same arguments as for (B.1) yield for $i\ne l$:

$$\begin{aligned} \int _{A_i} \left| \left( f'_0(\sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma )-f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{2N/(N+2)} =o\left( \delta _l^{-\frac{2N}{N+2}}\right) . \end{aligned}$$

Finally,

$$\begin{aligned} \begin{aligned}&\int _{\Omega \backslash B(0,\rho )} \left| \left( f'_0(\sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma )-f'_0(U_{\delta _l,\xi _l})\right) \Psi _l^h\right| ^{\frac{2N}{(N+2)}}\\&\quad ={\left\{ \begin{array}{ll} O\left( \delta _l^{\frac{N(N-2)}{N+2}}\right) \left( O\left( \sigma ^{\frac{4N}{N+2}}\right) +\sum \limits _{i=1}^{k}O\left( \delta _i^{\frac{4N}{N+2}}\right) \right) \quad &{} \text {if} \,\, h=1,2,\dots ,N,\\ O\left( \delta _l^{\frac{N(N-4)}{N+2}}\right) \left( O\left( \sigma ^{\frac{4N}{N+2}}\right) +\sum \limits _{i=1}^{k}O\left( \delta _i^{\frac{4N}{N+2}}\right) \right) \quad &{} \text {if} \,\, h=0. \end{array}\right. } \end{aligned} \end{aligned}$$

Then (i) follows. $\square$

Lemma B.3

There holds

$$\begin{aligned} \left\| \iota _\mu ^*\left( \sum \limits _{i=1}^{k}(-1)^{i-1}f_0(U_{\delta _i,\xi _i})+(-1)^kf_0(V_\sigma )\right) -V_{\varepsilon ,\lambda ,\zeta }\right\| _\mu \le \sum \limits _{i=1}^{k}O(\mu \delta _i)+O\left( \left( \mu \sigma ^{\frac{N-2}{2}}\right) ^{\frac{1}{2}}\right) \end{aligned}$$

as $\mu ,\sigma ,\delta _i\rightarrow 0$ uniformly for $\xi _i$ in a compact subset of $\Omega$.

Proof

It is similar to [6, Lemma A.4]. $\square$

Lemma B.4

For $\epsilon \rightarrow 0$, the following estimates hold uniformly for $0<\mu <\overline{\mu }$ and $(\lambda ,\zeta )\in {{\mathcal {O}}}_\eta$:

$$\begin{aligned} \begin{aligned}&\Vert (f'_\varepsilon (V_{\varepsilon ,\lambda ,\zeta })-f'_0(V_{\varepsilon ,\lambda ,\zeta }))\phi \Vert _{2N/(N+2)} =O(\varepsilon )\Vert \phi \Vert _\mu ,\\&\Vert f_\varepsilon (V_{\varepsilon ,\lambda ,\zeta })-f_0(V_{\varepsilon ,\lambda ,\zeta })\Vert _{2N/(N+2)} =O(\varepsilon ),\\&\Vert f_0(V_{\varepsilon ,\lambda ,\zeta })-\left( \sum \limits _{i=1}^{k}(-1)^{i-1}f_0(U_{\delta _i,\xi _i})+(-1)^k f_0(V_\sigma )\right) \Vert _{2N/(N+2)} = O\left( \varepsilon ^\frac{N+2}{2(N-2)}\right) . \end{aligned}\end{aligned}$$

Proof

The first two are from [3]. The last one can be proved as (4.5) in [26]. $\square$

Lemma B.5

Let $k\ge 1$. Assume without loss of generality that $1\le i<j\le k$.Then the following estimates hold uniformly for $(\lambda ,\zeta )\in {{\mathcal {O}}}_\eta$:

(i):

For $\epsilon \rightarrow 0$:

$$\begin{aligned} \int _{\Omega }|\nabla P V_\sigma |^2-\mu \frac{|P V_\sigma |^2}{|x|^2}=S_\mu ^{\frac{N}{2}}+o(\varepsilon ). \end{aligned}$$

(ii):

For $\epsilon \rightarrow 0$:

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla P V_\sigma \nabla PU_{\delta _i,\xi _i}-\mu \frac{P V_\sigma PU_{\delta _i,\xi _i}}{|x|^2}\\&\quad = {\left\{ \begin{array}{ll} C_0^{2^*}(\frac{\overline{\lambda }}{\lambda _k})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{(1+|y|^2)^{\frac{N+2}{2}}}\frac{1}{(1+|\zeta _k|^2)^{\frac{N-2}{2}}}\cdot \varepsilon +o(\varepsilon ) \quad &{} \text {if} ~~i=k,\\ o(\varepsilon )\quad &{} \text {if} ~~i\ne k. \end{array}\right. } \end{aligned} \end{aligned}$$

(iii):

For $\epsilon \rightarrow 0$:

$$\begin{aligned} \mu \int _{\Omega }\frac{|PU_{\delta _i,\xi _i}|^2}{|x|^2}=\mu C_0^2\int _{{\mathbb {R}}^N}\frac{1}{|y|^2(1+|y-\zeta _i|^2)^{N-2}}+o(\varepsilon ). \end{aligned}$$

(iv):

For $\epsilon \rightarrow 0$:

$$\begin{aligned} \mu \int _{\Omega }\frac{PU_{\delta _i,\xi _i}PU_{\delta _j,\xi _j}}{|x|^2} =o(\varepsilon ),~~~i\ne j. \end{aligned}$$

(v):

For $\epsilon \rightarrow 0$:

$$\begin{aligned} \int _{\Omega }|\nabla P U_{\delta _i,\xi _i}|^2 = {\left\{ \begin{array}{ll} S_0^{\frac{N}{2}}-C_0^{2^*}H(0,0)\lambda _1^{N-2}\int _{{\mathbb {R}}^N} \frac{1}{(1+|z|^2)^{\frac{N+2}{2}}}\cdot \varepsilon +o(\varepsilon ) \quad &{} \text {if} \,\,i=1,\\ S_0^{\frac{N}{2}}+o(\varepsilon )\quad &{} \text {if} \,\,i\ne 1. \end{array}\right. } \end{aligned}$$

(vi):

For $\epsilon \rightarrow 0$:

$$\begin{aligned}&\int _{\Omega }\nabla P U_{\delta _i,\xi _i}\nabla P U_{\delta _j,\xi _j}\\&\quad = {\left\{ \begin{array}{ll} C_0^{2^*}(\frac{\lambda _{i+1}}{\lambda _i})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{(1+|y|^2)^{\frac{N+2}{2}}}\frac{1}{(1+|\zeta _i|^2)^{\frac{N-2}{2}}}\cdot \varepsilon +o(\varepsilon ) \quad &{} \text {if} ~~j=i+1,\\ o(\varepsilon )\quad &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

(i) and (v) follow from Lemma A.2.

Now we prove (ii). Using (2.4), integration by parts yields

$$\begin{aligned} \int _{\Omega }\nabla P V_\sigma \nabla PU_{\delta _i,\xi _i}-\mu \frac{PV_\sigma PU_{\delta _i,\xi _i}}{|x|^2} = \int _{\Omega }V_\sigma ^{2^*-1}(U_{\delta _i,\xi _i}-\varphi _{\delta _i,\xi _i})+\mu \int _\Omega \frac{\varphi _\sigma (U_{\delta _i,\xi _i}-\varphi _{\delta _i,\xi _i})}{|x|^2}. \end{aligned}$$

(B.2)

It is easy to show, using (2.3) and (2.4), that

$$\begin{aligned} \begin{aligned} \int _{\Omega }V_\sigma ^{2^*-1}\varphi _{\delta _i,\xi _i}&\le C\delta _i^{\frac{N-2}{2}}\int _{\Omega } \frac{\sigma ^{\frac{N+2}{2}}}{(\sigma ^2|x|^{\beta _1}+|x|^{\beta _2})^{\frac{N+2}{2}}}\\&\le C\sigma ^{\frac{\overline{\mu }}{\sqrt{\overline{\mu }-\mu }}}\delta _i^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N} \frac{1}{(|y|^{\beta _1}+|y|^{\beta _2})^{\frac{N+2}{2}}}=O(\sigma ^{\frac{N-2}{2}}\delta _i^{\frac{N-2}{2}}), \end{aligned} \end{aligned}$$

(B.3)

and

$$\begin{aligned} \mu \int _\Omega \frac{\varphi _\sigma (U_{\delta _i,\xi _i}-\varphi _{\delta _i,\xi _i})}{|x|^2} \le \mu \int _\Omega \frac{\varphi _\sigma U_{\delta _i,\xi _i}}{|x|^2}\le O(\mu \sigma ^{\frac{N-2}{2}}). \end{aligned}$$

(B.4)

On the other hand,

$$\begin{aligned} \int _{\Omega }V_\sigma ^{2^*-1} U_{\delta _i,\xi _i}=\bigcup \limits _{j=1}^{k+1}\int _{A_j}V_\sigma ^{2^*-1} U_{\delta _i,\xi _i}+O(\sigma ^{\frac{N+2}{2}}\delta _i^{\frac{N-2}{2}}). \end{aligned}$$

(B.5)

If $i=k,j=k+1$, then

$$\begin{aligned} \begin{aligned} \int _{A_{k+1}}V_\sigma ^{2^*-1} U_{\delta _k,\xi _k}&= C_\mu ^{2^*-1} C_0\sigma ^{\frac{N+2}{2}}\delta _k^{\frac{N-2}{2}}\int _{A_{k+1}}\frac{1}{({\sigma ^2|x|^{\beta _1}} +|x|^{\beta _2})^{\frac{N+2}{2}}}\frac{1}{(\delta _k^2+|x-\xi _k|^2)^{\frac{N-2}{2}}}\\&= C_\mu ^{2^*-1} C_0\frac{\sigma ^{\frac{\overline{\mu }}{\sqrt{\overline{\mu }-\mu }}}}{\delta _k^{\frac{N-2}{2}}} \int _{\frac{A_{k+1}}{\sigma ^{\frac{\sqrt{\overline{\mu }}}{\sqrt{\overline{\mu }-\mu }}}}}\frac{1}{({|y|^{\beta _1}} +|y|^{\beta _2})^{\frac{N+2}{2}}}\frac{1}{\left( 1+\left| \frac{\sigma ^{\frac{\sqrt{\overline{\mu }}}{\sqrt{\overline{\mu }-\mu }}}}{\delta _k}y-\zeta _k\right| ^2\right) ^{\frac{N-2}{2}}}\\&= C_0^{2^*}\left( \frac{\overline{\lambda }}{\lambda _k}\right) ^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{(1+|y|^2)^{\frac{N+2}{2}}}\frac{1}{(1+|\zeta _k|^2)^{\frac{N-2}{2}}}\cdot \varepsilon +o(\varepsilon ). \end{aligned} \end{aligned}$$

(B.6)

If $i\ne k$ or $j\ne k+1$, then similar arguments as for (B.6) yield

$$\begin{aligned} \int _{A_j}V_\sigma ^{2^*-1} U_{\delta _i,\xi _i}=o(\varepsilon ). \end{aligned}$$

(B.7)

Then we conclude by (B.2)–(B.7).

Next (iii) follows from:

$$\begin{aligned} \begin{aligned} \mu \int _{\Omega }\frac{|PU_{\delta _i,\xi _i}|^2}{|x|^2}&=\mu \int _{\Omega }\frac{|U_{\delta _i,\xi _i}|^2}{|x|^2}+O(\mu \delta _i^{N-2})\\&=\mu C_0^2\int _{\frac{\Omega }{\delta _i}}\frac{1}{|y|^2(1+|y-\zeta _i|^2)^{N-2}}+O(\mu \delta _i^{N-2})\\&=\mu C_0^2\int _{{\mathbb {R}}^N}\frac{1}{|y|^2(1+|y-\zeta _i|^2)^{N-2}}+o(\varepsilon ). \end{aligned} \end{aligned}$$

Similar arguments imply (iv).

For the proof of (vi), without loss of generality let $1\le i<j\le k$. Then

$$\begin{aligned} &\int _{\Omega }\nabla P U_{\delta _i,\xi _i}\nabla P U_{\delta _j,\xi _j} =\int _{\Omega }U_{\delta _j,\xi _j}^{2^*-1}U_{\delta _i,\xi _i}+o(\varepsilon )\\&\quad = {\left \{\begin{array}{ll} C_0^{2^*}(\frac{\lambda _{i+1}}{\lambda _i})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{(1+|y|^2)^{\frac{N+2}{2}}}\frac{1}{(1+|\zeta _i|^2)^{\frac{N-2}{2}}}\cdot \varepsilon +o(\varepsilon ) \quad &{} \text {if} \,\,j=i+1,\\ o(\varepsilon )\quad &{} \text {otherwise}. \end{array}\right.} \end{aligned}$$

(B.8)

$\square$

Lemma B.6

Let $k\ge 1$. For $\epsilon \rightarrow 0$ there holds, uniformly for $(\lambda ,\zeta )\in {{\mathcal {O}}}_\eta$, setting $\lambda _{k+1}=\overline{\lambda }$:

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left| \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right| ^{2^*}\\&\quad = kS_0^{\frac{N}{2}}+S_\mu ^{\frac{N}{2}}-2^*C_0^{2^*}H(0,0)\lambda _1^{N-2}\int _{{\mathbb {R}}^N} \frac{1}{(1+|z|^2)^{\frac{N+2}{2}}}\cdot \varepsilon \\&\qquad -2^*C_0^{2^*}\sum \limits _{i=1}^{k}(\frac{\lambda _{i+1}}{\lambda _i})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{|y|^{N-2}(1+|y-\zeta _i|^2)^{\frac{N+2}{2}}}\cdot \varepsilon \\&\qquad -2^*C_0^{2^*}\sum \limits _{i=1}^{k}(\frac{\lambda _{i+1}}{\lambda _i})^{\frac{N-2}{2}} \int _{{\mathbb {R}}^N}\frac{1}{(1+|y|^2)^{\frac{N+2}{2}}}\frac{1}{(1+|\zeta _i|^2)^{\frac{N-2}{2}}}\cdot \varepsilon +o(\varepsilon ), \end{aligned} \end{aligned}$$

Proof

It is easy to see that

$$\begin{aligned} \int _{\Omega }\left| \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right| ^{2^*} =\bigcup \limits _{j=1}^{k+1}\int _{A_j}|\sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma |^{2^*} +O(\delta _1^N). \end{aligned}$$

Observe that

$$\begin{aligned} \begin{aligned} \int _{A_k}V_\sigma U_{\delta _k,\xi _k}^{2^*-1}&= C_\mu C_0^{2^*-1}\sigma ^{\frac{N-2}{2}}\delta _k^{\frac{N+2}{2}}\int _{A_k}\frac{1}{({\sigma ^2|x|^{\beta _1}} +|x|^{\beta _2})^{\frac{N-2}{2}}}\frac{1}{(\delta _k^2+|x-\xi _k|^2)^{\frac{N+2}{2}}}\\&= C_\mu C_0^{2^*-1}\frac{\sigma ^{\frac{N-2}{2}}}{\delta _k^{\sqrt{\overline{\mu }-\mu }}}\int _{\frac{A_k}{\delta _k}}\frac{1}{({(\frac{\sigma }{\delta _k})^2|y|^{\beta _1}} +|y|^{\beta _2})^{\frac{N-2}{2}}}\frac{1}{(1+|y-\zeta _k|^2)^{\frac{N+2}{2}}}\\&= C_0^{2^*}(\frac{\overline{\lambda }}{\lambda _k})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{|y|^{N-2}(1+|y-\zeta _k|^2)^{\frac{N+2}{2}}}\cdot \varepsilon +o(\varepsilon ), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \int _{A_j}V_\sigma U_{\delta _i,\xi _i}^{2^*-1}=o(\varepsilon ), ~\text {if}~i\ne k,~\text {or}~j\ne k. \end{aligned}$$

From [26], for $1\le i<j\le k$ we also have

$$\begin{aligned}&\int _{A_l}U_{\delta _j,\xi _j}U_{\delta _i,\xi _i}^{2^*-1}+o(\varepsilon )\\&\quad ={\left\{ \begin{array}{ll} C_0^{2^*}(\frac{\lambda _{i+1}}{\lambda _i})^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}\frac{1}{|y|^{N-2}(1+|y-\zeta _i|^2)^{\frac{N+2}{2}}}\cdot \varepsilon +o(\varepsilon ) \quad &{} \text {if} \,\,j=i+1,i=l,\\ o(\varepsilon )\quad &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Using (B.6), (B.8) and the above three equalities, the proof of (B.9) is contained in Lemma 6.2 in [26]. $\square$

Lemma B.7

For $\mu , \sigma , \delta _i \rightarrow 0$ there holds, uniformly in compact subsets of $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left| \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right| ^{2^*}\ln \left| \sum \limits _{i=1}^{k}(-1)^{i-1}PU_{\delta _i,\xi _i}+(-1)^k PV_\sigma \right| \\&\quad =-\frac{N-2}{2}\ln \sigma \cdot \int _{{\mathbb {R}}^N} V_1^{2^*}-\frac{N-2}{2}\ln (\delta _1 \delta _2\dots \delta _k)\cdot \int _{{\mathbb {R}}^N} U_{1,0}^{2^*}\\&\qquad +\int _{{\mathbb {R}}^N} V_1^{2^*}\ln V_1+k\int _{{\mathbb {R}}^N} U_{1,0}^{2^*}\ln U_{1,0}+o(1). \end{aligned} \end{aligned}$$

Proof

The proof is similar to the one of [6, Lemma A.9]. $\square$