Concentrated solution for some non-local and non-variational singularly perturbed problems

Abstract

In the present paper, we prove the existence of concentrated solution for the following non-local and non-variational singularly perturbed problem,

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\varepsilon ^2A(\varepsilon ^{-n}||u||^q_{L^q})\Delta u+ V(x)u=|u|^{p-1}u, &{}x\in \mathbb {R}^n,\\ 0<u\in H^1(\mathbb {R}^n), \lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \end{aligned}$$

(0.1)

where $n\ge 1$, $1<p<2^*-1$, $2\le q<2^*$, $A: (0, +\infty )\rightarrow [a, +\infty )$, $V(x): \mathbb {R}^n\rightarrow \mathbb {R}$ are two continuous functions, $a>0$ and $\varepsilon$ is a positive and small parameter. Problem (0.1) can be seen as a model to study the vibration of nonlinear string or bacteria’s density balance law in $\mathbb {R}^n$. The existence is based on the well-known Lyapunov–Schmidt reduction method. In order to make Lyapunov–Schmidt reduction method work well, existence and so-called non-degeneracy result of positive solutions for following problem will be needed.

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -A(\Vert U\Vert ^q_{L^q})\Delta U(y)+ \alpha U(y)=U^{p}(y), &{}y\in \mathbb {R}^n,\\ 0<U\in H^1(\mathbb {R}^n). \end{array} \right. \end{aligned}$$

(0.2)

In Sect. 2, existence and non-degeneracy results for positive solutions of problem (0.2) are proved. Compared to the standard Schrödinger equation, our results imply that the non-local term $A(\Vert U\Vert _{L^q}^q)$ has no effect on the non-degeneracy result of positive solutions of problem (0.2) when the solution U satisfies $\Vert U\Vert ^q_{L^q}A'(\Vert U\Vert ^q_{L^q})\ne \frac{2}{n}A(\Vert U\Vert ^q_{L^q})$ and that the non-local term $A(\Vert U\Vert _{L^q}^q)$ has significant effect on the existence result. Since the linearized operator L[U] for problem (0.2) is non-self-adjoint, our non-degeneracy result is a result about the Kernel space of the adjoint operator of linearized operator L[U] for problem (0.2) rather than the Kernel space of L[U] itself, which is essentially different from common non-local and self-adjoint equations, such as Kirchhoff equation, fractional Laplace equations and Choquard equations. Moreover, in our non-degeneracy result, we only assume that the solution U satisfies $\Vert U\Vert _{L^q}^qA'(\Vert U\Vert _{L^q}^q)\ne \frac{2}{n}A(\Vert U\Vert _{L^q}^q)$. In our existence result of the concentrated solution, we only assume that $A\in C^{1,1}_{loc}$. These two assumptions imply that A can be a non-monotonous or unbounded function.

Appendix

Let $B_{\alpha }(0)$ be the ball of $\mathbb {R}^n$ defined in (4.51), $S_\varepsilon$ be the operator defined in (4.6) and

$$\begin{aligned} S_1=\inf \left\{ \frac{\Vert u\Vert ^2}{\Vert u\Vert ^2_{L^2}}:0\not \equiv u\in H^1(\mathbb {R}^n)\right\} , \end{aligned}$$

we have,

Lemma A.1

Let $\delta$ be the positive constant chosen in Lemma 4.2. Then there exists a positive constant $\varepsilon _5$ such that for any $\varepsilon \in (0,\varepsilon _5]$ and $z\in B_{\varepsilon ^{1-\delta }}(0)$, we have

$$\begin{aligned} \Vert S_\varepsilon (U_{\varepsilon ,z})\Vert \le \varepsilon ^{2-3\delta } \end{aligned}$$

(5.1)

where $\varepsilon _5$ depends only on $|V|_{C^2(\mathbb {R}^n)}$, $\tau$, $C_1$, a, n, $\Vert U\Vert$, $S_1$ and $\delta$.

Proof

From the definition of $S_\varepsilon$, we have

$$\begin{aligned} \langle S_\varepsilon (U_{\varepsilon ,z}),\psi \rangle = \int _{\mathbb {R}^n}(V(\varepsilon y+z)-V(0))U\psi \left( y+\frac{z}{\varepsilon }\right) dy. \end{aligned}$$

Let $C_1$ and $\tau$ be the constants chosen in Theorem B given in Sect. 2. Recalling that $A=A(\Vert U\Vert ^q_{L^q})\ge a>0$ is a constant, we let $c=\max \{A^{\frac{n-1}{4}}C_1,\sqrt{A}\}$ and $\omega _n$ be the volume of $B_1(0)\subseteq \mathbb {R}^n$. It is easy to check that

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0}\frac{2\sqrt{c}\Vert V\Vert _{L^{\infty }(\mathbb {R}^n)}}{\sqrt{S_1}}\sqrt{\frac{c^3n\omega _n}{2\sqrt{\tau }}}\frac{2}{\varepsilon ^{2-3\delta }} e^{-\frac{\sqrt{\tau }}{c\varepsilon ^\delta }}=0. \end{aligned}$$

Therefore, there exists a constant $\varepsilon _2>0$, for any $\varepsilon$ satisfying $0<\varepsilon \le \varepsilon _2$, we have

$$\begin{aligned} \frac{2\sqrt{c}\Vert V\Vert _{L^{\infty }(\mathbb {R}^n)}}{\sqrt{S_1}}\sqrt{\frac{c^3n\omega _n}{2\sqrt{\tau }}}e^{-\frac{\sqrt{\tau }}{c\varepsilon ^\delta }}\le \frac{\varepsilon ^{2-3\delta }}{2}. \end{aligned}$$

(5.2)

From Theorem B given in Sect. 2 and (2.3), for any y satisfying $|y|\rightarrow \infty$, we have

$$\begin{aligned} |U(y)|\le ce^{-\frac{\sqrt{\tau }}{c}|y|}|y|^{-\frac{n-1}{2}}. \end{aligned}$$

This implies that for sufficiently small $\varepsilon >0$,

$$\begin{aligned} \left( \int _{\mathbb {R}^n\setminus B_{\varepsilon ^{-\delta }}(0)}U^2(y)dy\right) ^{\frac{1}{2}}\le c\sqrt{n\omega _n}\left( \int _{\varepsilon ^{-\delta }} ^{\infty }e^{-\frac{2\sqrt{\tau }r}{c}}dr\right) ^{\frac{1}{2}} =\sqrt{\frac{c^3n\omega _n}{2\sqrt{\tau }}} e^{-\frac{\sqrt{\tau }}{c\varepsilon ^\delta }}. \end{aligned}$$

(5.3)

It follows from (5.2), (5.3) and Hölder inequality that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^n\setminus B_{\varepsilon ^{-\delta }}(0)}|V(\varepsilon y+z)-V(0)|U\left| \psi \left( y+\frac{z}{\varepsilon }\right) \right| dy\\&\quad \le \frac{2\sqrt{c}\Vert V\Vert _{L^{\infty }(\mathbb {R}^n)}}{\sqrt{S_1}} \left( \int _{\mathbb {R}^n\setminus B_{\varepsilon ^{-\delta }}(0)}U^2(y)dy\right) ^{\frac{1}{2}}\Vert \psi \Vert \le \frac{\varepsilon ^{2-3\delta }}{2}\Vert \psi \Vert . \end{aligned} \end{aligned}$$

(5.4)

Moreover, since $\delta >0$, we can check that there exists a constant $\varepsilon _3$ such that for any $\varepsilon \in (0,\varepsilon _3]$,

$$\begin{aligned} \frac{4\Vert U\Vert |V|_{C^2(\mathbb {R}^n)}}{S_1}\varepsilon ^{\delta }\le \frac{1}{2}. \end{aligned}$$

(5.5)

Since $V\in C^2$ and 0 is the critical point of V, we see that there exists a constant $\varepsilon _4$, for any $\varepsilon \in (0,\varepsilon _4]$ and $z\in B_{\varepsilon ^{1-\delta }}(0)$,

$$\begin{aligned} \Vert V(\varepsilon y+z)-V(0)\Vert _{L^{\infty }(B_{\varepsilon ^{-\delta }}(0))}\le |V|_{C^2(\mathbb {R}^n)}|\varepsilon y+z|^2\le 4|V|_{C^2(\mathbb {R}^n)}\varepsilon ^{2-2\delta }. \end{aligned}$$

(5.6)

It follows from (5.5), (5.6) and Hölder inequality that for any $\varepsilon \in (0,\min \{\varepsilon _3,\varepsilon _4\}]$ and $z\in B_{\varepsilon ^{1-\delta }}(0)$,

$$\begin{aligned} \begin{aligned} \int _{B_{\varepsilon ^{-\delta }}(0)}|V(\varepsilon y+z)-V(0)|U\left| \psi \left( y+\frac{z}{\varepsilon }\right) \right| dy&\le \frac{4\Vert U\Vert |V|_{C^2(\mathbb {R}^n)}}{S_1}\varepsilon ^{2-2\delta }\Vert \psi \Vert \\&\le \frac{4\Vert U\Vert |V|_{C^2(\mathbb {R}^n)}}{S_1}\varepsilon ^{\delta }\varepsilon ^{2-3\delta }\Vert \psi \Vert \\&\le \frac{\varepsilon ^{2-3\delta }}{2}\Vert \psi \Vert . \end{aligned}\end{aligned}$$

(5.7)

Let $\varepsilon _5=\min \{\varepsilon _2,\varepsilon _3,\varepsilon _4\}$, from (5.4) and (5.7), for any $\varepsilon \in (0,\varepsilon _5]$ and $z\in B_{\varepsilon ^{1-\delta }}(0)$,

$$\begin{aligned} |\langle S_\varepsilon (U_{\varepsilon ,z}),\psi \rangle |\le \int _{\mathbb {R}^n}\left| (V(\varepsilon y+z)-V(0))U\psi \left( y+\frac{z}{\varepsilon }\right) \right| dy\le \varepsilon ^{2-3\delta }\Vert \psi \Vert . \end{aligned}$$

(5.8)

From the definition of norm, we get the desired estimate (5.1). This completes the proof of Lemma A.1. $\square$

Letting $B_{\varepsilon }$ be a ball of $H^1(\mathbb {R}^n)$ defined in (4.52), $R_{\varepsilon ,z}$ and $\{R^{(j)}_{\varepsilon ,z}\}_{j=1}^3$ be the operators defined in (4.8), (4.9), (4.10) and (4.11), we have,

Lemma A.2

Let $\delta$ be the positive constant chosen in Lemma 4.2. There exist positive constants $\varepsilon _6$ and $c_1$ such that for any $\varepsilon \in (0,\varepsilon _6]$ and any $\phi _{\varepsilon }, \phi _{1,\varepsilon }\in B_{\varepsilon }$,

$$\begin{aligned}&\Vert R^{(1)}_{\varepsilon ,z}(\phi _\varepsilon )\Vert \le c_1\Vert \phi _\varepsilon \Vert ^{\min \{p,2\}}, \end{aligned}$$

(5.9)

$$\begin{aligned}&\Vert R^{(1)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(1)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le c_1(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\min \{p-1,1\}}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \end{aligned}$$

(5.10)

$$\begin{aligned}&\Vert R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c_1\Vert \phi _\varepsilon \Vert ^{2}, \text {if}\ q=2,\\ c_1\Vert \phi _{\varepsilon }\Vert ^{\min \{q-1,2\}}, \text {if}\ q>2, \end{array} \right. \end{aligned}$$

(5.11)

$$\begin{aligned}&\Vert R^{(2)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c_1(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q=2,\\ c_1(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\min \{q-2,1\}}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q>2, \end{array} \right. \end{aligned}$$

(5.12)

$$\begin{aligned}&\Vert R^{(3)}_{\varepsilon ,z}(\phi _\varepsilon )\Vert \le c_1\Vert \phi _\varepsilon \Vert ^2, \end{aligned}$$

(5.13)

$$\begin{aligned}&\Vert R^{(3)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(3)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le c_1(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \end{aligned}$$

(5.14)

and thus,

$$\begin{aligned} \Vert R_{\varepsilon ,z}(\phi _\varepsilon )\Vert \le c_1\Vert \phi _\varepsilon \Vert ^{\mu _1}, \end{aligned}$$

(5.15)

and

$$\begin{aligned} \Vert R_{\varepsilon ,z}(\phi _{1,\varepsilon })-R_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le c_1(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\mu _2}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert . \end{aligned}$$

(5.16)

where $c_1$ is a positive constant, independent of $\varepsilon$ and z,

$$\begin{aligned} \mu _1=\left\{ \begin{array}{l@{\quad }l@{\quad }l} \min \{p,2\}, \text {if}\ q=2,\\ \min \{p,q-1,2\}, \text {if}\ q>2, \end{array} \right. \mu _2=\left\{ \begin{array}{l@{\quad }l@{\quad }l} \min \{p-1,1\}, \text {if}\ q=2,\\ \min \{p-1,q-2,1\}, \text {if}\ q>2. \end{array} \right. \end{aligned}$$

(5.17)

Proof

From a similar argument adopted by O. Rey ([58]), we have

$$\begin{aligned} \Vert R^{(1)}_{\varepsilon ,z}(\phi _\varepsilon )\Vert \le c_{(1)}\Vert \phi _\varepsilon \Vert ^{\min \{p,2\}} \end{aligned}$$

and

$$\begin{aligned} \Vert R^{(1)}_{\varepsilon ,z}(\phi _\varepsilon )-R^{(1)}_{\varepsilon ,z}(\phi _{1,\varepsilon })\Vert \le c_{(2)}(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\min \{p-1,1\}}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert . \end{aligned}$$

Therefore, it suffices to show the inequalities (5.11) $\sim$ (5.14). We divide the proof of Lemma A.2 into following four steps.

Step 1. The estimate of $R^{(2)}_{\varepsilon ,z}(\phi _\varepsilon )$. From mean value theorem of elementary calculus, there exists $\theta _1\in [0,1]$ such that

$$\begin{aligned} A(\Vert U_{\varepsilon ,z}+\phi _\varepsilon \Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q}) =qA'(\Vert U_{\varepsilon ,z}+\theta _1\phi _\varepsilon \Vert ^q_{L^q})\int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2} U_{\varepsilon ,z}\phi _{\varepsilon } dx. \end{aligned}$$

This implies that

$$\begin{aligned} \begin{aligned}&A(\Vert U_{\varepsilon ,z}+\phi _\varepsilon \Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q}) -qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U_{\varepsilon ,z}^{q-1}\phi _{\varepsilon } dx\\&\quad =q(A'(\Vert U_{\varepsilon ,z}+\theta _1\phi _\varepsilon \Vert ^q_{L^q})-A'(\Vert U_{\varepsilon ,z}\Vert _{L^q}^q)) \int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2} U_{\varepsilon ,z}\phi _{\varepsilon }dx\\&\qquad +qA'(\Vert U_{\varepsilon ,z}\Vert _{L^q}^q)\int _{\mathbb {R}^n} (|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2}-U_{\varepsilon ,z}^{q-2})U_{\varepsilon ,z}\phi _{\varepsilon } dx. \end{aligned}\end{aligned}$$

(5.18)

From mean value theorem of elementary calculus, there exists $\theta _2\in [0,1]$ such that

$$\begin{aligned} |U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^q-U^q_{\varepsilon ,z}=q|U_{\varepsilon ,z}+ \theta _1\theta _2\phi _\varepsilon |^{q-2}\theta _1U_{\varepsilon ,z}\phi _\varepsilon . \end{aligned}$$

From Hölder inequality, we see that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \left| \int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^q-U^q_{\varepsilon ,z}dx \right| \le c\Vert \phi _\varepsilon \Vert . \end{aligned}$$

(5.19)

Since $A\in C^{1,1}_{loc}([0,+\infty ))$, from (5.19), we see that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \begin{aligned} |A'(\Vert U_{\varepsilon ,z}+\theta _1\phi _\varepsilon \Vert ^q_{L^q})-A'(\Vert U_{\varepsilon ,z}\Vert _{L^q}^q)|&\le c \left| \int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _{\varepsilon }|^q-U_{\varepsilon ,z}^qdx \right| \le c\Vert \phi _{\varepsilon }\Vert . \end{aligned}\end{aligned}$$

This implies that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \begin{aligned} \left| (A'(\Vert U_{\varepsilon ,z}+\theta _1\phi _\varepsilon \Vert ^q_{L^q})-A'(\Vert U_{\varepsilon ,z}\Vert _{L^q}^q)) \int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2}U_{\varepsilon ,z}\phi _{\varepsilon } dx \right| \le c\Vert \phi _\varepsilon \Vert ^2. \end{aligned} \end{aligned}$$

(5.20)

Therefore, if $q=2$, it is easy to see that

$$\begin{aligned} \begin{aligned}&A(\Vert U_{\varepsilon ,z}+\phi _\varepsilon \Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q}) -qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U_{\varepsilon ,z}^{q-1}\phi _{\varepsilon } dx\\&\quad =q(A'(\Vert U_{\varepsilon ,z}+\theta _1\phi _\varepsilon \Vert ^q_{L^q})-A'(\Vert U_{\varepsilon ,z}\Vert _{L^q}^q)) \int _{\mathbb {R}^n}|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2} U_{\varepsilon ,z}\phi _{\varepsilon }dx. \end{aligned}\end{aligned}$$

(5.21)

This, together with (5.20), implies that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \begin{aligned} \left| A(\Vert U_{\varepsilon ,z}+\phi _\varepsilon \Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q}) -qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U_{\varepsilon ,z}^{q-1}\phi _{\varepsilon } dx \right| \le c\Vert \phi _\varepsilon \Vert ^2 \end{aligned} \end{aligned}$$

(5.22)

if $q=2$. From pp. 122 of Ambrosetti and Malchiodi [3], we get the following elementary inequality,

$$\begin{aligned} ||a+b|^{p_1}-a^{p_1}|\le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c|b|^{p_1},\ \text {if}\ 0<p_1\le 1,\\ c(|b|+|b|^{p_1}), \text {if}\ p_1>1, \end{array} \right. \end{aligned}$$

(5.23)

for any $a,b\in \mathbb {R}$ with $|a|\le 1$ where c is a constant, depending only on $p_1$. Therefore, if $q>2$, we have

$$\begin{aligned} ||U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2}-U_{\varepsilon ,z}^{q-2}|\le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c|\phi _\varepsilon |^{q-2}, \text {if}\ 2<q\le 3,\\ c(|\phi _\varepsilon |+|\phi _\varepsilon |^{q-2}), \text {if}\ q>3, \end{array} \right. \end{aligned}$$

(5.24)

where c is a constant, depending only on q and $\Vert U\Vert _{C^0(\mathbb {R}^n)}$. It follows from Hölder inequality that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \left| \int _{\mathbb {R}^n}(|U_{\varepsilon ,z}+\theta _1\phi _\varepsilon |^{q-2}-U_{\varepsilon ,z}^{q-2})U_{\varepsilon ,z}\phi _{\varepsilon } dx \right| \le c\Vert \phi _\varepsilon \Vert ^{\min \{q-1,2\}}. \end{aligned}$$

(5.25)

Putting (5.20) and (5.25) into (5.18) , we see that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \left| A(\Vert U_{\varepsilon ,z}+\phi _\varepsilon \Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q}) -qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U_{\varepsilon ,z}^{q-1}\phi _{\varepsilon } dx \right| \le c\Vert \phi _{\varepsilon }\Vert ^{\min \{q-1,2\}} \end{aligned}$$

(5.26)

if $q>2$. It follows from the definition of $R^{(2)}_{\varepsilon ,z}$, (5.22) and (5.26) that there exists a constant $c_{(3)}$, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \Vert R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c_{(3)}\Vert \phi _\varepsilon \Vert ^{2}, \text {if}\ q=2,\\ c_{(3)}\Vert \phi _{\varepsilon }\Vert ^{\min \{q-1,2\}}, \text {if}\ q>2. \end{array} \right. \end{aligned}$$

(5.27)

Step 2 . The estimate of $\Vert R^{(2)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert$. It follows from the definition of $R^{(2)}_{\varepsilon ,z}$ that

$$\begin{aligned}\begin{aligned} R^{(2)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon }) =&-(A(\Vert U_{\varepsilon ,z}+\phi _{1,\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q})\\&-qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U^{q-1}_{\varepsilon ,z}(\phi _{1,\varepsilon }-\phi _{\varepsilon })dy)\Delta U_{\varepsilon ,z}. \end{aligned}\end{aligned}$$

Adopting a similar argument of Step 1, we can show that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \begin{aligned}&\left| A(\Vert U_{\varepsilon ,z}+\phi _{1,\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q} -qA'(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})\int _{\mathbb {R}^n}U^{q-1}_{\varepsilon ,z}(\phi _{1,\varepsilon }-\phi _{\varepsilon })dy)\right| \\&\quad \le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q=2,\\ c(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\min \{q-2,1\}}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q>2. \end{array} \right. \end{aligned}\end{aligned}$$

(5.28)

Therefore, there exists a constant $c_{(4)}$, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \begin{aligned} \Vert R^{(2)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(2)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le \left\{ \begin{array}{l@{\quad }l@{\quad }l} c_{(4)}(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q=2,\\ c_{(4)}(\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )^{\min \{q-2,1\}}\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert , \text {if}\ q>2. \end{array} \right. \end{aligned}\end{aligned}$$

(5.29)

Step 3. The estimate of $R^{(3)}_{\varepsilon ,z}(\phi _\varepsilon )$. From (5.22) and (5.26), we see that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} |A(\Vert U_{\varepsilon ,y_\varepsilon }+\phi _\varepsilon \Vert _{L^q}^q)-A(\Vert U_{\varepsilon ,y_\varepsilon }\Vert _{L^q}^q)|\le c\Vert \phi _\varepsilon \Vert . \end{aligned}$$

(5.30)

It follows from the definition of $R^{(3)}_{\varepsilon ,z}$ that there exists a constant $c_{(5)}$, independent of $\varepsilon$ and z, such that

$$\begin{aligned} \Vert R^{(3)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le c|A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})|\Vert \phi _{\varepsilon }\Vert \le c_{(5)}\Vert \phi _{\varepsilon }\Vert ^2. \end{aligned}$$

Step 4 . The estimate of $\Vert R^{(3)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(3)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert$. From the definition of $R^{(3)}_{\varepsilon ,z}$, we have

$$\begin{aligned} \begin{aligned}&R^{(3)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(3)}_{\varepsilon ,z}(\phi _{\varepsilon })\\&\quad =-(A(\Vert U_{\varepsilon ,z}+\phi _{1,\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q}))\Delta \phi _{1,\varepsilon }\\&\qquad -(A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}\Vert ^q_{L^q})) \Delta (\phi _{1,\varepsilon }-\phi _{\varepsilon }). \end{aligned}\end{aligned}$$

(5.31)

It follows from (5.28) that there exists a constant c, independent of $\varepsilon$ and z, such that

$$\begin{aligned} |A(\Vert U_{\varepsilon ,z}+\phi _{1,\varepsilon }\Vert ^q_{L^q})-A(\Vert U_{\varepsilon ,z}+\phi _{\varepsilon }\Vert ^q_{L^q})|\le c\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert . \end{aligned}$$

(5.32)

It follows from (5.30), (5.32) and (5.31) that there exists a constant $c_{(6)}$, independent of $\varepsilon$ and z, such that

$$\begin{aligned}\begin{aligned} \Vert R^{(3)}_{\varepsilon ,z}(\phi _{1,\varepsilon })-R^{(3)}_{\varepsilon ,z}(\phi _{\varepsilon })\Vert \le c_{(6)}((\Vert \phi _{1,\varepsilon }\Vert +\Vert \phi _{\varepsilon }\Vert )\Vert \phi _{1,\varepsilon }-\phi _{\varepsilon }\Vert . \end{aligned}\end{aligned}$$

Let $c_1=\max \{c_{(k)}:k=1,2,\ldots ,6 \}$, we get the desired estimates and complete the proof of Lemma A.2. $\square$