Non-degeneracy of solution for critical Lane–Emden systems with linear perturbation

Abstract

In this paper, we consider the following elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), &{}\quad {\hbox {in}}\; \Omega , \\ -\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &{}\quad {\hbox {in}}\;\Omega , \\ u=v=0,&{}\quad {\hbox {on}}\; \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\ge 3$, $\epsilon$ is a small parameter, $\alpha$, $\beta _1$ and $\beta _2$ are real numbers, (p, q) is a pair of positive numbers lying on the critical hyperbola

$$\begin{aligned} \begin{aligned} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{aligned} \end{aligned}$$

We first revisited the blowing-up solutions constructed in Kim and Pistoia (J Funct Anal 281(2):58, 2021) and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.

Appendices

Appendix A. Preliminaries results

Let G be the Green’s function of the Laplacian $-\Delta$ in $\Omega$ with Dirichlet boundary condition. And H be its regular part, then $G(x,y)=S(x,y)-H(x,y)$ with $S(x,y) = \frac{\gamma _{N}}{|x-y|^{N-2}}$.

In addition, we introduce a function $\tilde{G}=\tilde{G}_{\Omega }:\Omega \times \Omega \rightarrow \mathbb {R}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _{x}\tilde{G}(x,y)=G^{p}(x,y), &{}\quad {\hbox {for}}\; x\in \Omega , \\ \tilde{G}=0,&{}\quad {\hbox {for}}\; x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

for each $y\in \Omega$, and its regular part $\tilde{H} = \tilde{H}_{\Omega }:\Omega \times \Omega \rightarrow \mathbb {R}$ by

$$\begin{aligned} \begin{aligned} \tilde{H}=\frac{\tilde{\gamma }_{N,p}}{|x-y|^{(N-2)p-2}}-\tilde{G}(x,y), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \tilde{\gamma }_{N,p}:=\frac{\gamma _{N}^{p}}{((N-2)p-2)(N-(N-2)p)}>0. \end{aligned} \end{aligned}$$

Lemma A.1

Let $\widehat{H}:\Omega \times \Omega \rightarrow \mathbb {R}$ be a smooth function such that

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _{x}\widehat{H}(x,y)=0, &{}\quad {\hbox {for}}\; x\in \Omega ,\\ \widehat{H}(x,y)=\frac{1}{|x-y|^{(N-2)p-2}},&{}\quad {\hbox {for}}\; x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

for any $y\in \Omega$. Then we have

$$\begin{aligned} P U_{1}(x)=U_{1}(x)-a_{N, p} \mu _{1}^{\frac{N p}{q+1}} \widehat{H}\left( x, P_{1}\right) +o\left( \mu ^{\frac{N p}{q+1}}\right) , \end{aligned}$$

and

$$\begin{aligned} P V_{1}(x)=V_{1}(x)-\left( \frac{b_{N, p}}{\gamma _{N}}\right) \mu _{1}^{\frac{N}{q+1}} H\left( x, P_{1}\right) +o\left( \mu ^{\frac{N}{q+1}}\right) , \end{aligned}$$

where $PU_1, PV_1$ are the same as in (1.6).

Proof

Lemma A.1 is proved in [13] by using comparison principle. We give a different proof. Indeed, we have

$$\begin{aligned} PV_{1}(x)=\int _{\Omega }G(x,y)U_{1}^{q}dy,\quad V_{1}(x)=\int _{\mathbb R^{N}}S(x,y)U_{1}^{q}dy. \end{aligned}$$

Thus

$$\begin{aligned} PV_{1}(x)-V_{1}(x)&=-\int _{\Omega ^{c}}S(x,y)U_{1}^{q}dy-\int _{\Omega }H(x,y)U_{1}^{q}dy:=\mathcal {A}_{1}+ \mathcal {A}_{2}. \end{aligned}$$

Since $dist(P_{1},\Omega )>\delta _{2}>0,$ we have

$$\begin{aligned} | \mathcal {A}_{1} |&\le C\int _{\Omega ^{c}}\frac{1}{|x-y|^{N-2}}\frac{\mu _{1}^{\frac{Npq}{q+1}}}{(1 + |y-P_{1}|)^{\frac{N(p+1)q}{q+1}} } \le C\mu _{1}^{\frac{Npq}{q+1}}. \end{aligned}$$

And

$$\begin{aligned} \mathcal {A}_{2} = \int _{\tilde{\Omega }} H(x, \mu _{1}y+P_{1} )U_{1,0}^{q}(y)dy\mu _{1}^{\frac{N}{q+1}}, \end{aligned}$$

where $\tilde{\Omega } = \mu _{1}^{-1}( \Omega -P_{1} )$. Since $H(x, \mu _{1}y+P_{1} )U_{1,0}^{q}(y) \le CU_{1,0}^{q}(y)$, by using the dominated convergence theorem, we have

$$\begin{aligned} \int _{\tilde{\Omega }} H(x, \mu _{1}y+P_{1} )U_{1,0}^{q}(y)dy = H(x,P_{1})\int _{\mathbb {R}^{N}}U_{1,0}^{q}(y)dy + o(1). \end{aligned}$$

So,

$$\begin{aligned} PV_{1}(x)-V_{1}(x) = \mu _{1}^{\frac{N}{q+1}}H(x,P_{1})\int _{\mathbb {R}^{N}}U_{1,0}^{q}(y)dy +o\left( \mu _{1}^{\frac{N}{q+1}}\right) . \end{aligned}$$

Moreover, $\displaystyle \int _{\mathbb {R}^{N}}U_{1,0}^{q}(y)dy = \frac{b_{N, p}}{\gamma _{N}}$.

$\square$

Similar, we can prove

Lemma A.2

$\frac{\partial P V_{1}}{\partial x_{i}}(x) = \frac{\partial V_{1}}{ \partial x_{i}}(x)-\left( \frac{b_{N, p}}{\gamma _{N}}\right) \mu _{1}^{\frac{N}{q+1}}\frac{\partial H}{ \partial x_{i}}\left( x, P_{1}\right) +o\left( \mu ^{\frac{N}{q+1}}\right) .$

Lemma A.3

(Theorem 2 in [14]) There exist positive constants $a_{N,p}$ and $b_{N,p}$ depending only on N and p such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \lim _{r\rightarrow \infty }r^{(N-2)p-2}U_{1,0}(r)=a_{N,p},\\ \lim _{r\rightarrow \infty }r^{N-2}V_{1,0}(r)=b_{N,p}, \end{array}\right. } \end{aligned}$$

where we wrote $U_{1,0}(x)= U_{1,0}(|x|)$, $V_{1,0}(x)= V_{1,0}(|x|)$ and $r = |x|$ by abusing notations. Furthermore,

$$\begin{aligned} b_{N,p}^{p}=a_{N,p}((N-2)p-2)(N-(N-2)p). \end{aligned}$$

Lemma A.4

(Theorem 1 in [6]])Set

$$\begin{aligned} (\Psi _{0},\Phi _{0}) = \bigg (x\cdot \nabla U_{1,0}+\frac{N U_{1,0}}{q+1},x\cdot \nabla V_{1,0}+\frac{N V_{1,0}}{p+1} \bigg ), \end{aligned}$$

and

$$\begin{aligned} (\Psi _{l},\Phi _{l}) = \left( \frac{\partial U_{1,0}}{\partial x_{l}},\frac{\partial V_{1,0}}{\partial x_{l}} \right) ,\,\,\,\hbox { for }l=1,\ldots ,N. \end{aligned}$$

Then the space of solutions to the linear system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \Psi =p V_{1,0}^{p-1}\Phi , &{}\quad {\hbox {in}}\; \mathbb {R}^{N}, \\ -\Delta \Phi =p U_{1,0}^{q-1}\Psi , &{}\quad {\hbox {in}}\; \mathbb {R}^{N}, \\ ( \Psi ,\Phi )\in \dot{W}^{2,\frac{p+1}{p}}( \mathbb {R}^{N} )\times \dot{W}^{2,\frac{q+1}{q}}( \mathbb {R}^{N} ), \end{array}\right. } \end{aligned}$$

is spanned by

$$\begin{aligned} \{ (\Psi _{0},\Phi _{0}),(\Psi _{1},\Phi _{1}) ,\ldots ,(\Psi _{N},\Phi _{N}) \}. \end{aligned}$$

Set

$$\begin{aligned} (\Psi _{1,0},\Phi _{1,0} ) = \left( \mu _{1}^{-\frac{N}{q+1}-1}\Psi _{0}(\mu _{1}^{-1}(x-P_{1})) , \mu _{1}^{-\frac{N}{p+1}-1}\Phi _{0}(\mu _{1}^{-1}(x-P_{1})) \right) , \end{aligned}$$

and

$$\begin{aligned} (\Psi _{1,l},\Phi _{1,l} ) = (\mu _{1}^{-\frac{N}{q+1}-1}\Psi _{l}(\mu _{1}^{-1}(x-P_{1})) , \mu _{1}^{-\frac{N}{p+1}-1}\Phi _{l}(\mu _{1}^{-1}(x-P_{1})) ), \end{aligned}$$

for $l=1,\ldots ,N$. Let the pair $( P\Psi _{1,l},P\Phi _{1,l} )$ be the unique smooth solution of the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta P\Psi _{1,l} =p V_{1}^{p-1}P\Phi _{1,l}, &{}\quad {\hbox {in}}\; \Omega , \\ -\Delta P\Phi _{1,l} =p U_{1}^{q-1}P\Psi _{1,l}, &{}\quad {\hbox {in}}\; \Omega , \\ P\Psi _{1,l}=P\Phi _{1,l}=0, &{} \quad {\hbox {in}}\; \partial \Omega , \end{array}\right. } \end{aligned}$$

for $l=1,\ldots ,N$. Then, we have the following Lemma.

Lemma A.5

(Lemma 2.10. in [13])

$$\begin{aligned} P \Psi _{1, l}(x)= {\left\{ \begin{array}{ll}\Psi _{1, l}(x)+\frac{Np}{q+1}a_{N, p} \mu _{1}^{\frac{N p}{q+1}-1} \widehat{H}\left( x, P_{1}\right) +o\left( \mu ^{\frac{N p}{q+1}-1}\right) , &{} \quad {\text {for}}\; l=0, \\ \Psi _{1,l}(x)+a_{N, p} \mu _{1}^{\frac{N p}{q+1}} \partial _{P_1, l} \widehat{H}\left( x, P_{1}\right) +o\left( \mu ^{\frac{N p}{q+1}}\right) , &{} \quad {\text {for}}\; l=1, \ldots , N,\end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} P \Phi _{1,l}(x)= {\left\{ \begin{array}{ll}\Phi _{1,l}(x)+\left( \frac{N}{q+1}\frac{b_{N, p}}{\gamma _{N}}\right) \mu _{1}^{\frac{N}{q+1}-1} H\left( x, P_{1}\right) +o\left( \mu ^{\frac{N}{q+1}-1}\right) , &{} \quad {\text {for}}\; l=0, \\ \Phi _{1,l}(x)+\left( \frac{b_{N, p}}{\gamma _{N}}\right) \mu _{1}^{\frac{N}{q+1}} \partial _{P_{1}, l} H\left( x, P_{1}\right) +o\left( \mu ^{\frac{N}{q+1}}\right) , &{} \quad {\text {for}}\; l=1, \ldots , N,\end{array}\right. } \end{aligned}$$

for $x \in \Omega$. Here, $\partial _{P_1, l} \widehat{H}(x, P_1)$ and $\partial _{P_1, l} H(x, P_1)$ stand for the l-th components of $\nabla _{P_1} \widehat{H}(x, P_1)$ and $\nabla _{P_1} H(x, P_1)$, respectively.

Recall that

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta PU_{d_{1},P_{1}} = PV_{1}^{p}, &{} \quad {\hbox {in}}\; \Omega ,\\ PU_{d_{1},P_{1}}=0,&{} \quad {\hbox {on}}\;\partial \Omega . \end{array}\right. } \end{aligned}$$

Let $\tilde{G}_{d_{1},P_{1}}:=\Omega \rightarrow \mathbb {R}^{N}$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \tilde{G}_{d_{1},P_{1}}(x) = d_{1}^{\frac{N}{q+1}}G(x,P_{1}), &{} \quad {\hbox {for}}\; x\in \Omega , \\ \tilde{G}=0,&{} \quad {\hbox {for}}\; x\in \partial \Omega . \end{array}\right. } \end{aligned}$$

and $\tilde{H}_{d_{1},P_{1}}:=\Omega \rightarrow \mathbb {R}^{N}$ be its regular part given by

$$\begin{aligned} \begin{aligned} \tilde{H}_{d_{1},P_{1}} = d_{1}^{\frac{N}{q+1}}\frac{\tilde{\gamma }_{N,p}}{|x-P_{1}|^{(N-2)p-2}}-\tilde{G}_{d_{1},P_{1}}(x). \end{aligned} \end{aligned}$$

Then we have the following Lemma.

Lemma A.6

(Lemma 2.12. in [13]) For any $x\in \Omega$, we have

$$\begin{aligned} P U_{{d_1}, {P_1}}(x)=\sum _{i=1}^{k} U_{i}(x)-\mu ^{\frac{N p}{q+1}}\left( \frac{b_{N, p}}{\gamma _{N}}\right) ^{p} \widetilde{H}_{{d_1}, {P_1}}(x)+o\left( \mu ^{\frac{N p}{q+1}}\right) . \end{aligned}$$

Appendix B. Some important estimation

Recall that $S(x,y) = \frac{\gamma _{N}}{|x-y|^{N-2}}$. And let $\epsilon _{n} \rightarrow 0$, as $n\rightarrow +\infty$.

Lemma B.1

$\frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}} }}{\partial x_i}(P_{1,k_{n}}) = o( \mu _{1,k_{n}}^{\frac{Np}{q+1}} )$.

Proof

We have

$$\begin{aligned} \begin{aligned} \frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}} }}{\partial x_i}(x) =\int _{\Omega } \frac{\partial G }{\partial x_i}(x,y)PV_{1,k_{n}}^{p}(y)dy, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{\partial U_{1,k_{n} }}{\partial x_i}(x) =\int _{\mathbb {R}^{N}} \frac{\partial S }{\partial x_i}(x,y)V_{1,k_{n}}^{p}(y)dy. \end{aligned} \end{aligned}$$

So

$$\begin{aligned}&\frac{\partial ( PU_{d_{1,k_{n}},P_{1,k_{n}} } -U_{1,k_{n} } ) }{\partial x_i} \\&\quad = -\int _{\Omega ^{c}} \frac{\partial S }{\partial x_i}(x,y)V_{1,k_{n}}^{p}(y)dy + \int _{\Omega } \frac{\partial S }{\partial x_i}(x,y) \big (pV_{1,k_{n}}^{p} - V_{1,k_{n}}^{p}\big )(y)dy \\&\qquad -\int _{\Omega }\frac{\partial H }{\partial x_i}(x,y)PV_{1,k_{n}}^{p}(y):=\mathcal {I}_1+\mathcal {I}_2+\mathcal {I}_3. \end{aligned}$$

It is easy to check that

$$\begin{aligned} \mathcal {I}_1&= -\mu _{k_{n}}^{\frac{Np}{q+1}}\int _{\Omega ^{c}} \frac{\partial S }{\partial x_i}(x,y)\frac{b_{N,p}^{p}}{\gamma ^{p}_{N}}S^{y}(y,P_{0}) + o\left( \mu _{k_{n}}^{\frac{Np}{q+1}}\right) , \\ \mathcal {I}_2&= \mu _{k_{n}}^{\frac{Np}{q+1}}\int _{\Omega } \frac{\partial S }{\partial x_i}(x,y)\frac{b_{N,p}^{p}}{\gamma _{N}^{p}}(G(y,P_{0})^{p} - S^{p}(y,P_{1}) )+o\left( \mu _{k_{n}}^{\frac{Np}{q+1}}\right) , \\ \mathcal {I}_3&= -\mu _{k_{n}}^{\frac{Np}{q+1}}\int _{\Omega }\frac{\partial H }{\partial x_i}(x,y)G^{p}(y,P_{0})dy+o\left( \mu _{k_{n}}^{\frac{Np}{q+1}}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}} }}{\partial x_i}(x) = \mu _{k_{n}}^{\frac{Np}{q+1}}\frac{\partial \widetilde{H}}{\partial x_{i}}(x,P_{0})+o\left( \mu _{k_{n}}^{\frac{Np}{q+1}} \right) . \end{aligned}$$

From [13], we get that $\frac{\partial \widetilde{H}}{\partial x_{i}}(P_{0},P_{0}) =0$. So $\frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}} }}{\partial x_i}(P_{1,k_{n}}) = o( \mu _{k_{n}}^{\frac{Np}{q+1}} )$. $\square$

Lemma B.2

For $x\in B_{\mu _{1,k_{n}}^{\alpha }}(P_{1,k_{n}})$, we have $\frac{\partial (PU_{d_{1,k_{n}},P_{1,k_{n}}} - U_{1,k_{n} })}{\partial x_i}(x) = O(\mu _{1,k_{n}}^{\frac{Np}{q+1}})$, as $n\rightarrow +\infty$, where $\alpha$ is a small fixed positive constant.

Proof

We have

$$\begin{aligned} \begin{aligned} \frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}} }}{\partial x_i}(x) =\int _{\Omega } \frac{\partial G }{\partial x_i}(x,y)PV_{1,k_{n}}^{p}(y)dy, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{\partial U_{1,k_{n} }}{\partial x_i}(x) =\int _{\mathbb {R}^{N}} \frac{\partial S }{\partial x_i}(x,y)V_{1,k_{n}}^{p}(y)dy. \end{aligned} \end{aligned}$$

So

$$\begin{aligned} \begin{aligned} \frac{\partial ( U_{1,k_{n} } -PU_{d_{1,k_{n}},P_{1,k_{n}}) }}{\partial x_i}&= \int _{\Omega ^{c}} \frac{\partial S }{\partial x_i}(x,y)V_{1,k_{n}}^{p}(y)dy + \int _{\Omega } \frac{\partial S }{\partial x_i}(x,y) \big (V_{1,k_{n}}^{p} - PV_{1,k_{n}}^{p}\big )(y)dy \\&\quad +\int _{\Omega }\frac{\partial H }{\partial x_i}(x,y)PV_{1,k_{n}}^{p}(y):=\mathcal {I}_1+\mathcal {I}_2+\mathcal {I}_3. \end{aligned} \end{aligned}$$

Direct computations shows that

$$\begin{aligned} \begin{aligned} |\mathcal {I}_1|&\le C\mu _{1,k_{n}}^{\frac{Np}{q+1}}\int _{\Omega ^{c}}\frac{1}{|x-y|^{N-1}}\frac{1}{ (1+ |y-P_{1,k_{n}}| )^{(N-2)p} } \le C\mu _{1,k_{n}}^{\frac{Np}{q+1}}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} | \mathcal {I}_2 |&\le C\int _{\Omega }\frac{1}{|x-y|^{N-1}}\bigg ( \frac{\mu _{1,k_{n}}^{-\frac{N(p-1)}{p+1}}}{( 1+\mu _{1,k_{n}}^{-1}|y-P_{1,k_{n}}|)^{ (N-2)(p-1) }}\mu _{1,k_{n}}^{\frac{N}{q+1}}+ O\left( \mu _{1,k_{n}}^{\frac{Np}{q+1}} \right) \bigg )dy \le C\mu _{1,k_{n}}^{\frac{Np}{q+1}}. \end{aligned} \end{aligned}$$

Since $x\in B_{\mu _{1,k_{n}}^{\alpha }}(P_{1,k_{n}})$ and $dist(P_{1,k_{n}},\Omega )>\delta _2$, we have $|\frac{\partial H }{\partial x_i}(x,y)|\le C$. Thus

$$\begin{aligned} \begin{aligned} |\mathcal {I}_3|&\le C\int _{\Omega }\frac{\mu _{1,k_{n}}^{-\frac{Np}{p+1}}}{\big ( 1 + \mu _{1,k_{n}}^{-1}| y-P_{1,k_{n}}|\big )^{(N-2)p}}dy \le C\mu _{1,k_{n}}^{\frac{Np}{q+1}}. \end{aligned} \end{aligned}$$

$\square$

Lemma B.3

For $x\in B_{\mu _{1,k_{n}}^{\alpha }}(P_{1,k_{n}})$, we have

$$\begin{aligned} \frac{\partial (P\Phi _{1,k_{n},0} - \Phi _{1,k_{n},0})}{\partial x_i}(x) = O\left( \mu _{1,k_{n}}^{\frac{N}{q+1}-1}\right) ,\quad \frac{\partial (P\Phi _{1,k_{n},j} - \Phi _{1,k_{n},j})}{\partial x_i}(x) = O\left( \mu _{1,k_{n}}^{\frac{N}{q+1}}\right) , \end{aligned}$$

for $j=1,\ldots ,N$, as $n\rightarrow +\infty$, where $\alpha$ is a small fixed positive constant.

Proof

We have

$$\begin{aligned} \begin{aligned}&\frac{\partial (P\Phi _{1,k_{n},0} - \Phi _{1,k_{n},0})}{\partial x_i}(x)\\&\quad = \int _{\Omega ^{c}} \frac{\partial S }{\partial x_i}(x,y)qU_{1,k_{n} }^{q-1}\Psi _{1,n,0}(y)dy +\int _{\Omega }\frac{\partial H }{\partial x_i}(x,y)qU_{1,k_{n} }^{q-1}\Psi _{1,n,0}(y)\\&\quad \le C\mu _{1,k_{n}}^{\frac{Npq}{q+1}-1}\int _{\Omega ^{c}}\frac{1}{|x-y|^{N-1}}\frac{1}{ (1+ |y-P_{1,k_{n}}| )^{\frac{N(p+1)q }{q+1}} }\\&\qquad +C\int _{\Omega }\frac{\mu _{1,k_{n}}^{-\frac{Np}{p+1}-1}}{\big ( 1 + \mu _{1,k_{n}}^{-1}| y-P_{1,k_{n}}|\big )^{\frac{N(p+1)q }{q+1}}}dy\\&\quad \le C\mu _{1,k_{n}}^{\frac{Npq}{q+1}-1}+C\mu _{1,k_{n}}^{\frac{N}{q+1}-1}.\\ \end{aligned} \end{aligned}$$

For $j=1,\ldots ,N$, we have

$$\begin{aligned} \begin{aligned}&\frac{\partial (P\Phi _{1,k_{n},j} - \Phi _{1,k_{n},j})}{\partial x_i}(x)\\&\quad = \int _{\Omega ^{c}} \frac{\partial S }{\partial x_i}(x,y)pU_{1,k_{n} }^{q-1}\Psi _{1,n,j}(y)dy +\int _{\Omega }\frac{\partial H }{\partial x_i}(x,y)pU_{1,k_{n} }^{q-1}\Psi _{1,n,j}(y)\\&\quad \le C\mu _{1,k_{n}}^{\frac{Npq}{q+1}}\int _{\Omega ^{c}}\frac{1}{|x-y|^{N-1}}\frac{1}{ (1+ |y-P_{1,k_{n}}| )^{\frac{N(p+1)q }{q+1}+1} }\\&\qquad +\int _{B_{\delta }(P_{1,k_{n}})}\frac{\partial H }{\partial x_i}(x,y)pU_{1,k_{n} }^{q-1}\Psi _{1,n,j}(y)+\int _{\Omega -B_{\delta }(P_{1,k_{n}})}\frac{\partial H }{\partial x_i}(x,y)pU_{1,k_{n} }^{q-1}\Psi _{1,n,j}(y)\\&\quad \le C\mu _{1,k_{n}}^{\frac{Npq}{q+1}}+O\left( \mu _{1,k_{n}}^{\frac{Npq}{q+1}} \right) +O\left( \mu _{1,k_{n}}^{\frac{N}{q+1}} \right) , \end{aligned} \end{aligned}$$

where $\delta$ is a fix small constant. $\square$

Lemma B.4

We have

$$\begin{aligned} \mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},0}(x) - \mu _{1,k_{n}} \Psi _{1,k_{n},0}(x) = O\big (\mu _{1,k_{n}}^{\frac{Np}{q+1}}\big ) \end{aligned}$$

and

$$\begin{aligned} \mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},j}(x) - \mu _{1,k_{n}} \Psi _{1,k_{n},j}(x) = O\left( \mu _{1,k_{n}}^{\frac{Np}{q+1} +1}\right) \end{aligned}$$

for $j\ne 0$.

Proof

for We have

$$\begin{aligned}&\mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},0}(x) - \mu _{1,k_{n}} \Psi _{1,k_{n},0}(x) \\&\quad =\int _{\Omega }G(x,y)p( \mu (PV_{1,k_{n}})^{p-1}P\Phi _{1,k_{n},0} ) - \int _{\mathbb {R}^{N}}s(x,y)p \mu (V_{1,k_{n}})^{p-1}\Phi _{1,k_{n},0} \\&\quad = -\int _{\Omega ^{c}}S(x,y)p (V_{1,k_{n}})^{p-1} \mu \Phi _{1,k_{n},0}-\int _{\Omega }H(x,y)p( (PV_{1,k_{n}})^{p-1} \mu _{1,k_{n}} P\Phi _{1,k_{n},0} )\\&\qquad + \int _{\Omega }S(x,y)p( (PV_{1,k_{n}})^{p-1}P\Phi _{1,k_{n},0} - \mu (V_{1,k_{n}})^{p-1}\Phi _{1,k_{n},0} ) \\&\quad =I_1+I_2+I_3. \end{aligned}$$

It is easy to check that $|I_{1}|+|I_{2}|+|I_{3}| = O(\mu _{1,k_{n}}^{\frac{Np}{q+1}})$. Simlarly, we can prove that $\mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},j}(x) - \mu _{1,k_{n}} \Psi _{1,k_{n},j}(x) = O(\mu _{1,k_{n}}^{\frac{Np}{q+1} +1})$. $\square$

Lemma B.5

We have

$$\begin{aligned}&\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}U^{q}_{1}\frac{\partial \sum _{j=1}^{N} b_{j,n}\mu _{1,k_{n}} (\widetilde{P\Psi }_{1,k_{n},j}-\Psi _{1,k_{n},j}) }{\partial x_{i}}(x)\\&\qquad + \int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}\sum _{j=1}^{N}qU_{1,k_{n}}^{q-1}\Psi _{1,k_{n},j}\frac{\partial (PU_{d_{1,k_{n}},P_{1,k_{n}}} - U_{1,k_{n}}) }{ \partial x_{i}} \\&\quad =-\frac{\mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}}{p+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}\sum _{j=1}^{N}b_{j,n}\frac{\partial ^{2}\tau }{\partial x_{i}\partial x_{j}}(P_{0})+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) . \end{aligned}$$

Proof

Firstly, we estimate $\frac{\partial \mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},j}}{\partial x_{i}}(x) - \frac{\partial \mu _{1,k_{n}} \Psi _{1,k_{n},j}}{\partial x_{i}}(x)$.

$$\begin{aligned}&\frac{\partial \mu _{1,k_{n}} \widetilde{P\Psi }_{1,k_{n},j}}{\partial x_{i}}(x) - \frac{\partial \mu _{1,k_{n}} \Psi _{1,k_{n},j}}{\partial x_{i}}(x) \\&\quad =\int _{\Omega }\frac{\partial G}{\partial x_{i}}(x,y)p\big ( \mu _{1,k_{n}}(PV_{1,k_{n}})^{p-1}P\Phi _{1,k_{n},j} \big ) - \int _{\mathbb {R}^{N}}\frac{\partial S}{\partial x_{i}}(x,y)p \mu _{1,k_{n}}(V_{1,k_{n}})^{p-1}\Phi _{1,k_{n},j}\\&\quad =-\int _{\Omega ^{c}}\frac{\partial S}{\partial x_{i}}(x,y)p (V_{1,k_{n}})^{p-1} \mu _{1,k_{n}}\Phi _{1,k_{n},j}-\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)p\big ( (PV_{1,k_{n}})^{p-1} \mu _{1,k_{n}} P\Phi _{1,k_{n},j} \big ) \\&\qquad + \int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)p\big ( \mu _{1,k_{n}}(PV_{1,k_{n}})^{p-1}P\Phi _{1,k_{n},j} - \mu _{1,k_{n}}(V_{1,k_{n}})^{p-1}\Phi _{1,k_{n},j} \big ) \\&\quad =I_1+I_2+I_3. \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_1 =\mu _{1,k_{n}}^{\frac{Np}{q+1} +1 }\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\Omega ^{c}}\frac{\partial S}{\partial x_{i}}(x,y)pS^{p-1}(P_{0},y)\frac{\partial S}{\partial x_{j}}(P_{0},y) +o\left( \mu _{1,k_{n}}^{\frac{Np}{q+1} +1 }\right) , \\ I_2= \mu _{1,k_{n}}^{\frac{Np}{q+1} +1 }\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)p G(P_{0},y)^{p-1} \frac{\partial G}{\partial x_{j}}(P_{0},y) )+o\left( \mu _{1,k_{n}}^{\frac{Np}{q+1} +1 }\right) . \end{aligned}$$

Now we rewrite $I_3$

$$\begin{aligned} I_{3}&=\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)\mu _{1,k_{n}} p\big ( (PV_{1,k_{n}})^{p-1} -(V_{1,k_{n}})^{p-1} \big )\Phi _{1,k_{n},j} \\&\quad +\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)\mu _{1,k_{n}} p (PV_{1,k_{n}})^{p-1} (P\Phi _{1,k_{n},j} - \Phi _{1,k_{n},j})\\&=I_{31}+I_{32}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_{32}=\mu _{1,k_{n}}^{\frac{Np}{q+1} +1 }\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p} \int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)pG^{p-1}(P_{0},y)\frac{\partial H}{\partial x_{j}}(P_{0},y) + o\left( \mu _{1,k_{n}}^{\frac{Np}{q+1} +1 } \right) . \end{aligned}$$

Now we rewrite $I_{31}$

$$\begin{aligned} I_{31}&=\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)\mu _{1,k_{n}} p\big ( (PV_{1,k_{n}})^{p-1} -(V_{1,k_{n}})^{p-1}\\&\quad + (p-1)V_{1,k_{n}}^{p-2}\frac{b_{N,p}}{\gamma _{N}}\mu ^{\frac{N}{q+1}}H(P_{0},y) )\Phi _{1,k_{n},j}\\&\quad - \int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)\mu ^{\frac{N}{q+1}+1}pV_{1,k_{n}}^{p-2}\frac{b_{N,p}}{\gamma _{N}}(p-1)V_{1,k_{n}}^{p-2}\frac{b_{N,p}}{\gamma _{N}}\Phi _{1,k_{n},j}\\&= I_{311}+I_{312}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_{311} =&\,-\mu _{1,k_{n}}^{\frac{Np}{q+1}+1}\int _{\Omega }\frac{\partial S}{\partial x_{i}}(P_{0},y)p\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}( G^{p-1}(P_{0},y) -S^{p-1}(P_{0},y)\\&+(p-1)S^{p-2}(P_{0},y)H(P_{0},y) )\frac{\partial S}{\partial x_{j}}(P_{0},y) + o\left( \mu _{1,k_{n}}^{\frac{Np}{q+1}+1} \right) . \end{aligned}$$

Now we estimate $\frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}}} }{\partial x_{i}} -\frac{\partial U_{1,k_{n}} }{\partial x_{i}}$,

$$\begin{aligned}&\frac{\partial PU_{d_{1,k_{n}},P_{1,k_{n}}} }{\partial x_{i}} -\frac{\partial U_{1,k_{n}} }{\partial x_{i}}(x) \\&\quad = \int _{\Omega }\frac{\partial G}{\partial x_{i}}(x,y)(PV_{1,k_{n}})^{p}-\int _{\mathbb {R}^{N}}\frac{\partial S}{\partial x_{i}}(x,y)V_{1,k_{n}}^{p} \\&\quad =-\int _{\Omega ^{c}}\frac{\partial S}{\partial x_{i}}(x,y)V_{1,k_{n}}^{p}-\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)(PV_{1,k_{n}})^{p}\\&\qquad +\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}) \\&\quad = I_{4}+I_{5}+I_{6}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_{4} =&\,-\int _{\Omega ^{c}}\frac{\partial S}{\partial x_{i}}(P_{0},y)V_{1,k_{n}}^{p}-\mu _{1,k_{n}}^{\frac{Np}{q+1}}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\\&\int _{\Omega ^{c}}\sum _{z=1}^{N}\frac{\partial ^2 S}{\partial x_{i}\partial x_{z}}(P_{0},y)(x-P_{0})_{z}S^{p}(P_{0},y) \\&+O\left( \mu _{1,k_{n}}^{\frac{Np}{q+1}}| x-P_{0}|^{2} \right) . \end{aligned}$$

and

$$\begin{aligned} I_{5} =&\,-\int _{\Omega }\frac{\partial H}{\partial x_{i}}(P_{0},y)(PV_{1,k_{n}})^{p}- \mu _{1,k_{n}}^{\frac{Np}{q+1}}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\\&\int _{\Omega }\sum _{z=1}^{N}\frac{\partial ^2 H}{\partial x_{i}\partial x_{z}}(P_{0},y)(x-P_{0})_{z}G^{p}(P_{0},y)\\&+O\left( \mu _{1,k_{n}}^{\frac{Np}{q+1}}| x-P_{0}|^{2} \right) . \end{aligned}$$

Thus

$$\begin{aligned}&\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{1})}q\mu \sum _{j=1}^{N}U_{1,k_{n}}^{q-1}\Psi _{1,k_{n},j}I_{4}dx \\&\quad =\mu ^{\frac{N(p+1)}{q+1}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q} \int _{\Omega ^{c}}\frac{\partial ^2 S}{\partial x_{j}\partial x_{z}}(P_{0},y)S^{p}(P_{0},y)+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1} \right) , \end{aligned}$$

$$\begin{aligned}&\int _{B_{\mu ^{\alpha }}(P_{1})}q\mu \sum _{j=1}^{N}U_{1,k_{n}}^{q-1}\Psi _{1,k_{n},j}I_{5}dx \\&\quad = \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p} \int _{\mathbb {R}^{N}}U_{0,1}^{q}\int _{\Omega }\frac{\partial ^2 H}{\partial x_{i}\partial x_{j}}(P_{0},y)G^{p}(P_{0},y)+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1} \right) . \end{aligned}$$

Now we estimate $I_{6}$.

$$\begin{aligned} I_{6}&= \int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y)((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p})dy \\&= \int _{\Omega }-\frac{\partial S}{\partial y_{i}}(x,y)\big ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}\big )dy \\&=-\int _{\partial \Omega }S(x,y)\big ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}\big )\nu _{i}ds+\int _{\Omega }S(x,y)\frac{\partial ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p})}{\partial y_{i}}dy \\&= -\int _{\Omega }S(x,y)\big ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}\big )\nu _{i}dy+\int _{\Omega }S(x,y)p\Big ( (PV_{1,k_{n}})^{p-1}\frac{\partial PV_{1,k_{n}}}{\partial y_{i}}\\&\quad -(V_{1,k_{n}})^{p-1}\frac{\partial V_{1,k_{n}}}{\partial y_{i}} \Big )dy = I_{61}+I_{62}. \end{aligned}$$

Direct computation shows that

$$\begin{aligned}&\int _{B_{\mu ^{\alpha }}(P_{0})}q\mu _{1,k_{n}} U_{1,k_{n}}^{q-1}(x)\Psi _{1,k_{n},j}(x) I_{61}dx\\&\quad = -\int _{\partial B_{\mu ^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}\int _{\partial \Omega }S(x,y)\big ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}\big )ds_{y}\nu _{j,x}ds_{x} \\&\qquad +\int _{B_{\mu ^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}\int _{\partial \Omega }\frac{\partial S}{\partial x_{j}}(x,y)\big ((PV_{1,k_{n}})^{p} - V_{1,k_{n}}^{p}\big )\nu _{i}ds_{y}dx \\&\quad = o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) +\mu _{1,k_{n}}^{\frac{N(p+1)}{(q+1)}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}dx\int _{\partial \Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y)(G^{p}(P_{0},y) \\&\qquad - S^{p}(P_{0},y))\nu _{i}ds_{y}. \end{aligned}$$

We rewrite $I_{62}$

$$\begin{aligned} I_{62}&=\int _{\Omega }S(x,y)p\big ( (PV_{1,k_{n}})^{p-1}\frac{\partial PV_{1,k_{n}}}{\partial y_{i}}-(V_{1,k_{n}})^{p-1}\frac{\partial V_{1,k_{n}}}{\partial y_{i}} \big )dy \\&= \int _{\Omega }S(x,y) p\big ((PV_{1,k_{n}})^{p-1} -V_{1,k_{n}}^{p-1} \big )\frac{\partial V_{1,k_{n}}}{\partial y_{i}}dy \\&\quad +\int _{\Omega }S(x,y) p(PV_{1,k_{n}})^{p-1} \left( \frac{\partial PV_{1,k_{n}}}{\partial y_{i}} -\frac{\partial V_{1,k_{n}}}{\partial y_{i}} \right) dy \\&= \int _{\Omega }S(x,y) p\Big ((PV_{1,k_{n}})^{p-1} -V_{1,k_{n}}^{p-1} + (p-1)\frac{b_{N,p}}{\gamma _{N}}V_{1,k_{n}}^{p-2}H(P_{0},y)\mu ^{\frac{N}{q+1}} \Big )\frac{\partial V_{1,k_{n}}}{\partial y_{i}}dy \\&\quad -\int _{\Omega }S(x,y) p(p-1)\frac{b_{N,p}}{\gamma _{N}}V_{1,k_{n}}^{p-2}H(P_{0},y)\mu ^{\frac{N}{q+1}} \frac{\partial V_{1,k_{n}}}{\partial y_{i}}dy \\&\quad +\int _{\Omega }S(x,y) p(PV_{1,k_{n}})^{p-1} \left( \frac{\partial PV_{1,k_{n}}}{\partial y_{i}} -\frac{\partial V_{1,k_{n}}}{\partial y_{i}} \right) dy \\&= I_{621}+I_{622}+I_{623}. \end{aligned}$$

By directly computing, we get that

$$\begin{aligned}&\int _{B_{\mu ^{\alpha }}(P_{0})}q\mu _{1,k_{n}} U_{1,k_{n}}^{q-1}(x)\Psi _{1,k_{n},j}(x) I_{621}dx \\&\quad =\int _{\partial B_{\mu ^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}(x)I_{621}v_{j}ds_{x}\\&\qquad -\int _{B_{\mu ^{\alpha }}(P_{0})}U_{1,k_{n}}^{q}\int _{\Omega }\mu _{1,k_{n}} \frac{\partial S}{\partial x_{j}}(x,y) p\big ((PV_{1,k_{n}}\big )^{p-1} -V_{1,k_{n}}^{p-1} \\&\qquad + (p-1)\mu ^{\frac{N}{q+1}}\frac{b_{N,p}}{\gamma _{N}}V_{1,k_{n}}^{p-2}H(P_{0},y) )\frac{\partial V_{1,k_{n}}}{\partial y_{i}}dy \\&\quad = -\mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}dx\int _{\Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y) p\big (G^{p-1}(P_{0},y) -S^{p-1}(P_{0},y) \\&\qquad + (p-1)\frac{b_{N,p}}{\gamma _{N}}S^{p-2}(P_{0},y)H(P_{0},y) \big ) \frac{\partial S}{\partial y_{i}}(P_{0},y)dy+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) +O\left( \mu _{1,k_{n}}^{Np-\alpha }\right) , \end{aligned}$$

$$\begin{aligned}&\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}q\mu _{1,k_{n}} U_{1,k_{n}}^{q-1}(x)\Psi _{1,k_{n},j}(x) I_{623}dx\\&\quad =-\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y) p(PV_{1,k_{n}})^{p-1} \left( \frac{\partial PV_{1,k_{n}}}{\partial y_{j}} -\frac{\partial V_{1,k_{n}}}{\partial y_{i}} \right) dy\\&\qquad +O\left( \mu ^{Np-\alpha }\right) \\&\quad =\mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}dx\int _{\Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y)pG(P_{0},y)^{p-1}\frac{\partial H}{\partial y_{i}}(P_{0},y)\\&\qquad +o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) +O\left( \mu _{1,k_{n}}^{Np-\alpha }\right) , \end{aligned}$$

and

$$\begin{aligned}&\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}q\mu _{1,k_{n}} U_{1,k_{n}}^{q-1}(x)\psi _{i}(x)I_{622}dx \\&\quad =-\int _{\partial B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}(x)I_{622}dx+\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}U_{1,k_{n}}^{q}(x) \\&\qquad \times \int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y) p(p-1)\frac{b_{N,p}}{\gamma _{N}}V_{1,k_{n}}^{p-2}H(P_{0},y)\mu _{1,k_{n}}^{\frac{N}{q+1}+1} \frac{\partial V_{1,k_{n}}}{\partial y_{j}}dy \\&\quad =O\left( \mu _{1,k_{n}}^{Np-\alpha }\right) +\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}U_{1,k_{n}}^{q}(x)\int _{\Omega }\frac{\partial S}{\partial x_{i}}(x,y) p(p-1)\frac{b_{N,p}}{\gamma _{N}}V_{1,k_{n}}^{p-2}H(P_{0},y)\mu _{1,k_{n}}^{\frac{N}{q+1}+1} \frac{\partial V_{1,k_{n}}}{\partial y_{j}}dy. \end{aligned}$$

So, we get

$$\begin{aligned} \int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}q\mu _{1,k_{n}} U_{1,k_{n}}^{q-1}(x)\psi _{i}(x)I_{622}dx+\int _{B_{\mu ^{\alpha }}(P_{0})}\mu _{1,k_{n}} U_{1,k_{n}}^{q}(x)I_{312}dx=O\left( \mu _{1,k_{n}}^{Np-\alpha }\right) . \end{aligned}$$

Thus, we get

$$\begin{aligned}&\int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}U^{q}_{1}\frac{\partial \sum _{j=1}^{N} b_{j,n}\mu _{1,k_{n}} (\widetilde{P\Psi }_{1,k_{n},j}-\Psi _{1,k_{n},j}) }{\partial x_{i}}(x)\\&\qquad + \int _{B_{\mu _{1,k_{n}}^{\alpha }}(P_{0})}\mu _{1,k_{n}}\sum _{j=1}^{N}qU_{1,k_{n}}^{q-1}\Psi _{1,k_{n},j}\frac{\partial (PU_{d_{1,k_{n}},P_{1,k_{n}}} - U_{1,k_{n}}) }{ \partial x_{i}} \\&\quad =\mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}\sum _{j=1}^{N}b_{j,n}\left( \int _{\Omega ^{c}}\frac{\partial S}{\partial x_{i}}(P_{0},y)pS^{p-1}(P_{0},y)\frac{\partial S}{\partial x_{j}}(P_{0},y)\right. \\&\qquad +\int _{\Omega }\frac{\partial H}{\partial x_{i}}(P_{0},y)p G(P_{0},y)^{p-1} \frac{\partial G}{\partial x_{j}}(P_{0},y) + \int _{\Omega }\frac{\partial S}{\partial x_{i}}(P_{0},y)pG^{p-1}(P_{0},y)\frac{\partial H}{\partial x_{j}}(P_{0},y) \\&\qquad -\int _{\Omega }\frac{\partial S}{\partial x_{i}}(P_{0},y)p\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}( G^{p-1}(P_{0},y)\\&\qquad -S^{p-1}(P_{0},y) +(p-1)S^{p-2}(P_{0},y)H(P_{0},y) )\frac{\partial S}{\partial x_{i}}(P_{0},y) \\&\qquad +\int _{\Omega ^{c}}\frac{\partial ^2 S}{\partial x_{j}\partial x_{i}}(P_{0},y)S^{p}(P_{0},y)+\int _{\Omega }\frac{\partial ^2 H}{\partial x_{i}\partial x_{j}}(P_{0},y)G^{p}(P_{0},y) \\&\qquad +\int _{\partial \Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y)(G^{p}(P_{0},y) - S^{p}(P_{0},y))\nu _{i}ds -\int _{\Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y) p(G^{p-1}(P_{0},y) -S^{p-1}(P_{0},y) \\&\qquad + (p-1)\frac{b_{N,p}}{\gamma _{N}}S^{p-2}(P_{0},y)H(P_{0},y) ) \frac{\partial S}{\partial y_{i}}(P_{0},y)dy \\&\qquad +\int _{\Omega }\frac{\partial S}{\partial x_{j}}(P_{0},y)pG(P_{0},y)^{p-1}\frac{\partial H}{\partial y_{i}}(P_{0},y))+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) \\&\quad =-\frac{\mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}}{p+1}\left( \frac{b_{N,p}}{\gamma _{N}}\right) ^{p}\int _{\mathbb {R}^{N}}U_{0,1}^{q}\sum _{j=1}^{N}b_{j,n}\frac{\partial ^{2}\tau }{\partial x_{i}\partial x_{j}}(P_{0})+o\left( \mu _{1,k_{n}}^{\frac{N(p+1)}{q+1}+1}\right) , \end{aligned}$$

the last equals sign follows from Lemma B.6. $\square$

Lemma B.6

We have that

$$\begin{aligned}&\frac{\partial ^{2} \tau }{\partial x_{i}\partial x_{j}}(x) \\&\quad = -(p+1)\bigg (\int _{\Omega }\frac{\partial ^{2} H}{\partial x_{i}\partial x_{j}}(x,y) G^{p}(x,y)dy-\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y) pG^{p-1}(x,y)\frac{\partial G}{\partial x_{j}}(x,y) dy \\&\qquad - \int _{\Omega }\frac{\partial H}{\partial x_{j}}(x,y) pG^{p-1}(x,y)\frac{\partial S}{\partial x_{j}}(x,y) - \int _{\Omega }\frac{\partial H}{\partial y_{i}}(x,y) pG^{p-1}(x,y)\frac{\partial S}{\partial x_{j}}(x,y)\bigg ). \end{aligned}$$

Proof

Since

$$\begin{aligned} H (x,z)&= \int _{\Omega }G(x,y)G^{p}(z,y)dy -\int _{\mathbb {R}^{N}}S(x,y)S^{p}(z,y)dy \\&=-\int _{\Omega ^{c}}S(x,y)S^{p}(z,y)dy - \int _{\Omega }H(x,y)G^{p}(z,y)dy+\int _{\Omega }S(x,y)(G^{p}(z,y) \\&\quad - S^{p}(z,y))dy, \end{aligned}$$

We have

$$\begin{aligned} \tau (x)&= -\int _{\Omega ^{c}}S^{p+1}(x,y)dy - \int _{\Omega }H(x,y)G^{p}(x,y)dy\\&\quad +\int _{\Omega }S(x,y)(G^{p}(x,y) - S^{p}(x,y))dy. \end{aligned}$$

So

$$\begin{aligned} \frac{\partial \tau }{\partial x_{i}}&=-\int _{\Omega ^{c}}(p+1)S^{p}(x,y)\frac{\partial S}{\partial x_{i}}(x,y)dy - \int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)G^{p}(x,y)dy \\&\quad -\int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial x_{i}}(x,y)dy+\int _{\Omega }\frac{\partial S}{\partial x_{i}}(G^{p}(x,y) - S^{p}(x,y) )dy \\&\quad +\int _{\Omega }\Gamma (x,y)(pG^{p-1}(x,y)\frac{\partial G}{\partial x_{i}}(x,y) - p S^{p-1}(x,y)\frac{\partial S}{\partial x_{i}}(x,y) )dy. \end{aligned}$$

Since

$$\begin{aligned}&\int _{\Omega }\frac{\partial S}{\partial x_{i}}(G^{p}(x,y) - S^{p}(x,y) )dy \\&\quad =\int _{\Omega }S^{p+1}(x,y)\nu _{i}ds + \int _{\Omega }\Gamma (x,y)(pG^{p-1}(x,y)\frac{\partial G}{\partial y_{i}}(x,y) - p S^{p-1}(x,y)\frac{\partial S}{\partial y_{i}}(x,y) )dy \end{aligned}$$

and

$$\begin{aligned} (p+1)\int _{\Omega ^{c}}(p+1)S^{p}(x,y)\frac{\partial S}{\partial x_{i}}(x,y)dy = \int _{\Omega }S^{p+1}(x,y)\nu _{i}ds, \end{aligned}$$

we get that

$$\begin{aligned} \frac{\partial \tau }{\partial x_{i}}&= -\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)G^{p}(x,y)dy- \int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial x_{i}}(x,y)dy \\&\quad -\int _{\Omega }S(x,y)pG^{p-1}(x,y)\left( \frac{\partial H}{\partial x_{i}}H(x,y) + \frac{\partial H}{\partial y_{i}}(x,y) \right) dy. \end{aligned}$$

Since

$$\begin{aligned}&\int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial S}{\partial x_{i}}(x,y)dy \\&\quad =-\int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial y_{i}}(x,y)dy - \int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial H}{\partial y_{i}}(x,y)dy \\&\quad =\int _{\Omega }\frac{\partial H}{\partial y_{i}}(x,y)G^{p}(x,y) - - \int _{\Omega }H(x,y)pG^{p-1}(x,y)\frac{\partial H}{\partial y_{i}}(x,y)dy, \end{aligned}$$

we get that

$$\begin{aligned} \frac{\partial \tau }{\partial x_{i}} = -(p+1)(\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)G^{p}(x,y)dy+\int _{\Omega }\frac{\partial H}{\partial y_{i}}(x,y)G^{p}(x,y)dy ). \end{aligned}$$

Thus

$$\begin{aligned} \frac{1}{p+1}\frac{\partial ^2 \tau }{\partial x_{j}\partial x_{i}}&= -\int _{\Omega }\frac{\partial ^2 H}{\partial x_{j}\partial x_{i}}(x,y)G^{p}(x,y)dy--\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial x_{j}}(x,y)dy\\&\quad -\int _{\Omega }\frac{\partial ^2 H}{\partial x_{j}\partial y_{i}}(x,y)G^{p}(x,y)dy-\int _{\Omega }\frac{\partial H}{\partial y_{i}}(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial x_{j}}(x,y)dy. \end{aligned}$$

Since

$$\begin{aligned}&-\int _{\Omega }\frac{\partial ^2 H}{\partial x_{j}\partial y_{i}}(x,y)G^{p}(x,y)dy \\&\quad =\int _{\Omega }\frac{\partial H}{\partial x_{j}}(x,y)pG^{p-1}(x,y)\frac{\partial G}{\partial y_{i}}(x,y)dy, \end{aligned}$$

we get that

$$\begin{aligned}&\frac{\partial ^{2} \tau }{\partial x_{i}\partial x_{j}}(x) \\&\quad = -(p+1)\bigg (\int _{\Omega }\frac{\partial ^{2} H}{\partial x_{i}\partial x_{j}}(x,y) G^{p}(x,y)dy-\int _{\Omega }\frac{\partial H}{\partial x_{i}}(x,y) pG^{p-1}(x,y)\frac{\partial G}{\partial x_{j}}(x,y) dy \\&\qquad - \int _{\Omega }\frac{\partial H}{\partial x_{j}}(x,y) pG^{p-1}(x,y)\frac{\partial S}{\partial x_{j}}(x,y) - \int _{\Omega }\frac{\partial H}{\partial y_{i}}(x,y) pG^{p-1}(x,y)\frac{\partial S}{\partial x_{j}}(x,y)\bigg ). \end{aligned}$$

$\square$