Proof. (i) By the definition of $E_1$ and Sobolev's imbedding theorem, we see that

\[ {I_m}(u) \ge \frac{1} {2}\| u \|_{{H^{1}}({\mathbb{R}^{N}})}^{2} - C\| u \|_{{H^{1}}({\mathbb{R}^{N}})}^{4} - C\| u \|_{{H^{1}}({\mathbb{R}^{N}})}^{\alpha + 2}, \]

then taking ${\rho _0},{\alpha _0} > 0$ small, (i) holds.

(ii) We have the following cases:

Case 1: $\alpha < 2$ or $a \le 0$. Fixing $u \in {H^{1}}({\mathbb {R}^{N}})\backslash \{ 0\}$, then for $t>0$, ${I_m}(tu) \to - \infty$ as $t \to +\infty$. Choosing ${t_0} > 0$ large to ensure that ${I_m}({t_0}u) < 0$, ${t_0}u$ is the desired ${\phi _0}$.

Case 2: $\alpha = 2$ and $a < a_\alpha ^ * = b$. Letting

(2.2)\begin{equation} C(u): ={-} b\int_{{\mathbb{R}^{N}}} {{E_1}({{({u^ + })}^{2}}){{({u^ + })}^{2}}} + a\int_{{\mathbb{R}^{N}}} {{{({u^ + })}^{4}}}, \end{equation}

we select a positive $u \in {H^{1}}({\mathbb {R}^{N}})\backslash \{ 0\}$ and set ${(u)^{t}}(x): = {t^{1/2}}u(t{x_1},{x_2},\ldots ,{x_N})$, we see from Plancherel's theorem that

\[ C({(u)^{t}}): = t\int_{{\mathbb{R}^{N}}} {\left( { - b\cdot\frac{{{t^{2}}\xi _1^{2}}} {{{t^{2}}\xi _1^{2} + |\xi '{|^{2}}}} + a} \right){{\left| {\mathcal{F}(|u{|^{2}})} \right|}^{2}}} , \]

where $\xi ' = ({\xi _2},\ldots ,{\xi _N})$. Since

\[ \mathop{\lim }\limits_{t \to + \infty } - b \cdot \frac{{{t^{2}}\xi _1^{2}}} {{{t^{2}}\xi _1^{2} + |\xi '{|^{2}}}} + a = a - b < 0, \]

we choose a large ${t_0} > 0$ to ensure that $C({(u)^{{t_0}}}) < 0$ and then select a large ${R_0} > 0$ such that ${I_m}({R_0}{(u)^{{t_0}}}) < 0$, ${R_0}{(u)^{{t_0}}}$ is the desired ${\phi _0}$.

Case 3: $\alpha \,{>}\, 2$ and $0 \,{<}\, a \,{<}\, a_\alpha ^{*} = {( {{b}/ {2}} )^{{\alpha }/ {2}}}{m^{{{(2 - \alpha ) }}/ {2}}}( {{{(\alpha \,{+}\, 2)}}/ {\alpha }} ){( {{{(\alpha - 2)}} /{\alpha }} )^{{{(\alpha \,{-}\, 2)}}/ {2}}}$. Denoting

\[ {G_a}(s): = \frac{m} {2}{s^{2}} + \frac{a} {{\alpha + 2}}{s^{\alpha + 2}} - \frac{b} {4}{s^{4}}, \]

we see that there exists ${s_a}>0$ such that ${G_a}({s_a})<0$ if and only if $a < a_\alpha ^ *$.

Letting

(2.3)\begin{equation} D(u): = \frac{m} {2}\int_{{\mathbb{R}^{N}}} {{u^{2}}} - \frac{b} {4}\int_{{\mathbb{R}^{N}}} {{E_1}({{({u^ + })}^{2}}){{({u^ + })}^{2}}} + \frac{a} {{\alpha + 2}}\int_{{\mathbb{R}^{N}}} {{{({u^ + })}^{\alpha + 2}}} \end{equation}

and defining

\[ {w_R}(x): = \begin{cases} {s_a}, & |x| \le R,\\ {s_a}(R + 1 - |x|), & R \le |x| \le R + 1,\\ 0, & |x| \ge R + 1, \end{cases} \]

for $t>0$, we denote ${(u)_t}(x): = u(t{x_1},{x_2},\ldots ,{x_N})$, we see that

(2.4)\begin{align} D({({w_R})_t}) &= {t^{ - 1}}\left[ \frac{m} {2}\int_{{\mathbb{R}^{N}}} {|{w_R}{|^{2}}} + \frac{a} {{\alpha + 2}}\int_{{\mathbb{R}^{N}}} {|{w_R}{|^{\alpha + 2}}}\right.\notag\\ &\quad -\left. \frac{b} {4}\int_{{\mathbb{R}^{N}}} {\frac{{{t^{2}}\xi _1^{2}}} {{{t^{2}}\xi _1^{2} + |\xi '{|^{2}}}}{{\left| {\mathcal{F}(|{w_R}{|^{2}})} \right|}^{2}}} \right].\end{align}

Note that

\[ \mathop{\lim }\limits_{t \to + \infty } \int_{{\mathbb{R}^{N}}} {\frac{{{t^{2}}\xi _1^{2}}} {{{t^{2}}\xi _1^{2} + |\xi '{|^{2}}}}{{\left| {\mathcal{F}(|{w_R}{|^{2}})} \right|}^{2}}} = \int_{{\mathbb{R}^{N}}} {{{| {{w_R}} |}^{4}}} , \]

then, for any $\varepsilon > 0$, there exists $T = T(\varepsilon ,R) > 0$ such that $t > T$ implies

(2.5)\begin{equation} \int_{{\mathbb{R}^{N}}} {\frac{{{t^{2}}\xi _1^{2}}} {{{t^{2}}\xi _1^{2} + |\xi '{|^{2}}}}{{\left| {\mathcal{F}(|{w_R}{|^{2}})} \right|}^{2}}} > \int_{{\mathbb{R}^{N}}} {{{| {{w_R}} |}^{4}}} - \varepsilon, \end{equation}

we see from (2.4) and (2.5) that for $R>1$,

\[ D({({w_R})_t}) \le {t^{ - 1}}\left[ {C{R^{N}}{G_a}({s_a}) + C{R^{N - 1}} + \frac{b} {4}\varepsilon } \right]. \]

Choosing $\varepsilon > 0$ small and $R>0$ large to make sure that

\[ C{R^{N}}{G_a}({s_a}) + C{R^{N - 1}} + \frac{b} {4}\varepsilon < 0, \]

then there exists ${t_R} > 0$ such that $D({({w_R})_{{t_R}}}) < 0$. Letting ${u_R}: = {({w_R})_{{t_R}}}$, we choose a large $R' > 0$ such that ${I_m}({u_R}(x/R')) < 0$, ${u_R}(x/R')$ is the desired ${\phi _0}$.

Proof. For any sequence $\{ {U_k}\} _{k = 1}^{\infty } \subset {S_m}$, we see that

\[ {b_m} = {I_m}({U_k}) = {I_m}({U_k}) - \frac{1} {N}{P_m}({U_k}) = \frac{1} {N}\int_{{\mathbb{R}^{N}}} {|\nabla {U_k}{|^{2}}} . \]

Thus, $\{ \nabla {U_k}\} _{k = 1}^{\infty }$ is bounded in ${L^{2}}({\mathbb {R}^{N}})$. Next, we claim that $\{ {U_k}\} _{k = 1}^{\infty }$ is also bounded in ${L^{2}}({\mathbb {R}^{N}})$. We divide the proof into two cases:

Case 1: $N=3$. Since ${P_m}({U_k}) = 0$, arguing as (2.9), we see that $\{ {U_k}\} _{k = 1}^{\infty }$ is bounded in ${L^{2}}({\mathbb {R}^{3}})$.

Case 2: $N=2$. We argue indirectly and assume that ${\lambda _k}: = {\| {{U_k}} \|_{{L^{2}}({\mathbb {R}^{2}})}} \to \infty$ as $k \to \infty$. Letting ${V_k}(x) = {U_k}({\lambda _k}x)$, we see that

(2.22)\begin{equation} {\| {{V_k}} \|_{{L^{2}}({\mathbb{R}^{2}})}} = 1\quad{\text{and}}\quad{\| {\nabla {V_k}} \|_{{L^{2}}({\mathbb{R}^{2}})}} = {\| {\nabla {U_k}} \|_{{L^{2}}({\mathbb{R}^{2}})}}. \end{equation}

In particular, $\{ {V_k}\} _{k = 1}^{\infty }$ is bounded in ${H^{1}}({\mathbb {R}^{2}})$. Next, we claim that for any $\{ {y_k}\} _{k = 1}^{\infty } \subset {\mathbb {R}^{2}}$, after taking a subsequence,

(2.23)\begin{equation} {{\tilde{V}}_k}(x): = {V_k}(x + {y_k}) \rightharpoonup 0\quad{\text{in}}\ {H^{1}}({\mathbb{R}^{2}}). \end{equation}

Assuming on the contrary that there exists $\{ {y_k}\} _{k = 1}^{\infty } \subset {\mathbb {R}^{2}}$, up to a subsequence,

(2.24)\begin{equation} {{\tilde{V}}_k}(x): = {V_k}(x + {y_k}) \rightharpoonup {{\tilde{V}}_0}(x) \ne 0\quad{\text{in }}{H^{1}}({\mathbb{R}^{2}}). \end{equation}

For any $\varphi \in {H^{1}}({\mathbb {R}^{2}})$, letting ${\varphi _k}(x) = \varphi ( {(x/{\lambda _k}) - {y_k}} )$. From $\langle {{I'_m}({U_k}),{\varphi _k}} \rangle = 0$, we have

(2.25)\begin{equation} \int_{{\mathbb{R}^{2}}} {\nabla {{\tilde{V}}_k} \cdot \nabla \varphi } + m\lambda _k^{2}\int_{{\mathbb{R}^{2}}} {{{\tilde{V}}_k}\varphi } - b\lambda _k^{2}\int_{{\mathbb{R}^{2}}} {{E_1}({{({{\tilde{V}}_k})}^{2}}){{\tilde{V}}_k}\varphi } + a\lambda _k^{2}\int_{{\mathbb{R}^{2}}} {{{({{\tilde{V}}_k})}^{\alpha + 1}}\varphi } = 0. \end{equation}

Multiplying (2.25) by $\lambda _k^{ - 2}$ and letting $k \to \infty$, we see that ${{\tilde {V}}_0} \ge 0$ satisfies

(2.26)\begin{equation} m{{\tilde{V}}_0} - b{E_1}({({{\tilde{V}}_0})^{2}}){{\tilde{V}}_0} + a{({{\tilde{V}}_0})^{\alpha + 1}} = 0. \end{equation}

In view of [Reference Pucci and Serrin39], we have the following Pohozaev's type identity:

(2.27)\begin{equation} \frac{m} {2}\int_{{\mathbb{R}^{2}}} {{{({{\tilde{V}}_0})}^{2}}} - \frac{b} {4}\int_{{\mathbb{R}^{2}}} {{E_1}({{({{\tilde{V}}_0})}^{2}}){{({{\tilde{V}}_0})}^{2}}} + \frac{a} {{\alpha + 2}}\int_{{\mathbb{R}^{2}}} {{{({{\tilde{V}}_0})}^{\alpha + 2}}} = 0, \end{equation}

that is, ${P_m}({{\tilde {V}}_0}) = 0$. From the definition of $b_m$,

(2.28)\begin{equation} \int_{{\mathbb{R}^{2}}} {|\nabla {{\tilde{V}}_0}{|^{2}}} \ge 2{b_m}. \end{equation}

Since ${U_k} \in {S_m}$, we have

(2.29)\begin{equation} \int_{{\mathbb{R}^{2}}} {|\nabla {{\tilde{V}}_k}{|^{2}}} = \int_{{\mathbb{R}^{2}}} {|\nabla {V_k}{|^{2}}} = \int_{{\mathbb{R}^{2}}} {|\nabla {U_k}{|^{2}}} = 2{b_m}. \end{equation}

(2.27), (2.28) and (2.29) imply that ${I_m}({{\tilde {V}}_0}) = {b_m}$. Using Ekeland's variational principle, there exist a Lagrange Multiplier $\lambda \in \mathbb {R}$ such that ${I'_m}({{\tilde {V}}_0}) = \lambda {P'_m}({{\tilde {V}}_0})$, by (2.26), ${{\tilde {V}}_0} = 0$, which contradicts (2.24). Thus, (2.23) holds. Since $\{ {y_k}\} _{k = 1}^{\infty }$ is arbitrary, we have

\[ \mathop{\lim}\limits_{k \to \infty } \mathop{\sup}\limits_{y \in {\mathbb{R}^{2}}} \int_{{B_2}(y)} {|{V_k}{|^{2}}} = 0, \]

by vanishing theorem,

(2.30)\begin{equation} \int_{{\mathbb{R}^{2}}} {|{V_k}{|^{p}}} \to 0\quad{\text{as }}k \to \infty {\text{ for }}2 < p <{+} \infty . \end{equation}

Multiplying (2.25) by $\lambda _k^{ - 2}$ and replacing $\varphi$ by ${{{\tilde {V}}_k}}$, we see from (2.30) that ${\| {{V_k}} \|_{{L^{2}}({\mathbb {R}^{2}})}} = o(1)$ as $k \to \infty$, which contradicts (2.22). Therefore, $\{ {U_k}\} _{k = 1}^{\infty }$ is bounded in ${L^{2}}({\mathbb {R}^{2}})$. From the above arguments of cases 1 and 2, we obtain that $\{ {U_k}\} _{k = 1}^{\infty }$ is bounded in ${H^{1}}({\mathbb {R}^{N}})$.

Next, we claim that there exist a sequence $\{ {x_k}\} _{k = 1}^{\infty } \subset {\mathbb {R}^{N}}$ and $R,\beta > 0$ such that

\[ \int_{{B_R}({x_k})} {|{U_k}{|^{2}}} \ge \beta . \]

Indeed, if not, by vanishing theorem,

(2.31)\begin{equation} \int_{{\mathbb{R}^{N}}} {|{U_k}{|^{p}}} \to 0\quad{\text{as }}k \to \infty {\text{ for }}2 < p < {2^ * }. \end{equation}

(2.31) and $\langle {{I'_m}({U_k}),{U_k}} \rangle = 0$ imply that ${\| {{U_k}} \|_{{H^{1}}({\mathbb {R}^{N}})}} = o(1)$, which contradicts ${I_m}({U_k}) = {b_m} > 0$. Letting ${{\bar {U}}_k}(x): = {U_k}(x + {x_k})$, up to a subsequence, there is a $\bar {U} \in {H^{1}}({\mathbb {R}^{N}})\backslash \{ 0\}$ such that ${{\bar {U}}_k} \rightharpoonup \bar {U}$ in ${H^{1}}({\mathbb {R}^{N}})$. We infer that $\bar {U}$ is a nontrivial solution to (2.1) and

\begin{align*} {b_m} &\le{I_m}(\bar{U}) = {I_m}(\bar{U}) - \frac{1} {N}{P_m}(\bar{U}) = \frac{1} {N}\int_{{\mathbb{R}^{N}}} {|\nabla \bar{U}{|^{2}}} \\ &\le\mathop{\underline{\lim}}\limits_{k \to \infty } \frac{1} {N}\int_{{\mathbb{R}^{N}}} {|\nabla {{\bar{U}}_k}{|^{2}}} = \mathop{\underline{\lim}}\limits_{k \to \infty } \left[ {{I_m}({{\bar{U}}_k}) - \frac{1} {N}{P_m}({{\bar{U}}_k})} \right] = {b_m}. \end{align*}

Hence, $\nabla {{\bar {U}}_k} \to \nabla \bar {U}$ in ${L^{2}}({\mathbb {R}^{N}})$. By Gagliardo–Nirenberg's inequality, we see that ${{\bar {U}}_k} \to \bar {U}$ in ${L^{q}}({\mathbb {R}^{N}})$ for $2 < q < {2^ * }$, noting that $\langle {{I'_m}({{\bar {U}}_k}) - {I'_m}(\bar {U}),{{\bar {U}}_k} - \bar {U}} \rangle = 0$, we have

(2.32)\begin{equation} {{\bar{U}}_k} \to \bar{U}\quad{\text{in }}{H^{1}}({\mathbb{R}^{N}}){\text{ as }}k \to \infty . \end{equation}

By lemma B.1, we have

(2.33)\begin{equation} {\| {{{\bar{U}}_k}} \|_{{L^{\infty} }({\mathbb{R}^{N}})}} = {\| {{U_k}} \|_{{L^{\infty} }({\mathbb{R}^{N}})}} \le C\ {\text{uniformly for }}k \in \mathbb{N}. \end{equation}

By Hölder's inequality, (2.33) and ${E_1} \in \mathcal {L}({L^{p}}({\mathbb {R}^{N}}),{L^{p}}({\mathbb {R}^{N}}))(1 < p < + \infty )$ [Reference Cipolatti15, lemma 2.1], we see that for any $y \in {\mathbb {R}^{N}}$,

(2.34)\begin{align} &{\| {{E_1}(|{{\bar{U}}_k}{|^{2}}){{\bar{U}}_k} + |{{\bar{U}}_k}{|^{\alpha} }{{\bar{U}}_k}} \|_{{L^{N}}({B_2}(y))}} \nonumber\\ &\quad \le {\| {{E_1}(|{{\bar{U}}_k}{|^{2}})} \|_{{L^{2N}}({\mathbb{R}^{N}})}}{\| {{{\bar{U}}_k}} \|_{{L^{2N}}({B_2}(y))}} + {\| {|{{\bar{U}}_k}{|^{\alpha + 1}}} \|_{{L^{N}}({B_2}(y))}}\nonumber\\ &\quad \le C\left({\| {{{\bar{U}}_k}} \|_{{L^{2}}({B_2}(y))}^{1/N} + \| {{{\bar{U}}_k}} \|_{{L^{2}}({B_2}(y))}^{2/N}} \right), \end{align}

where $C$ is independent of $y \in {\mathbb {R}^{N}}$. Choosing $p=2$, $q=2N$ and $R=1$ in theorem 8.17 of [Reference Gilbarg and Trudinger28], by (2.34) and Young's inequality, we have

\[ \mathop{\sup}\limits_{{B_1}(y)} {{\bar{U}}_k} \le C\left( {{{\| {{{\bar{U}}_k}} \|}_{{L^{2}}({B_2}(y))}} + \| {{{\bar{U}}_k}} \|_{{L^{2}}({B_2}(y))}^{1/N}} \right) \]

where $C$ is independent of $y \in {\mathbb {R}^{N}}$ and $k \in N$. Letting $|y| \to + \infty$, we see from (2.32) that

(2.35)\begin{equation} \mathop{\lim}\limits_{|x| \to \infty } {{\bar{U}}_k}(x) = 0\ {\text{uniformly for }}k \in \mathbb{N}. \end{equation}

Multiplying (2.1) by ${U_k}$ and integrating, we see from Plancherel's theorem that

\[ m\int_{{\mathbb{R}^{N}}} {U_k^{2}} \le C\left( {\int_{{\mathbb{R}^{N}}} {U_k^{4}} + \int_{{\mathbb{R}^{N}}} {U_k^{\alpha + 2}} } \right) \le C(U_k^{2}(0) + U_k^{\alpha} (0))\int_{{\mathbb{R}^{N}}} {U_k^{2}}, \]

which implies that ${{\bar {U}}_k}( - {x_k}) = {U_k}(0) \ge C > 0$, then by (2.35), $\{ {x_k}\} _{k = 1}^{\infty }$ must be bounded, that is, $|{x_k}| \le {r_0}$. By (2.32) and Sobolev's imbedding theorem, we see that for any $\delta > 0$, there exists ${R_\delta }>0$ such that for all $k \in \mathbb {N}$,

(2.36)\begin{equation} \int_{{\mathbb{R}^{N}}\backslash {B_{{R_\delta } + {r_0}}}} {U_k^{p}} \le \int_{{\mathbb{R}^{N}}\backslash {B_{{R_\delta }}}} {\bar{U}_k^{p}} \le \delta, \end{equation}

where $2 \le p < {2^ * }$. By the boundedness of $\{ {U_k}\} _{k = 1}^{\infty }$ in ${H^{1}}({\mathbb {R}^{N}})$, there exists $U \in {H^{1}}({\mathbb {R}^{N}})$ such that ${U_k} \rightharpoonup U$ in ${H^{1}}({\mathbb {R}^{N}})$, ${U_k} \to U$ in $L_{{\text {loc}}}^{p}({\mathbb {R}^{N}})(1 \le p < {2^ * })$. Combining with (2.36), we see that ${U_k} \to U$ in ${L^{p}}({\mathbb {R}^{N}})$ with $2 \le p < {2^ * }$. Recalling that $\left \langle {{I'_m}({U_k}) - {I'_m}(U),{U_k} - U} \right \rangle = 0$, we conclude that ${U_k} \to U$ in ${H^{1}}({\mathbb {R}^{N}})$. This completes the proof that $S_m$ is compact in ${H^{1}}({\mathbb {R}^{N}})$.

Proof. For the sake of simplicity, we only give the proof corresponding to the case $N=2$, the case $N=3$ is similar. For each $\delta > 0$, ${s_i},{s_j} \in [0,1]$ with $1 \le i,j \le k$, $i \ne j$, we see from Appendix A that

(3.5)\begin{align} &\int_{{\mathbb{R}^{2}}} {{E_1}(|\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}})|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}}\nonumber\\ &\quad = \int_{{\mathbb{R}^{2}}} {(\mathcal{K} * |\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}})|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}} \nonumber\\ &\quad = \frac{1} {{2\pi }}\int_{{\mathbb{R}^{2}}} {\left( {\frac{{ - x_1^{2} + x_2^{2}}} {{|x{|^{4}}}} * |\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}}} \right)|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}} +\frac{1} {2}\int_{{\mathbb{R}^{2}}} {\left( {{\delta _0} * |\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}}} \right)|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}} \nonumber\\ &\quad = (I) + (II), \end{align}

where ${\delta _0}$ is the Dirac measure on ${\mathbb {R}^{2}}$ giving unit mass to the point $0$. Choosing $\theta > 0$ small but fixed, we see from Hardy–Littlewood–Sobolev's inequality [Reference Lieb and Loss32, theorem 4.3] that,

(3.6)\begin{align} |(I)| &\le C{\varepsilon ^{\theta} }\int_{{B_{2\beta /\varepsilon }}(z_*^{j}/\varepsilon )} \left[ {\int_{{B_{2\beta /\varepsilon }}(z_*^{i}/\varepsilon )} {\frac{1} {{|x - y{|^{2 - \theta }}}}|\gamma _\delta ^{i}(s_i){|^{2}}(x - (z_*^{i}/\varepsilon ))} \,{\rm d}y} \right]\notag\\ &\quad \times | \gamma_\delta ^{j}(s_j){|^{2}}(x - (z_*^{j}/\varepsilon )) \,{\rm d}x \nonumber\\ &\le C{\varepsilon ^{\theta} }\left\| {\gamma _\delta ^{i}(s_i)} \right\|_{{L^{8/(2 + \theta )}}({\mathbb{R}^{2}})}^{2}\left\| {\gamma _\delta ^{j}(s_j)} \right\|_{{L^{8/(2 + \theta )}}({\mathbb{R}^{2}})}^{2}. \end{align}

Since $\gamma _\delta ^{i}([0,1])\ (1\le i \le k)$ is compact in ${H^{1}}({\mathbb {R}^{2}})$ and $2 < {8}/ {{2 + \theta }} < + \infty$ for $\theta > 0$ small, we get $(I) = o(1)$. Moreover,

(3.7)\begin{align} (II) &=\frac{1}{2}\int_{{\mathbb{R}^{2}}} {\left[ {\int_{{\mathbb{R}^{2}}} {{\delta _0}(x - y)|\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}}(y)} \,{\rm d}y} \right]|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}(x)} \,{\rm d}x \nonumber\\ &=\frac{1} {2} \int_{{\mathbb{R}^{2}}} {|\gamma _{\varepsilon ,\delta }^{i}({s_i}){|^{2}}|\gamma _{\varepsilon ,\delta }^{j}({s_j}){|^{2}}} = 0. \end{align}

Lemma 2.3(i) and (3.5)–(3.7) show that for each $\delta > 0$,

\begin{align*} {{\tilde c}_{\varepsilon ,\delta }}&: = \mathop{\max }\limits_{s \in {{[0,1]}^{k}}} {J_\varepsilon }({\gamma _{\varepsilon ,\delta }}(s)) = \sum\limits_{i = 1}^{k} {\mathop{\max }\limits_{{s_i} \in [0,1]} {J_\varepsilon }(\gamma _{\varepsilon ,\delta }^{i}({s_i}))} + o(1) \\ &= \sum\limits_{i = 1}^{k} {\mathop{\max }\limits_{t \in [0,1]} {I_{{m_i}}}(\gamma _\delta ^{i}(t))} + o(1) = \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} + o(1), \end{align*}

(i) holds. Moreover, by (3.3), (ii) is obvious.

Proof. We only prove (i). (ii) follows with minor modifications. For notational simplicity, we write $\varepsilon$ for ${\varepsilon _j}$ and still use $\varepsilon$ after taking a subsequence. By the definition of $X_\varepsilon ^{{d_0}}$ and the compactness of ${S_{{m_i}}}$ and ${({\mathcal {M}^{i}})^{\beta } }$, we see that there exist ${W^{i}} \in {S_{{m_i}}}$, ${\{ z_\varepsilon ^{i}\} _{\varepsilon > 0}} \subset {({\mathcal {M}^{i}})^{\beta } }$ such that for $\varepsilon > 0$ small and $1 \le i \le k$,

(3.9)\begin{equation} {\left\| {{u_\varepsilon } - \sum\limits_{i = 1}^{k} {\varphi (\varepsilon x - z_\varepsilon ^{i}){W^{i}}(x - (z_\varepsilon ^{i}/\varepsilon ))} } \right\|_{{H_\varepsilon }}} \le 2{d_0}\quad{\text{and}}\quad z_\varepsilon ^{i} \to {z^{i}} \in {({\mathcal{M}^{i}})^{\beta} }\text{ as }\varepsilon \to 0. \end{equation}

Step 1: We claim that

(3.10)\begin{equation} \mathop{\lim }\limits_{\varepsilon \to 0} \mathop{\sup }\limits_{y \in {A_\varepsilon }} \int_{{B_1}(y)} {|{u_\varepsilon }{|^{2}}} = 0, \end{equation}

where ${A_\varepsilon } = \mathop \cup \nolimits _{i = 1}^{k} ({B_{3\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /2\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))$.

Assuming on the contrary that there exists $r>0$ such that

\[ \mathop{\underline{\lim}}\limits_{\varepsilon \to 0} \mathop{\sup }\limits_{y \in {A_\varepsilon }} \int_{{B_1}(y)} {|{u_\varepsilon }{|^{2}}} = 2r > 0, \]

then there exists ${y_\varepsilon } \in {A_\varepsilon }$ such that for $\varepsilon > 0$ small,

(3.11)\begin{equation} \int_{{B_1}({y_\varepsilon })} {|{u_\varepsilon }{|^{2}}} \ge r > 0. \end{equation}

Letting ${v_\varepsilon }(x): = {u_\varepsilon }(x + {y_\varepsilon })$, up to a subsequence, there exists $v \in {H^{1}}({\mathbb {R}^{N}})\backslash \{ 0\}$ such that ${v_\varepsilon } \rightharpoonup v$ in ${H^{1}}({\mathbb {R}^{N}})$ and $\varepsilon {y_\varepsilon } \to {x_0}$ with ${x_0}$ in the closure of $\mathop \cup \nolimits _{i = 1}^{k} ({B_{3\beta }}({z^{i}})\backslash {B_{\beta /2}}({z^{i}}))$. Moreover, we see that $v$ satisfies (2.1) with $m$ replaced by $V(x_0)$. Since

\[ {c_{V({x_0})}} \le {I_{V({x_0})}}(v) - \frac{1} {N}{P_{V({x_0})}}(v) = \frac{1} {N}\int_{{\mathbb{R}^{N}}} {|\nabla v{|^{2}}} , \]

then for $R>0$ large,

(3.12)\begin{align} \mathop{\underline {\lim } }\limits_{\varepsilon \to 0} \frac{1} {N}\int_{{B_R}({y_\varepsilon })} {|\nabla {u_\varepsilon }{|^{2}}} &= \mathop{\underline {\lim } }\limits_{\varepsilon \to 0} \frac{1} {N}\int_{{B_R}(0)} {|\nabla {v_\varepsilon }{|^{2}}} \nonumber\\ & \ge \frac{1} {N}\int_{{B_R}(0)} {|\nabla v{|^{2}}} \ge \frac{1} {{2N}}\int_{{\mathbb{R}^{N}}} {|\nabla v{|^{2}}} \ge \frac{1} {2}{c_{V({x_0})}} > 0. \end{align}

By (3.9) and Sobolev's imbedding theorem, we have

\begin{align*} \frac{1} {N}\int_{{B_R}({y_\varepsilon })} {|\nabla {u_\varepsilon }{|^{2}}} &\le C\sum\limits_{i = 1}^{k} {\int_{{B_R}({y_\varepsilon })} {|\nabla (\varphi (\varepsilon x - z_\varepsilon ^{i}){W^{i}}(x - (z_\varepsilon ^{i}/\varepsilon ))){|^{2}}} } + C{d_0}\\ &\le C\sum\limits_{i = 1}^{k} {\int_{{B_R}({y_\varepsilon } - (z_\varepsilon ^{i}/\varepsilon ))} {|\nabla {W^{i}}{|^{2}}} } + C{d_0} + o(1) = C{d_0} + o(1), \end{align*}

where $o(1) \to 0$ as $\varepsilon \to 0$ and we have used $|{y_\varepsilon } - (z_\varepsilon ^{i}/\varepsilon )| \ge \beta /2\varepsilon$. This leads to a contradiction for $d_0$ small. Hence, (3.10) holds.

Since

\[ \mathop{\sup }\limits_{y \in {A_\varepsilon }} \int_{{B_1}(y)} {|{u_\varepsilon }{|^{2}}} \ge \mathop{\sup }\limits_{y \in {\mathbb{R}^{N}}} \int_{{B_1}(y)} {|{\eta _\varepsilon }{u_\varepsilon }{|^{2}}} , \]

where ${\eta _\varepsilon } \in C_c^{\infty } ({\mathbb {R}^{N}},[0,1])$ such that ${\eta _\varepsilon }(x) = 1$ for $x \in \mathop \cup \nolimits _{i = 1}^{k} ({B_{(3\beta /\varepsilon ) - 2}}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{(\beta /2\varepsilon ) + 2}}(z_\varepsilon ^{i}/\varepsilon ))$, ${\text {supp}}\,{\eta _\varepsilon } \subset \mathop \cup \nolimits _{i = 1}^{k} ({B_{(3\beta /\varepsilon ) - 1}}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{(\beta /2\varepsilon ) + 1}}(z_\varepsilon ^{i}/\varepsilon ))$, $|\nabla {\eta _\varepsilon }| \le C$ and $C>0$ is independent of $\varepsilon$. By (3.10) and the boundedness of ${\{ {\eta _\varepsilon }{u_\varepsilon }\} _{\varepsilon > 0}}$ in ${H^{1}}({\mathbb {R}^{N}})$, we derive from vanishing theorem that for any $q \in (2,{2^ * })$,

(3.13)\begin{equation} \mathop{\lim }\limits_{\varepsilon \to 0} \int_{\mathop{\bigcup}\nolimits_{i = 1}^{k} ({B_{2\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))} {|{u_\varepsilon }{|^{q}}} \to 0. \end{equation}

Step 2: Let ${u_{\varepsilon ,1}}(x): = \sum \nolimits _{i = 1}^{k} {u_{\varepsilon ,1}^{i}(x)} : = \sum \nolimits _{i = 1}^{k} {\varphi (\varepsilon x - z_\varepsilon ^{i}){u_\varepsilon }(x)}$, ${u_{\varepsilon ,2}}(x): = {u_\varepsilon }(x) - {u_{\varepsilon ,1}}(x)$, by (3.13), we see that

(3.14)\begin{align} &\int_{{\mathbb{R}^{N}}} {|\nabla {u_\varepsilon }{|^{2}}} \ge \int_{{\mathbb{R}^{N}}} {|\nabla u_{\varepsilon ,1}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {|\nabla u_{\varepsilon ,2}{|^{2}}} + o(1), \end{align}

(3.15)\begin{align} &\int_{{\mathbb{R}^{N}}} {V(\varepsilon x)|{u_\varepsilon }{|^{2}}} \ge \int_{{\mathbb{R}^{N}}} {V(\varepsilon x)|u_{\varepsilon ,1}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {V(\varepsilon x)|u_{\varepsilon ,2}{|^{2}}} , \end{align}

(3.16)\begin{align} &\int_{{\mathbb{R}^{N}}} {|{u_\varepsilon }{|^{\alpha + 2}}} = \int_{{\mathbb{R}^{N}}} {|{u_{\varepsilon ,1}}{|^{\alpha + 2}}} + \int_{{\mathbb{R}^{N}}} {|{u_{\varepsilon ,2}}{|^{\alpha + 2}}} + o(1), \end{align}

(3.17)\begin{align} &{Q_\varepsilon }({u_{\varepsilon ,1}}) = 0,{\quad{}}{Q_\varepsilon }({u_{\varepsilon ,2}}) = {Q_\varepsilon }({u_\varepsilon }) \ge 0. \end{align}

Moreover, we deduce from ${E_1} \in \mathcal {L}({L^{p}}({\mathbb {R}^{N}}),{L^{p}}({\mathbb {R}^{N}}))\ (1 < p < + \infty )$, the boundedness of ${\{ {u_\varepsilon }\} _{\varepsilon > 0}}$ in ${H^{1}}({\mathbb {R}^{N}})$, Hölder's inequality and (3.13) that

(3.18)\begin{equation} \int_{{\mathbb{R}^{N}}} {{E_1}({u_{\varepsilon ,1}}{u_{\varepsilon ,2}})|{u_\varepsilon }{|^{2}}} = o(1) \end{equation}

and arguing as in the proof of lemma 3.1(i), we see from (3.13) that

(3.19)\begin{equation} \int_{{\mathbb{R}^{N}}} {{E_1}(|{u_{\varepsilon ,1}}{|^{2}})|{u_{\varepsilon ,2}}{|^{2}}} = o(1). \end{equation}

Hence,

(3.20)\begin{equation} \int_{{\mathbb{R}^{N}}} {{E_1}(|{u_\varepsilon }{|^{2}})|{u_\varepsilon }{|^{2}}} = \int_{{\mathbb{R}^{N}}} {{E_1}(|{u_{\varepsilon ,1}}{|^{2}})|{u_{\varepsilon ,1}}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {{E_1}(|{u_{\varepsilon ,2}}{|^{2}})|{u_{\varepsilon ,2}}{|^{2}}} + o(1). \end{equation}

From (3.14)–(3.17), (3.20), we infer that

(3.21)\begin{equation} {J_\varepsilon }({u_\varepsilon }) \ge {J_\varepsilon }(u_{\varepsilon ,1}) + {J_\varepsilon }(u_{\varepsilon ,2}) + o(1). \end{equation}

From (3.9), it follows

\begin{align*} {\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}} &\le {\left\| {{u_{\varepsilon ,1}} - \sum\limits_{i = 1}^{k} {\varphi (\varepsilon x - z_\varepsilon ^{i}){W^{i}}(x - (z_\varepsilon ^{i}/\varepsilon ))} } \right\|_{{{H_\varepsilon }\left({\mathop{\bigcup}\nolimits_{i = 1}^{k} {B_{2\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )} \right)}}} + 2{d_0} \\ &\le {\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }\left( {\mathop{\bigcup}\nolimits_{i = 1}^{k} ({B_{2\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))} \right)}} + 4{d_0}\\ &\le C{\| {{u_\varepsilon }} \|_{{H_\varepsilon }\left({\mathop{\bigcup}\nolimits_{i = 1}^{k} ({B_{2\beta /\varepsilon }} (z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))} \right)}} + 4{d_0}\\ &\le C{\left\| {\sum\limits_{i = 1}^{k} {\varphi (\varepsilon x - z_\varepsilon ^{i}){W^{i}}(x - (z_\varepsilon ^{i}/\varepsilon ))} } \right\|_{{H_\varepsilon }\left({\mathop{\bigcup}\nolimits_{i = 1}^{k} ({B_{2\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))} \right)}} + C{d_0} \\ & \le C\sum\limits_{i = 1}^{k} {{{\| {\varphi (\varepsilon x - z_\varepsilon ^{i}){W^{i}}(x - (z_\varepsilon ^{i}/\varepsilon ))} \|}_{{H^{1}}({B_{2\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon ))}}} + C{d_0} \\ & \le C\sum\limits_{i = 1}^{k} {{{\| {{W^{i}}} \|}_{{H^{1}}({B_{2\beta /\varepsilon }}(0)\backslash {B_{\beta /\varepsilon }}(0))}}} + C{d_0} = C{d_0} + o(1), \end{align*}

where $o(1) \to 0$ as $\varepsilon \to 0$. Thus, $\mathop {\overline {\lim }}\nolimits _{\varepsilon \to 0} {\| {u_{\varepsilon ,2}} \|_{{H_\varepsilon }}} \le C{d_0}$.

Moreover, since $\langle {{J'_\varepsilon }({u_\varepsilon }),{u_{\varepsilon ,2}}} \rangle \to 0$ as $\varepsilon \to 0$ and $\langle {{Q'_\varepsilon }({u_\varepsilon }),{u_{\varepsilon ,2}}} \rangle = \langle {Q'_\varepsilon }({u_{\varepsilon ,2}}), {u_{\varepsilon ,2}} \rangle \ge 0$, we get

\[ \int_{{\mathbb{R}^{N}}} {\nabla {u_\varepsilon } \cdot \nabla {u_{\varepsilon ,2}}} + \int_{{\mathbb{R}^{N}}} {V(\varepsilon x){u_\varepsilon }{u_{\varepsilon ,2}}} \le \frac{b} {4}\int_{{\mathbb{R}^{N}}} {{E_1}(|{u_\varepsilon }{|^{2}}){u_{\varepsilon }}{u_{\varepsilon ,2}}} + C\int_{{\mathbb{R}^{N}}} {|{u_\varepsilon }{|^{\alpha} }{u_\varepsilon }{u_{\varepsilon ,2}}}. \]

Using Hölder's inequality and Sobolev's imbedding theorem, we see from (3.13), (3.18), (3.19) and ${E_1} \in \mathcal {L}({L^{p}}({\mathbb {R}^{N}}),{L^{p}}({\mathbb {R}^{N}})) (1 < p < + \infty )$ that

\[ \| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}^{2} \le C\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}^{4} + C\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}^{\alpha + 2} + o(1). \]

Choosing ${d_0} > 0$ small, we see that ${\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}} = o(1)$, by (3.21),

(3.22)\begin{equation} {J_\varepsilon }({u_\varepsilon }) \ge {J_\varepsilon }(u_{\varepsilon ,1}) + o(1). \end{equation}

Step 3: For each $1 \le i \le k$, letting $\tilde {w}_\varepsilon ^{i}(x): = u_{\varepsilon ,1}^{i}(x + (z_\varepsilon ^{i}/\varepsilon )): = \varphi (\varepsilon x){u_\varepsilon }(x + (z_\varepsilon ^{i}/\varepsilon ))$, up to a subsequence, as $\varepsilon \to 0$, $\exists {{\tilde {w}}^{i}} \in {H^{1}}({\mathbb {R}^{N}})$ such that $\tilde {w}_\varepsilon ^{i} \rightharpoonup {{\tilde {w}}^{i}}$ in ${H^{1}}({\mathbb {R}^{N}})$. Next, we claim that

(3.23)\begin{equation} \tilde{w}_\varepsilon ^{i} \to {{\tilde{w}}^{i}}{\text{ in }}{L^{q}}({\mathbb{R}^{N}})\quad{\text{for }}q \in (2,{2^ * }). \end{equation}

If not, by vanishing theorem, $\exists r > 0$ such that

\[ \mathop{\underline{\lim}}\limits_{\varepsilon \to 0} \mathop{\sup }\limits_{x \in {\mathbb{R}^{N}}} \int_{{B_1}(x)} {|\tilde{w}_\varepsilon ^{i} - {{\tilde{w}}^{i}}{|^{2}}} = 2r > 0, \]

then for $\varepsilon > 0$ small, $\exists x_\varepsilon ^{i} \in {\mathbb {R}^{N}}$ such that

(3.24)\begin{equation} \int_{{B_1}(x_\varepsilon ^{i})} {|\tilde{w}_\varepsilon ^{i} - {{\tilde{w}}^{i}}{|^{2}}} \ge r > 0. \end{equation}

There are two cases:

Case 1: ${\{ x_\varepsilon ^{i}\} _{\varepsilon > 0}}$ is bounded, that is, $|x_\varepsilon ^{i}| \le {R_i}$ for some ${R_i} > 0$, then for $\varepsilon > 0$ small,

\[ \int_{{B_{{R_i} + 1}}(0)} {|\tilde{w}_\varepsilon ^{i} - {{\tilde{w}}^{i}}{|^{2}}} \ge r > 0, \]

which contradicts $\tilde {w}_\varepsilon ^{i} \to {{\tilde {w}}^{i}}$ in $L_{{\text {loc}}}^{2}({\mathbb {R}^{N}})$.

Case 2: ${\{ x_\varepsilon ^{i}\} _{\varepsilon > 0}}$ is unbounded, by (3.24),

(3.25)\begin{equation} \mathop{\underline{\lim }}\limits_{\varepsilon \to 0} \int_{{B_1}(x_\varepsilon ^{i})} {|\varphi (\varepsilon x){u_\varepsilon }(x + (z_\varepsilon ^{i}/\varepsilon )){|^{2}}} \ge r > 0. \end{equation}

Since $\varphi (x) = 0$ for $|x| \ge 2\beta$, we see that $|{x_\varepsilon ^{i}}| \le 3\beta /\varepsilon$ for $\varepsilon > 0$ small. Furthermore, if $|{x_\varepsilon ^{i}}| \ge \beta /2\varepsilon$, then ${x_\varepsilon ^{i}} \in {B_{3\beta /\varepsilon }}(0)\backslash {B_{\beta /2\varepsilon }}(0)$, by (3.10),

\begin{align*} &\mathop{\underline{\lim}}\limits_{\varepsilon \to 0} \int_{{B_1}(x_\varepsilon ^{i})} {|\varphi (\varepsilon x){u_\varepsilon }(x + (z_\varepsilon ^{i}/\varepsilon )){|^{2}}} \\ &\quad \le \mathop{\underline {\lim}}\limits_{\varepsilon \to 0} \mathop{\sup}\limits_{z \in {B_{3\beta /\varepsilon }}(0)\backslash {B_{\beta /2\varepsilon }}(0)} \int_{{B_1}(z)} {|{u_\varepsilon }(x + (z_\varepsilon ^{i}/\varepsilon )){|^{2}}} \\ &\quad =\mathop{\underline{\lim}}\limits_{\varepsilon \to 0} \mathop {\sup }\limits_{y \in {B_{3\beta /\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )\backslash {B_{\beta /2\varepsilon }}(z_\varepsilon ^{i}/\varepsilon )} \int_{{B_1}(y)} {|{u_\varepsilon }{|^{2}}} \le \mathop{\underline{\lim}}\limits_{\varepsilon \to 0} \mathop{\sup}\limits_{y \in {A_\varepsilon }} \int_{{B_1}(y)} {|{u_\varepsilon }{|^{2}}} = 0, \end{align*}

which contradicts (3.25). Thus, $|{x_\varepsilon ^{i}}| \le \beta /2\varepsilon$ for $\varepsilon > 0$ small. Up to a subsequence, $\varepsilon x_\varepsilon ^{i} \to {x^{i}} \in \overline {{B_{\beta /2}}(0)}$ and $\bar {w}_\varepsilon ^{i}(x): = \tilde {w}_\varepsilon ^{i}(x + x_\varepsilon ^{i}) \rightharpoonup {{\bar {w}}^{i}}$ in ${H^{1}}({\mathbb {R}^{N}})$, by (3.25), ${{\bar {w}}^{i}} \ne 0$ and satisfies (2.1) with $m$ replaced by $V({z^{i}} + {x^{i}})$. Arguing as in step 1, we get a contradiction for ${d_0} > 0$ small. Thus, (3.23) follows.

Similar to the argument in lemma 3.1(i), we have

\[ {J_\varepsilon }({u_{\varepsilon ,1}}) = \sum\limits_{i = 1}^{k} {{J_\varepsilon }(u_{\varepsilon ,1}^{i}(x))} + o(1). \]

Recalling that $z_\varepsilon ^{i} \to {z^{i}}$ and $\tilde {w}_\varepsilon ^{i}(x) = u_{\varepsilon ,1}^{i}(x + (z_\varepsilon ^{i}/\varepsilon ))$, by (3.22), (3.23), we obtain

(3.26)\begin{equation} \sum\limits_{i = 1}^{k} {{I_{V({z^{i}})}}({{\tilde{w}}^{i}})} \le \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} . \end{equation}

For any $\psi \in C_c^{\infty } ({\mathbb {R}^{N}})$, letting $\psi (x - (z_\varepsilon ^{i}/\varepsilon ))$ as a test function for ${J'_\varepsilon }({u_\varepsilon })$. Since for $\varepsilon > 0$ small, ${\text {supp}}\,\psi (x - (z_\varepsilon ^{i}/\varepsilon )) \subset \Lambda /\varepsilon$, we observe that $\left \langle {{Q'_\varepsilon }({u_\varepsilon }),\psi (x - (z_\varepsilon ^{i}/\varepsilon ))} \right \rangle = 0$ and recalling ${\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}} = o(1)$, we see that ${{{\tilde {w}}^{i}}}$ is a solution of (2.1) with $m$ replaced by ${V({z^{i}})}$. Moreover, thanks to (3.19), (3.23), ${E_1} \in \mathcal {L}({L^{p}}({\mathbb {R}^{N}}),{L^{p}}({\mathbb {R}^{N}}))\ (1 < p < + \infty )$ and $\langle {{J'_\varepsilon }({u_\varepsilon }),u_{\varepsilon ,1}^{i}} \rangle \to 0$, ${\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}} \to 0$ as $\varepsilon \to 0$, we have

\begin{align*} &\int_{{\mathbb{R}^{N}}} {|\nabla {{\tilde{w}}^{i}}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {V({z^{i}})|{{\tilde{w}}^{i}}{|^{2}}} \\ &\quad\le\mathop{\underline {\lim } }\limits_{\varepsilon \to 0} \left[ {\int_{{\mathbb{R}^{N}}} {|\nabla \tilde{w}_\varepsilon ^{i}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {V(\varepsilon x + z_\varepsilon ^{i})|\nabla \tilde{w}_\varepsilon ^{i}{|^{2}}} } \right] \\ &\quad\le \mathop{\underline {\lim } }\limits_{\varepsilon \to 0} \left[ {\frac{b} {4}\int_{{\mathbb{R}^{N}}} {{E_1}(|\tilde{w}_\varepsilon ^{i}{|^{2}})|\tilde{w}_\varepsilon ^{i}{|^{2}}} - \frac{a} {{\alpha + 2}}\int_{{\mathbb{R}^{N}}} {|\tilde{w}_\varepsilon ^{i}{|^{\alpha + 2}}} } \right] \\ &\quad=\frac{b} {4}\int_{{\mathbb{R}^{N}}} {{E_1}(|{{\tilde{w}}^{i}}{|^{2}})|{{\tilde{w}}^{i}}{|^{2}}} - \frac{a} {{\alpha + 2}}\int_{{\mathbb{R}^{N}}} {|{{\tilde{w}}^{i}}{|^{\alpha + 2}}} \\ & \quad = \int_{{\mathbb{R}^{N}}} {|\nabla {{\tilde{w}}^{i}}{|^{2}}} + \int_{{\mathbb{R}^{N}}} {V({z^{i}})|{{\tilde{w}}^{i}}{|^{2}}} , \end{align*}

then as $\varepsilon \to 0$,

(3.27)\begin{equation} \int_{{\mathbb{R}^{N}}} {V(\varepsilon x + z_\varepsilon ^{i})|\tilde{w}_\varepsilon ^{i}{|^{2}}} \to \int_{{\mathbb{R}^{N}}} {V({z^{i}})|{{\tilde{w}}^{i}}{|^{2}}} \quad{\text{and}}\quad\int_{{\mathbb{R}^{N}}} {|\nabla \tilde{w}_\varepsilon ^{i}{|^{2}}} \to \int_{{\mathbb{R}^{N}}} {|\nabla {{\tilde{w}}^{i}}{|^{2}}} . \end{equation}

By (3.9), (3.23) and ${\| {{u_{\varepsilon ,2}}} \|_{{H_\varepsilon }}} = o(1)$, ${{\tilde {w}}^{i}} \ne 0$ for ${d_0} > 0$ small. Thus

(3.28)\begin{equation} {I_{V({z^{i}})}}({{\tilde{w}}^{i}}) \ge {c_{V({z^{i}})}}. \end{equation}

Since ${z^{i}} \in {({\mathcal {M}^{i}})^{\beta } } \subset {\Lambda ^{i}}$, (3.26) and (3.28) imply that $V({z^{i}}) = {m_i}$ and ${z^{i}} \in {\mathcal {M}^{i}}$. At this point, it is clear that $\exists {U^{i}} \in {S_{{m_i}}}$ and ${y^{i}} \in {\mathbb {R}^{N}}$ such that ${{\tilde {w}}^{i}}(x) = {U^{i}}(x - {y^{i}})$. Since

\[ {m_i}\int_{{\mathbb{R}^{N}}} {|\tilde{w}_\varepsilon ^{i}{|^{2}}} \le \int_{{\mathbb{R}^{N}}} {V(\varepsilon x + z_\varepsilon ^{i})|\tilde{w}_\varepsilon ^{i}{|^{2}}}, \]

by (3.27), $\tilde {w}_\varepsilon ^{i} \to {{\tilde {w}}^{i}}$ in ${H^{1}}({\mathbb {R}^{N}})$. Therefore,

\[ {\left\| {{u_\varepsilon } - \sum\limits_{i = 1}^{k} {\varphi (\varepsilon x - (z_\varepsilon ^{i} + \varepsilon {y^{i}})){U^{i}}(x - ((z_\varepsilon ^{i}/\varepsilon ) + {y^{i}}))} } \right\|_{{H_\varepsilon }}} \to 0\quad{\text{as }}\varepsilon \to 0. \]

This completes the proof.

Proof. Assuming on the contrary that, there exist $d \in (0,{d_0})$, $\{ {\varepsilon _j}\} _{j = 1}^{\infty }$, $\{ {\rho _j}\} _{j = 1}^{\infty }$ with ${\varepsilon _j}$, ${\rho _j} \to 0$, ${R_{{\varepsilon _j}}} \ge {R_0}/{\varepsilon _j}$ and ${u_j} \in J_{{\varepsilon _j}}^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} + {\rho _j}} \cap (X_{{\varepsilon _j}}^{{d_0}}\backslash X_{{\varepsilon _j}}^{d}) \cap H_0^{1}({B_{{R_{{\varepsilon _j}}}}}(0))$ such that

\[ {\| {{J'_{{\varepsilon _j}}}({u_j})} \|_{ * ,{\varepsilon _j},{R_{{\varepsilon _j}}}}} \to 0\quad{\text{as }}j \to \infty . \]

By lemma 3.3(i), for each $1 \le i \le k$, we find $\{ y_j^{i}\} _{j = 1}^{\infty } \subset {\mathbb {R}^{N}}$, ${z^{i}} \in {\mathcal {M}^{i}}$, ${U^{i}} \in {S_{{m_i}}}$ such that

\[ \mathop{\lim }\limits_{j \to \infty } |{\varepsilon _j}y_j^{i} - {z^{i}}| = 0\quad{\text{and}}\quad\mathop{\lim }\limits_{j \to \infty } {\left\| {{u_j} - \sum\limits_{i = 1}^{k} {\varphi ({\varepsilon _j}x - {\varepsilon _j}y_j^{i}){U^{i}}(x - y_j^{i})} } \right\|_{{H_{{\varepsilon _j}}}}} = 0, \]

which gives that ${u_j} \in X_{{\varepsilon _j}}^{d}$ for large $j$. This contradicts that ${u_j} \notin X_{{\varepsilon _j}}^{d}$.

Proof. First, there is a ${T'_0} > 0$ such that for each $1 \le i \le k$, $z_ * ^{i} \in {\mathcal {M}^{i}}$ mentioned in (3.2) and $u \in {H^{1}}({\mathbb {R}^{N}})$,

(3.29)\begin{equation} {\| {\varphi (\varepsilon x - z_ * ^{i})u(x - (z_ * ^{i}/\varepsilon ))} \|_{{H_\varepsilon }}} \le {T'_0}{\| {u(x)} \|_{{H^{1}}({\mathbb{R}^{N}})}}, \end{equation}

By lemma 2.3(ii), we define

\begin{align*} {\alpha _\delta } &= \frac{1} {4}\min \left\{ {\sum\limits_{i = 1}^{k} {{c_{{m_i}}}} - \sum\limits_{i = 1}^{k} {{I_{{m_i}}}(\gamma _\delta ^{i}({s_i}))} :{s_i} \in [0,1],\sum\limits_{i = 1}^{k} {{{\left\| {\gamma _\delta ^{i}({s_i}) - U_ * ^{i}} \right\|}_{{H^{1}}({\mathbb{R}^{N}})}}} \ge k \delta } \right\}\\ & > 0, \end{align*}

we have

(3.30)\begin{equation} \sum\limits_{i = 1}^{k} {I_{{m_i}}}(\gamma _\delta ^{i}({s_i})) \ge \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} - 2{\alpha _\delta }{\text{ implies }}\sum\limits_{i = 1}^{k} {{{\left\| {\gamma _\delta ^{i}({s_i}) - U_ * ^{i}} \right\|}_{{H^{1}}({\mathbb{R}^{N}})}}} \le k \delta . \end{equation}

Moreover, since $\gamma _\delta ^{i}([0,1])$ is compact in ${H^{1}}({\mathbb {R}^{N}})$, similar to the proof of lemma 3.1(i), we obtain

\[ \mathop{\max }\limits_{s \in {{[0,1]}^{k}}} \left| {{J_\varepsilon }({\gamma _{\varepsilon ,\delta }}(s)) - \sum\limits_{i = 1}^{k} {{I_{{m_i}}}(\gamma _\delta ^{i}({s_i}))} } \right| \to 0\quad{\text{as }}\varepsilon \to 0. \]

Hence, there exists an $\varepsilon _\delta >0$ such that

(3.31)\begin{equation} \mathop{\max}\limits_{s \in {{[0,1]}^{k}}} \left| {{J_\varepsilon }({\gamma _{\varepsilon ,\delta }}(s)) - \sum\limits_{i = 1}^{k} {{I_{{m_i}}}(\gamma _\delta ^{i}({s_i}))} } \right| \le {\alpha _\delta } \end{equation}

for all $\varepsilon \in (0,{\varepsilon _\delta })$. Thus, if $\varepsilon \in (0,{\varepsilon _\delta })$ and ${J_\varepsilon }({\gamma _{\varepsilon ,\delta }}(s)) \ge \sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - {\alpha _\delta }$, by (3.30) and (3.31), we have

\[ \sum\limits_{i = 1}^{k} {{{\left\| {\gamma _\delta ^{i}({s_i}) - U_ * ^{i}} \right\|}_{{H^{1}}({\mathbb{R}^{N}})}}} \le k \delta, \]

by (3.29), we have

\begin{align*} &{\left\| {\sum\limits_{i = 1}^{k} {\gamma _{\varepsilon ,\delta }^{i}({s_i})} - \sum\limits_{i = 1}^{k} {\varphi (\varepsilon x - z_ * ^{i})U_ * ^{i}(x - (z_ * ^{i}/\varepsilon ))} } \right\|_{{H_\varepsilon }}}\\ &\quad \le \sum\limits_{i = 1}^{k} {{{\left\| {\varphi (\varepsilon x - z_*^{i})((\gamma _\delta ^{i}({s_i})(x - (z_*^{i}/\varepsilon ))) - U_ * ^{i}(x - (z_ * ^{i}/\varepsilon )))} \right\|}_{{H_\varepsilon }}}}\\ &\quad \le {T'_0}\sum\limits_{i = 1}^{k} {{{\left\| {\gamma _\delta ^{i}({s_i})(x) - U_ * ^{i}(x)} \right\|}_{{H^{1}}({\mathbb{R}^{N}})}}} \le k {T'_0}\delta :={T_0}\delta. \end{align*}

Thus, ${\gamma _{\varepsilon ,\delta } }(s) \in X_\varepsilon ^{{T_0}\delta }$.

Proof. Assuming on the contrary that there always exist $\varepsilon > 0$ small, ${R_\varepsilon } > {R_0}/\varepsilon$ and $\gamma (\varepsilon ,{R_\varepsilon }) > 0$ such that

(3.32)\begin{equation} {\| {{J'_\varepsilon }(u)} \|_{ * ,\varepsilon ,{R_\varepsilon }}} \ge \gamma (\varepsilon ,{R_\varepsilon }) > 0 \end{equation}

for $u \in J_\varepsilon ^{{{\tilde c}_{\varepsilon ,{\delta _1}}} + \varepsilon } \cap X_\varepsilon ^{{d_0}} \cap H_0^{1}({B_{{R_\varepsilon }}}(0))$.

Letting $Y$ be a pseudo-gradient vector field for ${{J'_{\varepsilon }}}$ in $H_0^{1}({B_{{R_\varepsilon }}}(0))$, that is, $Y:H_0^{1}({B_{{R_\varepsilon }}}(0)) \to H_0^{1}({B_{{R_\varepsilon }}}(0))$ is a locally Lipschitz continuous vector field such that for every $u \in H_0^{1}({B_{{R_\varepsilon }}}(0))$,

(3.33)\begin{equation} {\| {Y(u)} \|_{{H_\varepsilon }}} \le 2{\| {{{J'}_\varepsilon }(u)} \|_{*,\varepsilon ,{R_\varepsilon }}}\quad{\text{and}}\quad\langle {J'_\varepsilon }(u),Y(u)\rangle \ge \| {{{J'}_\varepsilon }(u)} \|_{*,\varepsilon ,{R_\varepsilon }}^{2}. \end{equation}

Letting ${\psi _1}$, ${\psi _2}$ be locally Lipschitz continuous functions in $H_0^{1}({B_{{R_\varepsilon }}}(0))$ such that $0 \le {\psi _1},{\psi _2} \le 1$ and

\begin{align*} {\psi _1}(u) &= \begin{cases} 1, & \displaystyle\sum_{i = 1}^{k} {{c_{{m_i}}}} - \dfrac{1}{2}{\bar \alpha} \le {J_\varepsilon }(u) \le {{\tilde c}_{\varepsilon ,{\delta _1}}},\\ 0, & {J_\varepsilon }(u) \le \displaystyle\sum_{i = 1}^{k} {{c_{{m_i}}}} - {{\bar \alpha}}{\text{ or }}{{\tilde c}_{\varepsilon ,{\delta _1}}} + \varepsilon \le {J_\varepsilon }(u), \end{cases}\\ {\psi _2}(u) &= \begin{cases} 1, & u \in X_\varepsilon ^{3{d_0}/4},\\ 0, & u \notin X_\varepsilon ^{{d_0}}. \end{cases} \end{align*}

Considering the following ordinary differential equations:

(3.34)\begin{equation} \begin{cases} \displaystyle\dfrac{{\rm d}}{{{\rm d}r}}\eta (r,u) ={-} \dfrac{{Y(\eta (r,u))}} {{{{\| {Y(\eta (r,u))} \|}_{{H_\varepsilon }}}}}{\psi _1}(\eta (r,u)){\psi _2}(\eta (r,u)), \\ \eta (0,u) = u. \end{cases} \end{equation}

By (3.33) and (3.34), we have

\begin{align*} \frac{{\rm d}} {{{\rm d}r}}{J_\varepsilon }(\eta (r,u)) &= \left\langle {{J'_\varepsilon }(\eta (r,u)),\frac{{\rm d}} {{{\rm d}r}}\eta (r,u)} \right\rangle \\ &=\left\langle {{J'_\varepsilon }(\eta (r,u)), - \frac{{Y(\eta (r,u))}}{{{{\| {Y(\eta (r,u))} \|}_{{H_\varepsilon }}}}}{\psi _1}(\eta (r,u)){\psi _2}(\eta (r,u))} \right\rangle \\ &\le \displaystyle - \frac{{{\psi _1}(\eta (r,u)){\psi _2}(\eta (r,u))}} {{{{\| {Y(\eta (r,u))} \|}_{{H_\varepsilon }}}}}\| {{J'_\varepsilon }(\eta (r,u))} \|_{*,\varepsilon ,{R_\varepsilon }}^{2}\\ &\le - \frac{1} {2}{\psi _1}(\eta (r,u)){\psi _2}(\eta (r,u)){\| {{J'_\varepsilon }(\eta (r,u))} \|_{*,\varepsilon ,{R_\varepsilon }}} \end{align*}

and combining with lemma 3.1(i), lemma 3.4, (3.32), (3.34) and the definition of ${\psi _1}$, ${\psi _2}$, it is standard to show that $\eta \in C([0,+\infty ) \times H_0^{1}({B_{{R_\varepsilon }}}(0)),H_0^{1}({B_{{R_\varepsilon }}}(0)))$ and satisfies that for $\varepsilon > 0$ small,

1. (i) $({\textrm {d}}/{{\textrm {d}r}}){J_{\varepsilon }}(\eta (r,u)) \le 0$ for each $r \in [0,+\infty )$ and $u \in H_0^{1}({B_{{R_\varepsilon }}}(0))$;

2. (ii) $({\textrm {d}}/{{\textrm {d}r}}){J_\varepsilon }(\eta (r,u)) \le -{\omega _{{d_1}}}/2$ if $\eta (r,u) \in \overline {J_\varepsilon ^{{{\tilde {c}}_{\varepsilon ,{\delta _1}}}}\backslash J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - ({1}/{2}){\bar {\alpha }}}} \cap \overline {X_\varepsilon ^{3{d_0}/4}\backslash X_\varepsilon ^{{d_0}/4}}$;

3. (iii) $({\textrm {d}}/{{\textrm {d}r}}){J_\varepsilon }(\eta (r,u)) \le -\gamma (\varepsilon ,{R_\varepsilon })/2$ if $\eta (r,u) \in \overline {J_\varepsilon ^{{{\tilde c}_{\varepsilon ,{\delta _1}}}}\backslash J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - ({1}/{2}){\bar {\alpha }}}} \cap X_\varepsilon ^{3{d_0}/4}$;

4. (iv) $\eta (r,u) = u$ if ${J_\varepsilon }(u) \le \sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - {\bar \alpha }$.

Setting ${r_1}: = {\omega _{{d_1}}}{d_0}/\gamma (\varepsilon ,{R_\varepsilon })$ and ${\xi _\varepsilon }(s): = \eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s))$, we have the following cases:

Case 1: ${\gamma _{\varepsilon ,{\delta _1}}}(s) \in J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - {\bar \alpha }}$. By (iv), we see that

(3.35)\begin{equation} \eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) = {\gamma _{\varepsilon ,{\delta _1}}}(s). \end{equation}

Case 2: ${\gamma _{\varepsilon ,{\delta _1}}}(s) \notin J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - {\bar \alpha }}$. By lemma 3.5 and the definition of ${{{\tilde c}_{\varepsilon ,{\delta _1}}}}$, we see that

\[ {\gamma _{\varepsilon ,{\delta _1}}}(s) \in \overline {J_\varepsilon ^{{\tilde c}_{\varepsilon ,{\delta _1}}}\backslash J_\varepsilon ^{\sum\nolimits_{i = 1}^{k} {{c_{{m_i}}}} - {\bar{\alpha}}}} \cap X_\varepsilon ^{{d_0}/4}. \]

Moreover, we have

(3.36)\begin{equation} \eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in X_\varepsilon ^{{d_0}}\quad{\text{for }}r \in [0,{r_1}]. \end{equation}

Indeed, if not, $\exists r' \in [0,{r_1}]$ such that $\eta (r',{\gamma _{\varepsilon ,{\delta _1}}}(s)) \notin X_\varepsilon ^{{d_0}}$. Denote

\[ r': = \sup \left\{ {r \in [0,r']:\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in X_\varepsilon ^{{d_0}}} \right\}, \]

then by (3.34) and the definition of ${\psi _2}$, we see $\eta (r',{\gamma _{\varepsilon ,{\delta _1}}}(s)) = \eta (r'',{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in X_\varepsilon ^{{d_0}}$, which leads to a contradiction.

Next, we divide case 2 into the following three subcases:

Case 2.1: $\eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - ({1}/{2}){\bar {\alpha }}}$;

Case 2.2: $\eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in J_\varepsilon ^{{\tilde c}_{\varepsilon ,{\delta _1}}}\backslash J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - ({1}/{2}){\bar {\alpha }}}$ and $\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \notin X_\varepsilon ^{3{d_0}/4}$ for some $r \in [0,{r_1}]$;

Case 2.3: $\eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in J_\varepsilon ^{{\tilde c}_{\varepsilon ,{\delta _1}}}\backslash J_\varepsilon ^{\sum \nolimits _{i = 1}^{k} {{c_{{m_i}}}} - ({1}/{2}){\bar {\alpha }}}$ and $\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in X_\varepsilon ^{3{d_0}/4}$ for all $r \in [0,{r_1}]$.

In case 2.2, denote

\[ {r_2}: = \inf \left\{ {r \in [0,{r_1}]:\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \notin X_\varepsilon ^{3{d_0}/4}} \right\} \]

and

\[ {r_3}: = \sup \left\{ {r \in [0,{r_2}]:\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in X_\varepsilon ^{{d_0}/4}} \right\}, \]

then by (3.34), ${r_2} - {r_3} \ge ({1} /{2}){d_0}$ and $\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)) \in \overline {X_\varepsilon ^{3{d_0}/4}\backslash X_\varepsilon ^{{d_0}/4}}$ for each $r \in [{r_3},{r_2}]$. By (i), (ii) and lemma 3.1(i), we obtain

\begin{align*} {J_\varepsilon }(\eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s))) &= {J_\varepsilon }({\gamma _{\varepsilon ,{\delta _1}}}(s)) + \int_0^{{r_1}} {\frac{{\rm d}}{{{\rm d}r}}{J_\varepsilon }(\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)))} \,{\rm d}s \\ &\le {{\tilde c}_{\varepsilon ,{\delta _1}}} + \int_{{r_3}}^{{r_2}} {\frac{{\rm d}} {{{\rm d}r}}{J_\varepsilon }(\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)))} \,{\rm d}s \\ &\le {{\tilde c}_{\varepsilon ,{\delta _1}}} - \frac{1} {4}{\omega _{{d_1}}}{d_0} = \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} - \frac{1} {4}{\omega _{{d_1}}}{d_0} + o(1), \end{align*}

where $o(1) \to 0$ as $\varepsilon \to 0$.

In case 2.3, by (iii) and the definition of $r_1$, we have

\begin{align*} {J_\varepsilon }(\eta ({r_1},{\gamma _{\varepsilon ,{\delta _1}}}(s))) &= {J_\varepsilon }({\gamma _{\varepsilon ,{\delta _1}}}(s)) + \int_0^{{r_1}} {\frac{{\rm d}}{{{\rm d}r}}{J_\varepsilon }(\eta (r,{\gamma _{\varepsilon ,{\delta _1}}}(s)))} \,{\rm d}s \\ &\le {{\tilde c}_{\varepsilon ,{\delta _1}}} - \frac{1} {2}{\omega _{{d_1}}}{d_0} = \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} - \frac{1}{2}{\omega _{{d_1}}}{d_0} + o(1). \end{align*}

To sum up, choosing $\bar \mu = \min \{ {{\bar {\alpha }}/2,{\omega _{{d_1}}}{d_0}/4} \} > 0$, we see that, for ${s \in {{[0,1]}^{k}}}$,

(3.37)\begin{equation} {J_\varepsilon }({\xi _\varepsilon }(s)) \le \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} - \bar \mu + o(1). \end{equation}

From (3.35) and (3.36), we have

(3.38)\begin{equation} {\| {{\xi _\varepsilon }(s)} \|_{{H_\varepsilon }}} \le C\quad{\text{for }}\varepsilon > 0{\text{ small and }}s \in {[0,1]^{k}}. \end{equation}

Letting $\psi \in C_c^{\infty } ({\mathbb {R}^{N}},[0,1])$ be such that $\psi (x) = 1$ for $x \in \Lambda$, $\psi (x) = 0$ for $x \notin {\Lambda ^{\beta } }$ and $|\nabla \psi | \le C/\beta$. We denote ${\xi _{\varepsilon ,1}}(s): = {\psi _\varepsilon }{\xi _\varepsilon }(s)$ and ${\xi _{\varepsilon ,2}}(s): = (1 - {\psi _\varepsilon }){\xi _\varepsilon }(s)$, where ${\psi _\varepsilon }(x) = \psi (\varepsilon x)$. From (3.37) and (3.38), we see that ${Q_\varepsilon }({\xi _\varepsilon }(s))$ is uniformly bounded, then for $\varepsilon >0$ small and $s \in {[0,1]^{k}}$,

(3.39)\begin{equation} \int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _\varepsilon }(s){|^{2}}} \le C\varepsilon . \end{equation}

By Gagliardo–Nirenberg's inequality, (3.38) and (3.39), we see that for each $2 < p < {2^ * }$,

(3.40)\begin{align} \int_{{\mathbb{R}^{N}}} {|{\xi _{\varepsilon ,2}}(s){|^{p}}} & \le C \left\| {\nabla {\xi _{\varepsilon ,2}}(s)} \right\|_{{L^{2}}({\mathbb{R}^{N}})}^{{{(p - 2)}} /{2}N}\left\| {{\xi _{\varepsilon ,2}}(s)} \right\|_{{L^{2}}({\mathbb{R}^{N}})}^{2 + {{(p - 2)}}/{2}(2 - N)} \nonumber\\ &\le C\left\| {{\xi _\varepsilon }(s)} \right\|_{{L^{2}}({{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )})}^{2 + {{(p - 2)}}/{2}(2 - N)} = o(1), \end{align}

where $o(1) \to 0$ as $\varepsilon \to 0$ uniformly for $s \in {[0,1]^{k}}$. By (3.38), (3.40), Hölder's inequality and ${E_1} \in \mathcal {L}({L^{p}}({\mathbb {R}^{N}}),{L^{p}}({\mathbb {R}^{N}})) (1 < p < + \infty )$, we get

\[ {J_\varepsilon }({\xi _\varepsilon }(s)) \ge {J_\varepsilon }({\xi _{\varepsilon ,1}}(s)) + {Q_\varepsilon }({\xi _\varepsilon }(s)) - {Q_\varepsilon }({\xi _{\varepsilon ,1}}(s)) + o(1). \]

Since for $A,B \ge 0$, ${(A + B - 1)_ + } \ge {(A - 1)_ + } + {(B - 1)_ + }$, it follows that

(3.41)\begin{align} {Q_\varepsilon }({\xi _\varepsilon }(s)) &= \left( {{\varepsilon ^{ - 1}}\displaystyle\int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _{\varepsilon ,1}}(s) + {\xi _{\varepsilon ,2}}(s){|^{2}}} - 1} \right)_ + ^{2} \nonumber\\ &\ge \left( {{\varepsilon ^{ - 1}}\displaystyle\int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _{\varepsilon ,1}}(s){|^{2}}} + {\varepsilon ^{ - 1}}\int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _{\varepsilon ,2}}(s){|^{2}}} - 1} \right)_ + ^{2} \nonumber\\ &\ge \left( {{\varepsilon ^{ - 1}}\displaystyle\int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _{\varepsilon ,1}}(s){|^{2}}} - 1} \right)_ + ^{2} + \left( {{\varepsilon ^{ - 1}}\displaystyle\int_{{\mathbb{R}^{N}}\backslash (\Lambda /\varepsilon )} {|{\xi _{\varepsilon ,2}}(s){|^{2}}} - 1} \right)_ + ^{2} \nonumber\\ &= {Q_\varepsilon }({\xi _{\varepsilon ,1}}(s)) + {Q_\varepsilon }({\xi _{\varepsilon ,2}}(s)). \end{align}

Thus,

(3.42)\begin{equation} {J_\varepsilon }({\xi _\varepsilon }(s)) \ge {J_\varepsilon }({\xi _{\varepsilon ,1}}(s)) + o(1). \end{equation}

Next, defining $\xi _{\varepsilon ,1}^{i}(s)(x) = {\xi _{\varepsilon ,1}}(s)(x)$ for $x \in {({\Lambda ^{i}})^{\beta } }/\varepsilon$, $\xi _{\varepsilon ,1}^{i}(s)(x) = 0$ for $x \notin {({\Lambda ^{i}})^{\beta } }/\varepsilon$ for each $1 \le i \le k$. Arguing as in the proof of lemma 3.1(i) and (3.42), we get

(3.43)\begin{equation} {J_\varepsilon }({\xi _{\varepsilon ,1}}(s)) \ge \sum\limits_{i = 1}^{k} {{J_\varepsilon }(\xi _{\varepsilon ,1}^{i}(s))} + o(1) = \sum\limits_{i = 1}^{k} {J_\varepsilon ^{i}(\xi _{\varepsilon ,1}^{i}(s))} + o(1). \end{equation}

Next, we introduce some notations as in [Reference Coti Zelati and Rabinowitz17]. For $s = ({s_1},\ldots ,{s_k}) \in {[0,1]^{k}}$, let ${0_i} = ({s_1},\ldots ,{s_{i - 1}},0,{s_{i + 1}},\ldots ,{s_k})$ and ${1_i} = ({s_1},\ldots ,{s_{i - 1}},1,{s_{i + 1}},\ldots ,{s_k})$. We see from lemma 3.1(ii) and (iv) in the proof of lemma 3.6 that ${\xi _\varepsilon }({0_i}) = {\gamma _{\varepsilon ,{\delta _1}}}({0_i})$ and ${\xi _\varepsilon }({1_i}) = {\gamma _{\varepsilon ,{\delta _1}}}({1_i})$, then by the definition of $\xi _{\varepsilon ,1}^{i}(s)$, we see that $J_\varepsilon ^{i}(\xi _{\varepsilon ,1}^{i}({0_i})) = J_\varepsilon ^{i}(0) = 0$ and $J_\varepsilon ^{i}(\xi _{\varepsilon ,1}^{i}({1_i})) = J_\varepsilon ^{i}(\gamma _{\varepsilon ,{\delta _1}}^{i}(1)) < 0$ for $\varepsilon > 0$ small by (3.3). Using the celebrated gluing method due to Coti Zelati and Rabinowitz [Reference Coti Zelati and Rabinowitz17, proposition 3.4], there exists ${{\bar s}_\varepsilon } \in {[0,1]^{k}}$ such that

(3.44)\begin{equation} J_\varepsilon ^{i}(\xi _{\varepsilon ,1}^{i}({{\bar{s}}_\varepsilon })) \ge c_{\varepsilon ,{\delta _1}}^{i}\quad {\text{for each }}1 \le i \le k. \end{equation}

(3.42), (3.43), (3.44) and lemma 3.2 yield

\[ \mathop{\max}\limits_{s \in {{[0,1]}^{k}}} {J_\varepsilon }({\xi _\varepsilon }(s)) \ge \sum\limits_{i = 1}^{k} {{c_{{m_i}}}} + o(1), \]

which contradicts (3.37) for $\varepsilon > 0$ small.