2. The functional setting

In the forthcoming sections, we apply a change of variables that transforms (1) into a rescaled equation we will work with, namely

(8) \begin{equation} -\Delta u + Q(\epsilon x)u -\Delta (u^{2})u = h(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}. \end{equation}

The energy functional corresponding to (8) is given by

(9) \begin{align} J_{\epsilon }(u) = \frac {1}{2}\int _{\mathbb{R}^N} (1+2|u|^2)|\nabla u |^2 dx+ \frac {1}{2}\int _{\mathbb{R}^N} Q(\epsilon x)|u|^2 dx - \int _{\mathbb{R}^N} H(u) dx, \end{align}

where $H(s)=\int _0^s h(t)dt$ . As observed in [Reference Severo27], there are some technical difficulties to apply variational methods directly for $J_{\epsilon }$ . The main difficulty is related to the fact that $J_{\epsilon }$ is not well defined in $H^{1}(\mathbb{R}^N)$ . Indeed, by a straightforward computation, if $u\in C_0^1(\mathbb{R}^N)$ is defined by

\begin{equation*} u(x)=|x|^{(2-N)/4}\ \ \mbox{for}\ \ x\in B_1-\{0\}, \end{equation*}

then $u\in H^{1}(\mathbb{R}^N)$ , but the function $|u|^2|\nabla u |^2 \not \in L^1(\mathbb{R}^N)$ . To bypass this difficulty, we use a change of variables developed in [Reference Severo27], which generalises the one found in [Reference Colin and Jeanjean8] and [Reference Liu, Wang and Wang21].

More precisely, we introduce the change of variables $v= f^{-1}(u)$ , where $f$ is such that

(10) \begin{align} \begin{array}{cllc} f(t)&=&-f(\!-\!t) & \;\;\mbox{ for }\;\; t \leq 0, \\[4pt] f'(t)&=&\dfrac {1}{(1+ 2|\, f(t)|^2)^{1/2}} & \;\;\mbox{ for }\;\; t \geq 0. \end{array} \end{align}

Therefore, using the above change of variables, we consider a new functional $\Psi _{\epsilon }$ given by

(11) \begin{equation} \Psi _{\epsilon }(v) = J_{\epsilon }(\,f(v))= \frac {1}{2}\int _{\mathbb{R}^N} |\nabla v |^2 dx + \frac {1}{2}\int _{\mathbb{R}^N} Q(\epsilon x)|\, f(v)|^2 dx\ - \int _{\mathbb{R}^N} H(\,f(v)) dx, \end{equation}

which is well defined in the Banach space $H^{1}({\mathbb{R}^N})$ endowed with the norm

(12) \begin{equation} \|u\|_{\epsilon } =\left [\displaystyle \int _{\mathbb{R}^N} |\nabla u|^{2} dx +\displaystyle \int _{\mathbb{R}^N}Q(\epsilon x)|u|^{2}dx \right ]^{1/2}. \end{equation}

A direct computation shows that $\Psi _{\epsilon }:H^{1}({\mathbb{R}^N}) \to \mathbb{R}$ is of class $C^1$ under conditions (2) and (5). Moreover,

(13) \begin{equation} \Psi _{\epsilon }'(v)w= \displaystyle \int _{\mathbb{R}^N}\nabla v\nabla w dx + Q(\epsilon x) \, f(v) \, f'(v)w dx - \displaystyle \int _{\mathbb{R}^N}h(\,f(v)) \, f'(v)w dx \ \text{for} \ \ v,w\in H^{1}(\mathbb{R}^N). \end{equation}

Thus, the critical points of $\Psi _{\epsilon }$ correspond exactly to the weak solutions of the problem

(14) \begin{align} \ \ \left \{ \begin{array}{l} -\Delta v + Q(\epsilon x) \, f(v) \, f'(v)=h(\,f(v)) \, f'(v) \ \ \mbox{in} \ \ \mathbb{R}^N \\[3pt] v \in H^{1}(\mathbb{R}^N). \end{array} \right . \end{align}

The result below is taken from [Reference Severo27], and it establishes a relationship between solutions of (14) with the ones of (8).

From the above Lemma 2.1, it is clear that to obtain a weak solution of (8), it is sufficient to obtain a critical point of the functional $\Psi _{\epsilon }$ in $H^{1}(\mathbb{R}^N)\cap L^{\infty }_{loc}(\mathbb{R}^N)$ .

We collect below some properties related to the change of variables $f \, :\, \mathbb{R}\to \mathbb{R}$ defined in (10). The result follows from [Reference Severo27].

The following additional properties of $f$ are already proved in [Reference Alves, Figueiredo and Severo2].

Using (2) we endow $H^{1}(\mathbb{R}^{N})$ with two equivalent norms, namely

\begin{equation*} \|u\|^{2}_{0}= \int _{\mathbb{R}^N} |\nabla v |^2 dx + \int _{\mathbb{R}^N} Q_0 |v|^2 dx \end{equation*}

and

\begin{equation*} \|u\|^{2}_{\infty }= \displaystyle \int _{\mathbb{R}^{N}}|\nabla u|^{2} \ dx + \displaystyle \int _{\mathbb{R}^{N}} Q_{\infty }u^{2} \ dx. \end{equation*}

We also define the functionals $\Psi _{0}$ and $\Psi _{\infty } \, : \, H^{1}(\mathbb{R}^{N}) \to \mathbb{R}$ by

(15) \begin{equation} \Psi _{0}(u)= \frac {1}{2}\int _{\mathbb{R}^N} |\nabla v |^2 dx + \frac {1}{2}\int _{\mathbb{R}^N} Q_{0}|\, f(v)|^2 dx\ - \int _{\mathbb{R}^N} H(\,f(v)) dx \end{equation}

and

(16) \begin{equation} \Psi _{\infty }(u)= \frac {1}{2}\int _{\mathbb{R}^N} |\nabla v |^2 dx + \frac {1}{2}\int _{\mathbb{R}^N} Q_{\infty }|\, f(v)|^2 dx\ - \int _{\mathbb{R}^N} H(\,f(v)) dx. \end{equation}

Our goal now is to prove the existence of a nonnegative solution for the equation:

(17) \begin{align} \ \ \left \{ \begin{array}{l} -\Delta v + Q_{0}f(v) \, f'(v)=h(\,f(v)) \, f'(v) \ \ \mbox{in} \ \ \mathbb{R}^N \\[3pt] v \in H^{1}(\mathbb{R}^N). \end{array} \right . \end{align}

which is the Euler–Lagrange equation corresponding to $\Psi _0$ . Furthermore, $\Psi _0$ is $C^{1}$ on $H^{1}(\mathbb{R}^{N})$ with

\begin{equation*} \Psi _0'(u)w =\left [\displaystyle \int _{\mathbb{R}^N}\nabla v\nabla w\ + Q_{0}f(v) \, f'(v)w\right ] dx- \displaystyle \int _{\mathbb{R}^N}h(\,f(v)) \, f'(v)wdx \,\, \mbox{for every} \,\, w \in H^{1}(\mathbb{R}^{N}). \end{equation*}

Consequently, critical points of $\Psi _{0}$ are precisely the weak solutions of (17). We denote by $\mathcal{W}_{0}$ the Nehari manifold corresponding to $\Psi _{0}$ , defined by

(18) \begin{equation} \mathcal{W}_{0}=\left \{u\in H^{1}(\mathbb{R}^{N}) \, : \,u \neq 0, \Psi _{0}'(u)u=0\right \}\!\!. \end{equation}

The first lemma is related to the fact that $\Psi _{0}$ has the mountain pass shape. The proof is found in [Reference Severo27, Lemma4.1]; see also [Reference Brezis4].

By means of Lemma 2.4, we are able to apply [Reference Willem33, Theorem 1.15] to ensure the existence of a Palais–Smale sequence $(u_{n})$ in $H^{1}(\mathbb{R}^{N})$ at the level $\zeta _0$ , that is, a sequence with the property:

(19) \begin{equation} \Psi _{0}(u_{n})\to \zeta _{0} \,\,\, \mbox{and}\,\,\, \Psi '_{0}(u_{n})\to 0, \end{equation}

where $\zeta _{0}$ is the minimax level of the mountain pass theorem applied to $\Psi _{0}$ , namely

\begin{equation*} \zeta _{0}=\displaystyle \inf _{\gamma \in \Upsilon }\displaystyle \max _{t\in [0,1]} \displaystyle \Psi _{0}(\gamma (t)), \end{equation*}

where $\Upsilon =\{\gamma \in C([0,1], H^{1}(\mathbb{R}^{N})) \, : \, \gamma (0)=0 \ \mbox{and} \ \Psi _{0}(\gamma (1))\lt 0\}$ .

The proof of the result below is a combination of [Reference Zhang and Ji34, Lemma 3.1] and [Reference Willem33, Lemma 4.1].

\begin{equation*} \int _{\mathbb{R}^{N}}H(\,f(u))dx \leq 2^{s/4}\int _{\mathbb{R}^{N}}|u|^{s/2}dx \end{equation*}

and again by (5), Lemma2.2 (ii), (iii) and (v), we have

\begin{equation*} \int _{\mathbb{R}^{N}} h(\,f(u)) \, f^{\prime }(u)udx \leq 2^{s/4}\int _{\mathbb{R}^{N}}|u|^{s/2}dx. \end{equation*}

Throughout the paper, $\rightharpoonup$ stands for weak convergence

Proof. Note that from (6), there is $C\gt 0$ such that

\begin{eqnarray*} C(1+\|u_{n}\|_0)\geq \Psi _{0}(u_{n})-\frac {1}{\vartheta }\Psi '_{0}(u_{n})u_{n}, \,\,\, \forall n \in \mathbb{N} \end{eqnarray*}

proving that $(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{N})$ . For a subsequence, $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and $u_{n}\to u$ in $L^{\mu }_{loc}(\mathbb{R}^{N})$ for $2\leq \mu \lt {2N}/{(N-2)}$ . By a reasoning as in [Reference Severo27] (see Remark2),

\begin{equation*} \Psi ^{\prime }_{0}(u_{n})\psi \to \Psi ^{\prime }_{0}(u)\psi , \end{equation*}

for every $\psi \in C^{\infty }_{0}(\mathbb{R}^{N})$ . By density, we obtain $\Psi _{0}'(u)\psi =0$ for every $\psi \in H^{1}(\mathbb{R}^{N})$ . Concluding, from (4) we obtain

\begin{eqnarray*} o_{n}(1)=\Psi _{0}'(u_{n})u^{-}_{n}= (u_n,u^{-}_{n})_0= - \|u^{-}_{n}\|^{2}_{0}. \end{eqnarray*}

Therefore,

\begin{equation*} \Psi _{0}(u_{n})=\Psi _{0}(u_{n}^{+})+o_{n}(1) \end{equation*}

and

\begin{equation*} \Psi _{0}'(u_{n})=\Psi _{0}'(u_{n}^{+})+o_{n}(1). \end{equation*}

Thus, $(u_{n})$ is a nonnegative sequence.

Next result is already present in [Reference Severo27].

Proof. By Lemma 2.6, there exists $u\in H^{1}(\mathbb{R}^{N})$ such that $\Psi _{0}'(u)=0$ and $u\geq 0$ . If $u\equiv 0$ , then $\{u_{n}\}$ does not converge to zero in $H^{1}(\mathbb{R}^{N})$ , because a contradiction argument would imply $\zeta _{0}=0$ . Thus, from Lemma 2.7, there is a sequence $( \, y_n)$ in $\mathbb{R}^N$ and $R, \alpha \gt 0$ such that

(20) \begin{eqnarray} \displaystyle \liminf _{n\to \infty }\int _{B_{R}( \, y_n)}|u_{n}|^{2}dx\gt a. \end{eqnarray}

Defining $\widetilde {u}_{n}(x) = u_{n}(x+x_n)$ , using the translation invariance of $\mathbb{R}^N$ , by a standard calculation, we conclude that $\|\widetilde {u}_{n}\|_0 = \|u_{n}\|_0$ , $\Psi _{0}({\widetilde u}_{n}) = \Psi _{0}(u_{n})$ and $\Psi _{0}'({\widetilde u}_{n})=0$ . Then, there is $\widetilde u$ such that ${\widetilde u_n}\rightharpoonup {\widetilde u}$ in $H^{1}(\mathbb{R}^{N})$ and as before it follows that $\Psi _{0}'({\widetilde u})= 0$ . Now, by (20), taking a subsequence and a sufficiently large $R\gt 0$ , we conclude that $\widetilde u$ is nontrivial. The proof is complete.

Observe that Lemmas 2.4–2.8, already proved for (17), are also valid for

(21) \begin{align} \ \ \left \{ \begin{array}{l} -\Delta v + Q_{\infty }f(v) \, f'(v)=h(\,f(v)) \,f'(v) \ \ \mbox{in} \ \ \mathbb{R}^N \\[3pt] v \in H^{1}(\mathbb{R}^N). \end{array} \right . \end{align}

In what follows, denote the mountain level corresponding to $\Psi _{\infty }$ defined in (16) by

\begin{equation*} \zeta _{\infty }=\displaystyle \inf _{\upsilon \in \Xi }\displaystyle \max _{t\in [0,1]} \displaystyle \Psi _{\infty }(\upsilon (t)), \end{equation*}

where $\Xi =\{\upsilon \in C([0,1], H^{1}(\mathbb{R}^{N})) \, : \, \upsilon (0)=0 \ \mbox{and} \ \Psi _{\infty }(\upsilon (1))\lt 0\}$ , as well as the Nehari manifold

\begin{equation*} \mathcal{W}_{\infty }=\left \{u\in H^{1}(\mathbb{R}^{N}) \, : \, u \neq 0, \Psi _{\infty }'(u)u=0 \right \}. \end{equation*}

Recalling $\Psi _0$ , $\zeta _0$ and $\mathcal{W}_{0}$ we denote by $Z_0$ and $Z_{\infty }$ the solutions of (17) and (21), respectively. In other words,

(22) \begin{equation} \Psi _0(Z_0)= \zeta _0, \quad \Psi '_0(Z_0)= 0 \end{equation}

and

(23) \begin{equation} \Psi _\infty (Z_\infty )= \zeta _\infty , \quad \Psi '_\infty (Z_\infty )= 0. \end{equation}

Since $0\lt Q_0\lt Q_\infty$ by (2), we have $\zeta _0\lt \zeta _\infty$ . We set

(24) \begin{equation} \sigma ^{*}= \left ( \frac {\zeta _\infty - \zeta _0}{2} \right )\!. \end{equation}

3. The equation with rescaled potential

We focus now on Equation (14). The functional $\Psi _\epsilon \, : \,\, H^{1}(\mathbb{R}^{N}) \to \mathbb{R}$ is $C^1$ and

\begin{equation*} \begin{array}{l} \Psi _{\epsilon }'(v)w= \displaystyle \int _{\mathbb{R}^N}\nabla v\nabla w dx + Q(\epsilon x) \, f(v) \, f'(v)w dx - \displaystyle \int _{\mathbb{R}^N}h(\,f(v)) \, f'(v)w dx, \end{array} \end{equation*}

for every $w \in H^{1}(\mathbb{R}^{N})$ .

Tracking a parallel with Lemma 2.4, we employ [Reference Willem33, Theorem 1.15] to ensure the existence of a Palais–Smale sequence $(u_{n})$ in $H^{1}(\mathbb{R}^{N})$ corresponding to $\Psi _\epsilon$ at the level $\zeta _{\epsilon }$ , that is, a sequence verifying

\begin{equation*} \Psi _{\epsilon }(u_{n})\to \zeta _{\epsilon } \,\,\, \mbox{and}\,\,\, \Psi '_{\epsilon }(u_{n})\to 0, \end{equation*}

where $\zeta _{\epsilon }$ is the minimax level of the mountain pass theorem with respect to $\Psi _{\epsilon }$ , that is,

\begin{equation*} \zeta _{\epsilon }=\displaystyle \inf _{\varrho \in \Lambda }\displaystyle \max _{t\in [0,1]} \displaystyle \Psi _{\epsilon }(\varrho (t)), \end{equation*}

where $\Lambda =\{\varrho \in C([0,1], H^{1}(\mathbb{R}^{N})) \, : \, \varrho (0)=0 \ \mbox{and} \ \Psi _{\epsilon }(\varrho (1))\lt 0\}$ .

By (4) and Lemma 2.6, without loss of generality, $(u_n)$ is bounded, and that for every $n$ , we have $u_{n} \geq 0$ . Therefore, for a subsequence, there is $u \in H^{1}(\mathbb{R}^{N})$ such that

\begin{equation*} u_{n}\rightharpoonup u \,\,\, \mbox{in} \,\,\, H^{1}(\mathbb{R}^{N}) \,\,\, \mbox{and}\,\,\, u_{n}(x)\to u(x) \,\,\, \mbox{a.e.}\,\,\, \mbox{in}\,\,\, \mathbb{R}^N. \end{equation*}

By analogy with Lemma 2.5, we deduce that $\zeta _{\epsilon }$ satisfies

\begin{equation*} \zeta _{\epsilon }= \displaystyle \inf _{u\in H^{1}(\mathbb{R}^{N}), u \neq 0}\sup _{t\geq 0}\Psi _{\epsilon }(tu)= \inf _{u\in \Pi _{\epsilon }}\Psi _{\epsilon }(u), \end{equation*}

where $\Pi _{\epsilon }$ is by definition the Nehari manifold with respect to $\Psi _{\epsilon }$ given by

(25) \begin{equation} \Pi _{\epsilon }=\left \{u\in H^{1}(\mathbb{R}^{N})\, : \, u \neq 0, \Psi '_{\epsilon }(u)u =0 \right \}. \end{equation}

Proof. By (2), we obtain $\zeta _0 \leq \zeta _{\epsilon }$ for every $\epsilon \gt 0$ . Let $\eta _{\epsilon } \gt 0$ such that $\eta _{\epsilon } Z_0 \in \Pi _{\epsilon }$ , here $Z_0$ is the one at (22). Then, using Lemma 2.3, item $(i)$ and item $(iii)$ , there is $R\gt 0$ such that

\begin{equation*} \displaystyle \int _{\mathbb{R}^{N}}|\nabla Z_0|^{2} dx \geq b\displaystyle \int _{B_{R}(0)}h(\,f(\eta _{\epsilon } b)) \, f'(\eta _{\epsilon } b)(\eta _{\epsilon } b)^{-1} dx- b\displaystyle \int _{B_{R}(0)}Q_{\infty }\,f(\eta _{\epsilon } b) \, f'(\eta _{\epsilon } b)(t_{\epsilon } b)^{-1}dx, \end{equation*}

where $b=\min \{Z_{0}(x) \, : \, x\in \overline {B_{R}(0)}\}$ . Thus, $(\eta _{\epsilon })$ is bounded in $\mathbb{R}$ , because otherwise, from Lemma 2.3 items $(i)$ and $(iii)$ , again we have $\displaystyle \lim _{\epsilon \to 0} \int _{\mathbb{R}^{N}}|\nabla Z_0|^{2} dx =\infty$ , which is a contradiction. Notice also that $\eta _{\epsilon }$ does not converge to $0$ , otherwise from (5) and Lemma 2.2, item $(iii)$ , given $\epsilon \gt 0$ , there is $C_{\epsilon }\gt 0$ such that

\begin{equation*} 0\leq \displaystyle \int _{\mathbb{R}^{N}}H(\,f(\eta _{\epsilon } Z_0)) dx\leq \epsilon \frac {\eta ^{2}_{\epsilon }}{2} \displaystyle \int _{\mathbb{R}^{N}} |Z_0|^{2} dx + \frac {\eta ^{s}_{\epsilon }}{s}C_{\epsilon } \displaystyle \int _{\mathbb{R}^{N}} |Z_0|^{s} dx \to 0. \end{equation*}

We then conclude that $\eta _{\epsilon }$ is bounded. Hence, for a subsequence, $\eta _{\epsilon }\to \eta _0\gt 0$ as $\epsilon \to 0$ .

By (3) and (22), since $Z_0\in \mathcal{W}_{0}$ and $\displaystyle \lim _{\epsilon \to 0} Q(\epsilon x)=Q(q_1)=0$ by (3), we have $\eta _0=1$ . Then

\begin{equation*} \zeta _0\leq \displaystyle \liminf _{\epsilon \to 0} \zeta _{\epsilon } \leq \displaystyle \limsup _{\epsilon \to 0} \zeta _{\epsilon } \leq \displaystyle \limsup _{\epsilon \to 0}\Psi _{\epsilon }(\eta _{\epsilon } Z_0)= \Psi _{0}( Z_0)=\zeta _0. \quad \end{equation*}

The next result is used to show that the functional $\Psi _{\epsilon }$ verifies the Palais–Smale condition below a suitable level.

Lemma 3.2. Let $(v_{n})$ be a Palais–Smale sequence in $H^{1}(\mathbb{R}^{N})$ with respect to $\Psi _{\epsilon }$ in the level $c$ with $c\leq \zeta _0 + \sigma ^*$ and $v_{n} \rightharpoonup 0$ in $H^{1}(\mathbb{R}^{N})$ , see (24) . Then

\begin{equation*} \displaystyle \lim _{n\to \infty }\sup _{y \in \mathbb{R}^{N}}\int _{B_1( \, y)}|v_n|^{2} dx=0. \end{equation*}

Proof. By contradiction, assume that there is a sequence $( \, y_n)$ in $\mathbb{R}^{N}$ and $\alpha \gt 0$ such that

\begin{equation*} \displaystyle \int _{B_{1}( \, y_n)}|v_n|^{2} dx \geq \alpha \ \ \mbox{for every} \ \ n\in \mathbb{N}. \end{equation*}

By the fact that $v_n\rightharpoonup 0$ in $H^{1}(\mathbb{R}^{N})$ , the sequence $( \, y_n)$ is unbounded. Hence, for a subsequence, we deduce

\begin{equation*} |y_n|\to \infty \ \ \mbox{as} \ \ n\to \infty . \end{equation*}

Defining $w_n(x)=v_n(x+y_n)$ , we obtain that $w_n \rightharpoonup w$ , for some $w \in H^{1}(\mathbb{R}^{N})$ with $w\neq 0$ and, using the translation invariance of $\mathbb{R}^N$ , by a standard calculation, we conclude that $(w_n)$ is a Palais–Smale sequence for $\Psi _{\epsilon }$ at the level $c$ . In particular,

\begin{equation*} \displaystyle \int _{\mathbb{R}^N} \left [ \nabla w_n\nabla \psi + Q(\epsilon x+ y_n) \, f(w_n) \, f'(w_n)\psi \right ]dx = \displaystyle \int _{\mathbb{R}^N}h(\,f(w_n)) \, f'(w_n)\psi dx + o_{n}(1) \ \ \mbox{for every} \ \ \psi \in H^{1}(\mathbb{R}^{N}). \end{equation*}

Letting $n \to \infty$ we have

\begin{equation*} \displaystyle \int _{\mathbb{R}^N} \left [ \nabla w\nabla \psi \ + Q_{\infty }f(w) \, f'(w)\psi \right ] dx = \displaystyle \int _{\mathbb{R}^N}h(\,f(w)) \, f'(w)\psi dx \ \ \mbox{for every} \ \ \psi \in H^{1}(\mathbb{R}^{N}). \end{equation*}

The conclusion is that $w \in H^{1}(\mathbb{R}^{N})$ is a nontrivial critical point of $\Psi _{\infty }$ . Furthermore,

\begin{eqnarray*} \zeta _{\infty }\leq \Psi _{\infty }(w) -\frac {1}{2} \Psi '_{\infty }(w)w = \displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(w)) \, f'(w)w - H(\,f(w))\right ] dx. \end{eqnarray*}

By the monotone increasing property (7), we deduce

\begin{eqnarray*} \zeta _{\infty } &\leq &\displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(w_n)) \, f'(w_n)w_n - H(\,f(w_n))\right ] dx \\[4pt] &=& \displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(v_n)) \, f'(v_n)v_n - H(\,f(v_n))\right ] dx \\[4pt] &= &\displaystyle \liminf _{n\to \infty } \left [\Psi _{\epsilon }(v_n) -\frac {1}{2} \Psi '_{\epsilon }(v_n)v _n \right ]=\displaystyle \liminf _{n\to \infty } \Psi _{\epsilon }(v_n) =c\leq \zeta _0 + \sigma ^{*}. \end{eqnarray*}

We arrive at a contradiction.

Proof. Let $(u_{n})$ be a sequence in $H^{1}(\mathbb{R}^{N})$ such that $\Psi _{\epsilon }(u_{n})\to c$ and $\Psi '_{\epsilon }(u_{n})\to 0$ . By a reasoning similar to the proof of Lemma 2.6, the sequence $(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{N})$ and there is $u \in H^{1}(\mathbb{R}^{N})$ such that, for a subsequence, $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and $\Psi '_{\epsilon }(u)=0$ . Defining $v_{n}= u_{n} - u$ , by Lemma 3.2, we obtain

\begin{equation*} v_{n}\to 0 \ \ \mbox{in} \ \ L^\mu (\mathbb{R}^{N}) \ \ \mbox{for} \ \ \mu \in \left ( 2, \displaystyle \frac {2N}{N-2} \right ), \end{equation*}

and therefore by Remark2, we obtain

\begin{equation*} \displaystyle \int _{\mathbb{R}^N} h(\,f(v_n)) \, f'(v_n)v_n dx \to 0. \end{equation*}

Since $\Psi '_{\epsilon }(v_n)v _n=o_n(1)$ we deduce $\|v_n\|^{2}_{\epsilon } =o_n(1)$ .

4. Existence of multiple solutions related to the potential wells

In the present part, we prove a result concerning the Nehari manifold $\Pi _{\epsilon }$ . By hypothesis (3), we pick $\rho _0$ , $r_0\gt 0$ verifying

\begin{align*} \overline {B_{\rho _0(q_i)}}\cap \overline {B_{\rho _0(q_j)}} = \emptyset \ \ \mbox{for} \ \ i\neq j \ \ \mbox{and} \ \ i, j \in \{1, 2, \ldots , \wp \}, \\[-25pt] \end{align*}

\begin{align*} \left ( \displaystyle \cup ^{\wp }_{i=1}B_{\rho _0(q_i)} \right ) \subset B_{r_0}(0), \end{align*}

and define the closure

\begin{equation*} \mathcal{U}_{\frac {\rho _0}{2}}= \displaystyle \cup ^{\wp }_{i=1}\overline {B_{\rho _0(q_i)}}. \end{equation*}

Define $\Gamma \, : \, \mathbb{R}^N \to \mathbb{R}^N$ by $\Gamma (x) = x$ for $\mid x \mid \leq r_0$ and $\Gamma (x) = \frac {r_0 x}{|x|}$ for $|x|\geq r_0$ . Define the function $\mathcal{P}_{\epsilon } : \Pi _{\epsilon } \to \mathbb{R}^N$ by

(26) \begin{equation} \mathcal{P}_{\epsilon }(u)=\frac {\displaystyle \int _{\mathbb{R}^N}\Gamma (\epsilon x)| u(x)|^{\mu } dx }{\displaystyle \int _{\mathbb{R}^N}|u(x)|^{\mu } dx }, \end{equation}

where $\mu \in \left ( 2, \displaystyle \frac {2N}{N-2} \right )$ .

Proof. By contradiction, assume that there is $\epsilon _n \to 0$ and $u_n \in \Pi _{\epsilon _n}$ such that

\begin{equation*} P_{\epsilon _n}(u_n) \not \in \mathcal{U}_{\frac {\rho _0}{2}}\ \ \mbox{and} \ \ \zeta _{\epsilon _n}\leq \Psi _{\epsilon _n}(u_n) \leq \zeta _0 + \frac {1}{n} \ \ \mbox{for every} \ \ n \in \mathbb{N}. \end{equation*}

By Lemma 3.1, we obtain $u_n \in \Pi _{\epsilon _n}$ and $\Psi _{\epsilon _n}(u_n) \to \zeta _0$ as $n \to \infty$ . Furthermore, as in the proof of Lemma 2.6, we deduce that $(u_n)$ is a bounded sequence in $H^{1}(\mathbb{R}^{N})$ .

By [Reference Brezis4, Lemma 7], there is another sequence $v_n \in H^{1}(\mathbb{R}^{N})$ such that

(27) \begin{eqnarray} \|u_n - v_n\|_{\epsilon _n} = o_n(1), \ \ \Psi _{\epsilon _n}(v_n)= \zeta _0 +o_n(1) \ \ \mbox{and} \ \ \|\Psi '_{\epsilon _n}(v_n)\|_{\epsilon _n} = o_n(1). \end{eqnarray}

Thus, for a subsequence, $P_{\epsilon _n}(v_n) \not \in \mathcal{U}_{\frac {\rho _0}{4}}.$

Claim. We claim that

$\boldsymbol{1}^{st}.$ there is a sequence $(x_n)$ in $\mathbb{R}^{N}$ such that $\epsilon _n x_n \to q_i$ such that $Q(q_i) = Q_0$ for some $i=1,\ldots ,\wp$ , see (3), and

$\boldsymbol{2}^{nd}.$ denoting $\widetilde {v}_{n}(x)= v_n(x+x_n)$ , there exists $\widetilde {v}\in H^{1}(\mathbb{R}^{N})$ such that

(28) \begin{eqnarray} \widetilde {v}_{n}\to \widetilde {v} \ \ \mbox{in} \ \ L^{\mu }(\mathbb{R}^{N}) \ \ \mbox{for} \ \ \mu \in \left ( 2, \displaystyle \frac {2N}{N-2} \right )\!. \end{eqnarray}

We postpone the proof of statements $\boldsymbol{1}^{st}$ and $\boldsymbol{2}^{nd}$ , but assuming they are true, we deduce

\begin{eqnarray*} \mathcal{P}_{\epsilon _n}(v_n)&=&\frac {\displaystyle \int _{\mathbb{R}^N}\Gamma (\epsilon _n x)| v_n(x)|^{\mu } dx}{ \|v_n\|^{\mu }_{L^\mu (\mathbb{R}^N)}} =\frac {\displaystyle \int _{\mathbb{R}^N}\Gamma (\epsilon _n x+ \epsilon _n x_n)| \widetilde {v}_n(x)|^{\mu } dx }{\|\widetilde v_n\|^{\mu }_{L^{\mu }(\mathbb{R}^N)}} \\[3pt] &=& \frac {\displaystyle \int _{\mathbb{R}^N} q_i| \widetilde {v}(x)|^{\mu } dx }{\|\widetilde v_n\|^{\mu }_{L^\mu (\mathbb{R}^N)}} + o_n(1)= q_i +o_n(1) \in \mathcal{U}_{\frac {\rho _0}{2}}\!. \end{eqnarray*}

Hence, $P_{\epsilon _n}(v_n) \in \mathcal{U}_{\frac {\rho _0}{4}}$ for a sufficiently large $n$ , which is a contradiction. The proof of the lemma is complete.

Proof of claims $\boldsymbol{1}^{st}$ and $\boldsymbol{2}^{nd}$ . The sequence $(u_n)$ is bounded in $H^{1}(\mathbb{R}^{N})$ ; thus by (27), the sequence $(v_n)$ is also bounded in $H^{1}(\mathbb{R}^{N})$ . Hence, for a subsequence, there is $v\in H^{1}(\mathbb{R}^{N})$ such that $v_n \rightharpoonup v$ in $H^{1}(\mathbb{R}^{N})$ . Since

\begin{equation*} \displaystyle \int _{\mathbb{R}^N}\left [|\nabla v_n|^{2}+ Q(\epsilon x) \, f(v_n) \, f'(v_n)v_n \right ]dx= \displaystyle \int _{\mathbb{R}^N}h(\,f(v_n)) \, f'(v_n)v_n dx +o_{n}(1), \end{equation*}

by (27) we deduce that $v_n \to 0$ in $L^{\mu }(\mathbb{R}^{N})$ .

Define the sequence in $L^{1}(\mathbb{R}^{N})$ by

\begin{equation*} \rho _n(x)=\frac {|v_n(x)|}{\|v_n\|^{\mu }_{L^\mu (\mathbb{R}^N)}}, \end{equation*}

where $\mu \in (2,2N / (N-2))$ . Notice that $\rho _n$ is bounded in $L^\mu (\mathbb{R}^N)$ . According to [Reference Lions14], we will show that only one of the statements (j), (jj) and (jjj) is true.

(j) Vanishing.

\begin{equation*} \displaystyle \lim _{n\to \infty }\sup _{y\in \mathbb{R}^{N}}\int _{B_{R}( \, y)}\rho _n dx = 0 \ \ \mbox{for every} \ \ R\gt 0. \end{equation*}

(jj) Dichotomy. Let $\mathcal{X}_{A}$ be the characteristic function on the set $A$ . There is a sequence $(x_n)$ in $\mathbb{R}^{N}$ , $0 \lt \alpha \lt 1$ , $R_*\gt 0$ and a sequence $R_n\to \infty$ such that $\rho _{n,1}(x)=\mathcal{X}_{B_{R_*}(x_n)}(x)\rho _n(x)$ and $\rho _{n,2}(x)=\mathcal{X}_{B^{c}_{R_*}(x_n)}(x)\rho _n(x)$ verify

\begin{equation*} \displaystyle \int _{\mathbb{R}^{N}}\rho _{1,n} dx \to \alpha \ \ \mbox{and } \ \ \displaystyle \int _{\mathbb{R}^{N}}\rho _{2,n} dx \to 1- \alpha . \end{equation*}

(jjj) Compactness. There is a sequence $(x_n)$ in $\mathbb{R}^{N}$ such that for every $\theta \gt 0$ , there is $R\gt 0$ such that

\begin{equation*} \displaystyle \int _{B_{R}(x_n)}\rho _n dx \geq 1- \theta \ \ \mbox{for every} \ \ n\in \mathbb{N}. \end{equation*}

(j) Vanishing does not happen. By contradiction, if it is true, by [Reference Lions14], we obtain $\rho _n \to 0$ in $L^{\mu }(\mathbb{R}^{N})$ for $2\lt \mu \lt 2N /(N-2)$ . This implies that $v_n \to 0$ in $L^{\mu }(\mathbb{R}^{N})$ , which is a contradiction.

(jj) Dichotomy is ruled out. We consider two cases with respect to the sequence $(x_n)$ , which may be either bounded or unbounded. We will deduce that these two cases do not occur, and we denote them by (1-jj) and (2-jj).

(1-jj) Assume $(x_n)$ is bounded. Since $\int _{\mathbb{R}^{N}}\rho _{1,n} dx \to \alpha \gt 0$ , we have

(29) \begin{equation} \int _{B_{R}(x_n)}|v_n|^{\mu } dx \geq \varkappa \end{equation}

for a sufficiently large $n$ and for some $\varkappa \gt 0$ . Since $(x_n)$ is bounded, we obtain

\begin{equation*} \int _{B_{R_0}(0)}|v|^{\mu } dx = \int _{B_{R_0}(0)}|v_n|^{\mu } dx \geq \varkappa \end{equation*}

for some $R_0\gt R$ , which implies that $v\neq 0$ . Define $\varphi \in C^{\infty }_{0}(\mathbb{R}^N)$ such that $0\leq \varphi (x) \leq 1$ , $\varphi (x) = 1$ for $x \in B_{R}(0)$ and $\varphi (x) = 0$ for $x \in \mathbb{R} - B_{2R}(0)$ . For $ \varphi _{R}(x)= \varphi (\frac {x}{R})$ , one obtains that $\Psi _{\epsilon _n}(v_n)(\varphi _{R}v_n)= o_n(1)$ . Letting $n\to \infty$ and subsequently $R\to \infty$ , we have $\Psi _{0} (v)v\leq 0$ . Then, there is $\varsigma _{0} \in (0,1]$ such that $\varsigma _{0} v\in \mathcal{W}_{0}$ , see (18). By (7), we obtain

\begin{eqnarray*} \zeta _0 \leq \Psi _{0}(\varsigma _{0}v)-\frac {1}{2} \Psi '_{0}(\varsigma _{0}v)\varsigma _{0} v & = & \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(\varsigma _{0} v)) \, f'(\varsigma _{0} v)\varsigma _{0} v - H(\,f(\varsigma _{0} v))\right ] dx \\[3pt] & \leq & \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f( v)) \, f'( v) v - H(\,f( v))\right ] dx. \end{eqnarray*}

By Fatou’s lemma, we deduce

\begin{eqnarray*} \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f( v)) \, f'( v) v - H(\,f( v))\right ] dx &\leq & \displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f( v_n)) \, f'( v_n) v_n - H(\,f( v_n))\right ] dx \\[3pt] &=&\displaystyle \liminf _{n\to \infty }\left [ \Psi _{\epsilon _n}(v_n) - \frac {1}{2} \Psi '_{\epsilon _n}(v_n)v_n \right ]= \zeta _0. \end{eqnarray*}

Hence, $\varsigma _{0}=1$ , $\Psi _0(v)=\zeta _0$ and

\begin{equation*} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f( v_n)) \, f'( v_n) v_n - H(\,f( v_n))\right ] dx= \displaystyle \int _{\mathbb{R}^{N}} \left [ \frac {1}{2} h(\,f( v)) \, f'( v) v - H(\,f( v))\right ] dx, \end{equation*}

implies

\begin{equation*} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}}h(\,f( v_n)) \, f'( v_n) v_n dx = \displaystyle \int _{\mathbb{R}^{N}} h(\,f( v)) \, f'( v) v dx \end{equation*}

and

\begin{equation*} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}} H(\,f(v_n)) dx = \displaystyle \int _{\mathbb{R}^{N}} H(\,f(v)) dx, \end{equation*}

therefore, $\|v_n-v\|^{2}_{0}= o_n(1)$ . For a subsequence, we obtain $v_n \to v$ in $L^{\mu }(\mathbb{R}^{N})$ . By assumption, $(x_n)$ is bounded. The definition of $\rho _{1,n}$ gives

\begin{equation*} \displaystyle \int _{\mathbb{R}^{N}}\rho _{1,n} dx \to 0, \end{equation*}

implying $\alpha =0$ , which is a contradiction. Hence, (1-jj) is false.

(2-jj) Now, assume that $(x_n)$ is unbounded. For a subsequence, we obtain

\begin{equation*} |x_n|\to \infty \ \ \mbox{as} \ \ n\to \infty . \end{equation*}

Define $w_{n}(x)= v_n (x+x_n)$ . Since $(v_n)$ is bounded in $H^{1}(\mathbb{R}^{N})$ , then $(w_{n})$ is bounded in $H^{1}(\mathbb{R}^{N})$ . For a subsequence, we obtain $w_n \rightharpoonup w$ . If $w=0$ , we deduce

\begin{equation*} \alpha + o_n(1)=\displaystyle \int _{\mathbb{R}^{N}}\rho _{1,n} dx= \displaystyle \int _{\mathbb{R}^{N}}\mathcal{X}_{B_{r_1}(x_n)}\frac {v_n(x)}{\|v_n\|^{\mu }_{L^\mu (\mathbb{R}^N)}} dx = \displaystyle \int _{\mathbb{R}^{N}}\mathcal{X}_{B_{r_1}(0)}\frac {w_n(x-x_n))}{\|w_n\|^{\mu }_{L^\mu (\mathbb{R}^N)}} dx = o_n(1), \end{equation*}

which is a contradiction with the fact that $\mathcal{X}_{B_{r_1}(0)}$ is the characteristic function of $B_{r_1}(0)$ . The conclusion is that $w_n \rightharpoonup w$ in $H^{1}(\mathbb{R}^{N})$ and $w \neq 0$ .

We distinguish two cases regarding the sequence $(\epsilon _n x_n)$ .

$\bullet$ Assume that $(\epsilon _n x_n)$ is unbounded. The sequence $(v_n)$ in $H^{1}(\mathbb{R}^{N})$ is a Palais–Smale sequence for $\Psi _{\epsilon }$ , then

(30) \begin{align} \displaystyle \int _{\mathbb{R}^N}\left [\nabla w_n\nabla \psi \ + Q(\epsilon x+ x_n) \, f(w_n) \, f'(w_n)\psi \right ]dx= \displaystyle \int _{\mathbb{R}^N}h(\,f(w_n)) \, f'(w_n)\psi dx +o_{n}(1) \ \ \mbox{for every} \ \ \psi \in H^{1}(\mathbb{R}^{N}). \end{align}

By the fact that $|(\epsilon _n x_n)|\to \infty$ as $n \to \infty$ and letting $n \to \infty$ in (30), we deduce

\begin{equation*} \displaystyle \int _{\mathbb{R}^N}\left [\nabla w\nabla \psi \ + Q_{\infty }f(w) \, f'(w)\psi \right ]dx= \displaystyle \int _{\mathbb{R}^N}h(\,f(w)) \, f'(w)\psi dx \ \ \mbox{for every} \ \ \psi \in H^{1}(\mathbb{R}^{N}). \end{equation*}

Therefore, $w \in H^{1}(\mathbb{R}^{N})$ is a nontrivial critical point of $\Psi _{\infty }$ and

\begin{eqnarray*} \zeta _{\infty }\leq \Psi _{\infty }(w) -\frac {1}{2} \Psi '_{\infty }(w)w = \displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(w)) \, f'(w)w - H(\,f(w))\right ] dx . \end{eqnarray*}

By (7), we have

\begin{eqnarray*} \zeta _{\infty } &\leq &\displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(w_n)) \, f'(w_n)w_n - H(\,f(w_n))\right ] dx \\[3pt] &=& \displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^N}\left [ \frac {1}{2} h(\,f(v_n)) \, f'(v_n)v_n - H(\,f(v_n))\right ] dx \\[3pt] &= &\displaystyle \liminf _{n\to \infty } \left [\Psi _{\epsilon }(v_n) -\frac {1}{2} \Psi '_{\epsilon }(v_n)v _n \right ]=\displaystyle \liminf _{n\to \infty } \Psi _{\epsilon }(v_n) =c\lt \zeta _0 + \sigma ^*, \end{eqnarray*}

where $\sigma ^*$ was defined at (24). Thus, we reach a contradiction.

$\bullet$ If $(\epsilon _n x_n)$ is bounded, define $\varphi \in C^{\infty }_{0}(\mathbb{R}^N)$ such that $0\leq \varphi (x) \leq 1$ , $ \varphi (x) = 1$ for $x \in B_{R}(0)$ and $0\leq \varphi (x) \leq 1$ , $ \varphi (x) = 0$ for $x \in \mathbb{R}^N - B_{2R}(0)$ . For $ \varphi _{R}(x)= \varphi (\frac {x}{R})$ , we have $\Psi '_{\epsilon _n}(w_n)(\varphi _{R}w_n)= o_n(1)$ . Letting $n\to \infty$ and after $R\to \infty$ , we deduce $\Psi _{0} (w)w\leq 0$ . Hence, there is $\varsigma _{0} \in (0,1]$ such that $\varsigma _{0} w\in \mathcal{W}_{0}$ ; see (18). By (7), we obtain

\begin{eqnarray*} \zeta _0 \leq \Psi _{0}(\varsigma _{0}w)-\frac {1}{2} \Psi '_{0}(\varsigma _{0}w)\varsigma _{0} v & = & \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(\varsigma _{0} w)) \, f'(\varsigma _{0} w)\varsigma _{0} w - H(\,f(\varsigma _{0} w))\right ] dx \\[3pt] & \leq & \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(w)) \, f'( w)\varsigma _{0} w - H(\,f( w))\right ] dx . \end{eqnarray*}

Fatou’s lemma implies

\begin{eqnarray*} \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(w)) \, f'( w)\varsigma _{0} w - H(\,f( w))\right ] dx &\leq & \displaystyle \liminf _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(w_n)) \, f'( w_n)\varsigma _{0} w_n - H(\,f( w_n))\right ] dx\\[3pt] &=&\displaystyle \liminf _{n\to \infty }\left [ \Psi _{\epsilon _n}(w_n) - \frac {1}{2} \Psi '_{\epsilon _n}(w_n)w_n \right ]= \zeta _0. \end{eqnarray*}

Thus, $\varsigma _{0}=1$ , $\Psi _0(w)=\zeta _0$ and

\begin{equation*} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(w_n)) \, f'( w_n)\varsigma _{0} w_n - H(\,f( w_n))\right ] dx = \displaystyle \int _{\mathbb{R}^{N}}\left [ \frac {1}{2} h(\,f(w)) \, f'( w)\varsigma _{0} w - H(\,f( w))\right ] dx, \end{equation*}

therefore,

(31) \begin{equation} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}} h(\,f(w_n)) \, f'( w_n)\varsigma _{0} w_n dx = \displaystyle \int _{\mathbb{R}^{N}} h(\,f(w)) \, f'( w)\varsigma _{0} w dx \end{equation}

and

(32) \begin{equation} \displaystyle \lim _{n\to \infty }\displaystyle \int _{\mathbb{R}^{N}} H(\,f( w_n)) dx = \displaystyle \int _{\mathbb{R}^{2=N}} H(\,f( w)) dx, \end{equation}

implying $\|w_n-w\|^{2}_{0}= o_n(1)$ . For a subsequence, we have $w_n \to w$ in $L^{s}(\mathbb{R}^{N})$ . The assumptions imply that $(\epsilon _n x_n)$ is bounded, and by definition of $\rho _{1,n}$ we obtain

\begin{equation*} \displaystyle \int _{\mathbb{R}^{N}}\rho _{1,n} dx \to 0, \end{equation*}

hence $\alpha =0$ , and we obtain a contradiction. In total, (2-jj) is also false.

Therefore, (j) Vanishing and (jj) Dichotomy are excluded, and then only (jjj) Compactness takes place. Hence, we can argue as in the previous step (31)–(32) and conclude that $\|v_n-v\|^{2}_{0}= o_n(1)$ , and in particular, for a subsequence, $\epsilon _n x_n\to q_i$ such that $Q(q_i)= Q_0$ for some $i=1, \ldots , \wp$ , see (3), and $v_n \to v$ in $L^{s}(\mathbb{R}^{N})$ . This completes the proof of items $\boldsymbol{1}^{st}$ and $\boldsymbol{2}^{nd}$ of the claim. The proof of Lemma 4.1 is complete.

5. Proof of Theorem1.1

Recall (13), (25) and (26) and define the following sets and quantities:

\begin{align*} \mathcal{U}^{i}_{\epsilon }=\{u \in \Pi _{\epsilon } \, : \, |\mathcal{P}_{\epsilon }(u)-q_i|\lt \rho _0\}, \\[-25pt] \end{align*}

\begin{align*} \partial \mathcal{U}^{i}_{\epsilon }=\{u \in \Pi _{\epsilon } \, : \, |\mathcal{P}_{\epsilon }(u)-q_i|=\rho _0\}, \\[-25pt]\end{align*}

\begin{align*} \alpha ^{i}_{\epsilon }=\inf _{u\in \mathcal{U}^{i}_{\epsilon }}\Psi _{\epsilon }(u) \end{align*}

and

\begin{align*} \widetilde {\alpha }^{i}_{\epsilon }=\inf _{u\in \partial \mathcal{U}^{i}_{\epsilon }}\Psi _{\epsilon }(u). \end{align*}

Proof. By (22), $Z_0 \in H^{1}(\mathbb{R}^{N})$ is the function such that $\Psi _0(Z_0)=\zeta _0$ and $\Psi '_0(Z_0)Z_0=0$ . Recall from (3) that $Q(q_i) = Q_0$ for $i=1,2,\ldots , \wp$ . For $i\in \{1,2,\ldots , \wp \}$ define

\begin{equation*} Z^{i}_{n}= Z_{0} \left (x - \frac {q_i}{\epsilon _n}\right )\!. \end{equation*}

Let $\varsigma _{\epsilon _n}\gt 0$ be such that $\varsigma _{\epsilon _n}Z^{i}_{n} \in \Pi _{\epsilon _n}$ . Then, by a change of variables and an argument analogous to the proof of Lemma 3.1, we deduce for a subsequence, that $\varsigma _{\epsilon _n}\to \varsigma _{0}$ and

\begin{equation*} \displaystyle \limsup _{n\to \infty } \Psi _{\epsilon _n} (\varsigma _{\epsilon _n}Z^{i}_{n})\leq \Psi _{0}(\varsigma _{0} Z_0)\leq \Psi _0(Z_0)=\zeta _0. \end{equation*}

Since $\epsilon _n$ is an arbitrary sequence and $Z^{i}_{n}\in \mathcal{U}^{i}_{\epsilon }$ , we have $\alpha ^{i}_{\epsilon } \lt \zeta _0 + \varkappa /2$ . The inequality $\alpha ^{i}_{\epsilon }\lt \widetilde {\alpha }^{i}_{\epsilon }$ follows from Lemma 4.1.