# Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN

## Abstract

In this article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear elliptic equations

$\left\{\begin{array}{c}-\Delta v+\lambda v=f\left(z\right){v}^{p-1}+h\left(z\right){v}^{q-1}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{ℝ}^{N};\hfill \\ v\in {H}^{1}\left({ℝ}^{N}\right),\hfill \end{array}\right\$

where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3.

AMS (MOS) subject classification: 35J20; 35J25; 35J65.

## 1 Introduction

For N ≥ 3, 1 ≤ q < 2 < p < 2* = 2N/(N - 2), we consider the semilinear elliptic equations

$\left\{\begin{array}{c}-\Delta v+\lambda v=f\left(z\right){v}^{p-1}+h\left(z\right){v}^{q-1}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{ℝ}^{N};\hfill \\ v\in {H}^{1}\left({ℝ}^{N}\right),\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left({E}_{\lambda }\right)$

where λ > 0.

Let f and h satisfy the following conditions:

(f 1) f is a positive continuous function in Nand lim|z| → ∞f(z) = f > 0.

(f 2) there exist k points a1, a2,..., akin Nsuch that

$f\left({a}^{i}\right)={f}_{\text{max}}=\underset{z\in {ℝ}^{N}}{\text{max}}f\left(z\right)\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{1}\le \text{i}\le k,$

and f < fmax.

(h 1) $h\in {L}^{\frac{p}{p-q}}\left({ℝ}^{N}\right)\cap {L}^{\infty }\left({ℝ}^{N}\right)$ and $h\gneqq 0$.

Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain

$\left\{\begin{array}{c}-\Delta u=ch\left(z\right){\left|u\right|}^{q-2}u+{\left|u\right|}^{p-2}u\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}\Omega ;\hfill \\ u\in {H}_{0}^{1}\left(\Omega \right),\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left({E}_{c}\right)$

have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2] $h\in C\left(\stackrel{̄}{\Omega }\right)$ and changes sign, 1 < q < 2 < p < 2*). They proved that this equation has at least two positive solutions for sufficiently small c > 0. More general results of Equation (E c ) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].

In this article, we consider the existence and multiplicity of positive solutions of Equation (E λ ) in N. For the case q = λ = 1 and f(z) ≡ 1 for all z N, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (E λ ) admits at least two positive solutions in N. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (E λ ) admits at least two positive solutions in N. For the case q = λ = 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [9] asserted that Equation (E λ ) admits at least four positive solutions in N, where f(z) 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small ${∥h∥}_{{H}^{-1}}>0$. Similarly, in Hsu and Lin [10], they have studied that there are at least four positive solutions of the general case -Δu + u = f(z)vp-1+ λh(z) vq-1in Nfor sufficiently small λ > 0.

By the change of variables

Equation (E λ ) is transformed to

$\left\{\begin{array}{c}-\Delta u+u=f\left(\epsilon z\right){u}^{p-1}+{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}^{q-1}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{ℝ}^{N};\hfill \\ u\in {H}^{1}\left({ℝ}^{N}\right),\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left({E}_{\epsilon }\right)$

Associated with Equation (E ε ), we consider the C1-functional J ε , for u H1 (N),

${J}_{\epsilon }\left(u\right)=\frac{1}{2}{∥u∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz-\frac{1}{q}\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}_{+}^{q}dz,$

where ${∥u∥}_{H}^{2}=\underset{{ℝ}^{N}}{\int }\left({\left|\Delta u\right|}^{2}+{\left|u\right|}^{2}\right)\phantom{\rule{2.77695pt}{0ex}}dz$ is the norm in H1 (N) and u+ = max{u, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (E ε ) are equivalent to the critical points of J ε . This article is organized as follows. First of all, we use the argument of Tarantello [11] to divide the Nehari manifold M ε into the two parts ${\mathbf{M}}_{\epsilon }^{+}$ and ${\mathbf{M}}_{\epsilon }^{-}$. Next, we prove that the existence of a positive ground state solution ${u}_{0}\in {\mathbf{M}}_{\epsilon }^{+}$ of Equation (E ε ). Finally, in Section 4, we show that the condition (f 2) affects the number of positive solutions of Equation (E ε ), that is, there are at least k critical points ${u}_{1},...,{u}_{k}\in {\mathbf{M}}_{\epsilon }^{-}$ of J ε such that ${J}_{\epsilon }\left({u}_{i}\right)={\beta }_{{}^{\epsilon }}^{i}\left(\left(\text{PS}\right)-\text{value}\right)$ for 1 ≤ ik.

Let

$S=\underset{\underset{{‖u‖}_{H}=1}{u\in {H}^{1}\left({ℝ}^{N}\right)}}{\mathrm{sup}}{‖u‖}_{{L}^{p}},$

then

${∥u∥}_{{L}^{p}}\le S{∥u∥}_{H}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}u\in {H}^{1}\left({ℝ}^{N}\right)\\left\{0\right\}.$
(1.1)

For the semilinear elliptic equations

$\left\{\begin{array}{c}-\Delta u+u=f\left(\epsilon z\right){u}^{p-1}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{ℝ}^{N};\hfill \\ u\in {H}^{1}\left({ℝ}^{N}\right),\hfill \end{array}\right\$
(E0)

we define the energy functional ${I}_{\epsilon }\left(u\right)=\frac{1}{2}{∥u∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz$, and

${\gamma }_{\epsilon }=\underset{u\in {\mathbf{N}}_{\epsilon }}{\text{inf}}{I}_{\epsilon }\left(u\right),$

where N ε = {u H1 (N) \ {0} | u+ 0 and $⟨{I}_{\epsilon }^{\prime }\left(u\right),u⟩=0$}. Note that

(i) if ff, we define ${I}_{\infty }\left(u\right)=\frac{1}{2}{∥u∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }{f}_{\infty }{u}_{+}^{p}{d}_{z}$ and

${\gamma }_{\infty }=\underset{u\in {\mathbf{N}}_{\infty }}{\text{inf}}{I}_{\infty }\left(u\right),$

where N = {u H1 (N) \ {0} | u+ 0 and $⟨{I}_{\infty }^{\prime }\left(u\right),u⟩=0$};

(ii) if ffmax, we define ${I}_{\text{max}}\left(u\right)=\frac{1}{2}{∥u∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }{f}_{\text{max}}{u}_{+}^{p}dz$ and

${\gamma }_{\text{max}}=\underset{u\in {\mathbf{N}}_{\text{max}}}{\text{inf}}{I}_{\text{max}}\left(u\right),$

where Nmax = {u H1 (N) \ {0} | u+ 0 and $⟨{I}_{\text{max}}^{\prime }\left(u\right),u⟩=0$}.

Lemma 1.1

${\gamma }_{\text{max}}=\frac{p-2}{2p}{\left({f}_{\text{max}}{S}^{p}\right)}^{-2/\left(p-2\right)}>0.$

Proof. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].

Our main results are as follows.

(I) Let Λ = ε2(p-q)/(p-2). Under assumptions (f 1) and (h 1), if

$0<\Lambda <{\Lambda }_{0}=\left(p-2\right){\left(\frac{2-q}{{f}_{\text{max}}}\right)}^{\frac{2-q}{p-2}}{\left[\left(p-q\right){S}^{2}\right]}^{\frac{q-p}{p-2}}{∥h∥}_{#}^{-1},$

where h# is the norm in ${L}^{\frac{p}{p-q}}\left({ℝ}^{N}\right)$, then Equation (E ε ) admits at least a positive ground state solution. (See Theorem 3.4)

(II) Under assumptions (f 1) - (f 2) and (h 1), if λ is sufficiently large, then Equation (E λ ) admits at least k + 1 positive solutions. (See Theorem 4.8)

## 2 The Nehari manifold

First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H1(N) for some functional J.

Definition 2.1 (i) For β , a sequence {u n } is a (PS) β -sequence in H1(N) for J if J(u n ) = β + o n (1) and J'(u n ) = o n (1) strongly in H-1 (N) as n → ∞, where H-1 (N) is the dual space of H1(N);

(ii) J satisfies the (PS) β -condition in H1(N) if every (PS) β -sequence in H1(N) for J contains a convergent subsequence.

Next, since J ε is not bounded from below in H1 (N), we consider the Nehari manifold

${\mathbf{M}}_{\epsilon }=\left\{u\in {H}^{1}\left({ℝ}^{N}\right)\\left\{0\right\}\left|{u}_{+}\not\equiv 0\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}⟨{{J}^{\prime }}_{\epsilon }\left(u\right),u⟩=0\right\right\},$
(2.1)

where

$⟨{J}_{\epsilon }^{\prime }\left(u\right),u⟩={∥u∥}_{H}^{2}-\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz-\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}_{+}^{q}dz.$

Note that M ε contains all nonnegative solutions of Equation (E ε ). From the lemma below, we have that J ε is bounded from below on M ε .

Lemma 2.2 The energy functional J ε is coercive and bounded from below on M ε .

Proof. For u M ε , by (2.1), the Hölder inequality $\left({p}_{1}=\frac{p}{p-q},{p}_{2}=\frac{p}{q}\right)$ and the Sobolev embedding theorem (1.1), we get

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(u\right)& =\left(\frac{1}{2}-\frac{1}{p}\right){∥u∥}_{H}^{2}-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ \ge \frac{{∥u∥}_{H}^{q}}{p}\left[\frac{p-2}{2}{∥u∥}_{H}^{2-q}-\frac{p-q}{q}{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}{∥h∥}_{#}{S}^{q}\right].\phantom{\rule{2em}{0ex}}\end{array}$

Hence, we have that J ε is coercive and bounded from below on M ε .

Define

${\psi }_{\epsilon }\left(u\right)=⟨{J}_{\epsilon }^{\prime }\left(u\right),u⟩.$

Then for u M ε , we get

$\begin{array}{ll}\hfill ⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩& =2{∥u∥}_{H}^{2}-p\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz-q{\underset{{ℝ}^{N}}{\int }\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ =\left(p-q\right)\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}_{+}^{q}dz-\left(p-2\right){∥u∥}_{H}^{2}\phantom{\rule{2em}{0ex}}\end{array}$
(2.2)
$=\left(2-q\right){∥u∥}_{H}^{2}-\left(p-q\right)\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz.$
(2.3)

We apply the method in Tarantello [11], let

$\begin{array}{ll}\hfill {\mathbf{M}}_{\epsilon }^{+}& =\left\{u\in {\mathbf{M}}_{\epsilon }\left|⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩>0\right\right\};\phantom{\rule{2em}{0ex}}\\ \hfill {\mathbf{M}}_{\epsilon }^{0}& =\left\{u\in {\mathbf{M}}_{\epsilon }\left|⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩=0\right\right\};\phantom{\rule{2em}{0ex}}\\ \hfill {\mathbf{M}}_{\epsilon }^{-}& =\left\{u\in {\mathbf{M}}_{\epsilon }\left|⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩<0\right\right\}.\phantom{\rule{2em}{0ex}}\end{array}$

Lemma 2.3 Under assumptions (f 1) and (h 1), if 0 < Λ (= ε2(p-q)/(p- 2)) < Λ0, then ${\mathbf{M}}_{\epsilon }^{0}=\varnothing$.

Proof. See Hsu and Lin [[10], Lemma 5].

Lemma 2.4 Suppose that u is a local minimizer for J ε on M ε and $u\notin {\mathbf{M}}_{\epsilon }^{0}$. Then ${J}_{\epsilon }^{\prime }\left(u\right)=0$ in H-1 (N).

Proof. See Brown and Zhang [[4], Theorem 2.3].

Lemma 2.5 We have the following inequalities.

(i) $\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){u}_{+}^{q}dz>0$ for each $u\in {\mathbf{M}}_{\epsilon }^{+}$;

(ii) ${∥u∥}_{H}<{\left(\frac{p-q}{p-2}\Lambda {∥h∥}_{#}{S}^{q}\right)}^{1/\left(2-q\right)}$ for each $u\in {\mathbf{M}}_{\epsilon }^{+}$;

(iii) ${∥u∥}_{H}>{\left[\frac{2-q}{\left(p-q\right){f}_{\text{max}}{S}^{p}}\right]}^{1/\left(p-2\right)}$ for each $u\in {\mathbf{M}}_{\epsilon }^{-}$;

(iv) If $0<\Lambda \left(={\epsilon }^{2\left(p-q\right)/\left(p-2\right)}\right)<\frac{q{\Lambda }_{0}}{2}$, then J ε (u) > 0 for each $u\in {\mathbf{M}}_{\epsilon }^{-}$.

Proof. (i) It can be proved by using (2.2).

(ii) For any $u\in {\mathbf{M}}_{\epsilon }^{+}\subset {\mathbf{M}}_{\epsilon }$, by (2.2), we apply the Hölder inequality $\left({p}_{1}=\frac{p}{p-q},{p}_{2}=\frac{p}{q}\right)$ to obtain that

$\begin{array}{ll}\hfill 0& <\left(p-q\right)\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){u}_{+}^{q}dz-\left(p-2\right){∥u∥}_{H}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(p-q\right)\Lambda {∥h∥}_{#}{S}^{q}{∥u∥}_{H}^{q}-\left(p-2\right){∥u∥}_{H}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

(iii) For any $u\in {\mathbf{M}}_{\epsilon }^{-}$, by (2.3), we have that

${∥u∥}_{H}^{2}<\frac{p-q}{2-q}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz\le \frac{p-q}{2-q}{S}^{p}{∥u∥}_{H}^{p}{f}_{\text{max}}.$

(iv) For any $u\in {\mathbf{M}}_{\epsilon }^{-}\subset {\mathbf{M}}_{\epsilon }$, by (iii), we get that

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(u\right)& =\left(\frac{1}{2}-\frac{1}{p}\right){∥u∥}_{H}^{2}-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){u}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ \ge \frac{{∥u∥}_{H}^{q}}{p}\left[\frac{p-2}{2}{∥u∥}_{H}^{2-q}-\frac{p-q}{q}\Lambda {∥h∥}_{#}{S}^{q}\right]\phantom{\rule{2em}{0ex}}\\ >\frac{1}{p}{\left[\frac{2-q}{\left(p-q\right){f}_{\text{max}}{S}^{p}}\right]}^{\frac{q}{p-2}}\left[\frac{p-2}{2}{\left[\frac{2-q}{\left(p-q\right){f}_{\text{max}}{S}^{p}}\right]}^{\frac{2-q}{p-2}}-\frac{p-q}{q}\Lambda {∥h∥}_{#}{S}^{q}\right].\phantom{\rule{2em}{0ex}}\end{array}$

Thus, if $0<\Lambda <\frac{q}{2}\left(p-2\right){\left(\frac{2-q}{{f}_{\text{max}}}\right)}^{\frac{2-q}{p-2}}{\left[\left(p-q\right){S}^{2}\right]}^{\frac{q-p}{p-2}}{∥h∥}_{#}^{-1}$, we get that J ε (u) ≥ d0 > 0 for some constant d0 = d0(ε, p, q, S, h # , fmax).

For u H1 (N) \ {0} and u + 0, let

$\stackrel{̄}{t}=\stackrel{̄}{t}\left(u\right)={\left[\frac{\left(2-q\right){∥u∥}_{H}^{2}}{\left(p-q\right)\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz}\right]}^{1/\left(p-2\right)}>0.$

Lemma 2.6 For each u H1 (N)\ {0} and u + 0, we have that

(i) if ${\int }_{{ℝ}^{N}}h\left(\epsilon z\right){u}_{+}^{q}dz=0$, then there exists a unique positive number ${t}^{-}={t}^{-}\left(u\right)>\stackrel{̄}{t}$ such that ${t}^{-}u\in {\mathbf{M}}_{\epsilon }^{-}$ and J ε (t-u) = supt ≥ 0J ε (tu);

(ii) if 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0 and ${\int }_{{ℝ}^{N}}h\left(\epsilon z\right){u}_{+}^{q}dz>0$, then there exist unique positive numbers ${t}^{+}={t}^{+}\left(u\right)<\stackrel{̄}{t}<{t}^{-}={t}^{-}\left(u\right)$ such that ${t}^{+}u\in {\mathbf{M}}_{\epsilon }^{+},{t}^{-}u\in {\mathbf{M}}_{\epsilon }^{-}$ and

${J}_{\epsilon }\left({t}^{+}u\right)=\underset{0\le t\le \stackrel{̄}{t}}{\text{inf}}{J}_{\epsilon }\left(tu\right),\phantom{\rule{1em}{0ex}}{J}_{\epsilon }\left({t}^{-}u\right)=\underset{t\ge \stackrel{̄}{t}}{\text{sup}}{J}_{\epsilon }\left(tu\right).$

Proof. See Hsu and Lin [[10], Lemma 7].

Applying Lemma 2.3 $\left({\mathbf{M}}_{\epsilon }^{0}=\varnothing \phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}0<\Lambda <{\Lambda }_{0}\right)$, we write ${\mathbf{M}}_{\epsilon }={\mathbf{M}}_{\epsilon }^{+}\cup {\mathbf{M}}_{\epsilon }^{-}$, where

$\begin{array}{c}{\mathbf{M}}_{\epsilon }^{+}=\left\{u\in {\mathbf{M}}_{\epsilon }|\left(2-q\right){∥u∥}_{H}^{2}-\left(p-q\right)\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz>0\right\},\\ {\mathbf{M}}_{\epsilon }^{-}=\left\{u\in {\mathbf{M}}_{\epsilon }|\left(2-q\right){∥u∥}_{H}^{2}-\left(p-q\right)\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz<0\right\}.\end{array}$

Define

${\alpha }_{\epsilon }=\underset{u\in {\mathbf{M}}_{\epsilon }}{\text{inf}}{J}_{\epsilon }\left(u\right);\phantom{\rule{1em}{0ex}}{\alpha }_{\epsilon }^{+}=\underset{u\in {\mathbf{M}}_{\epsilon }^{+}}{\text{inf}}{J}_{\epsilon }\left(u\right);\phantom{\rule{1em}{0ex}}{\alpha }_{\epsilon }^{-}=\underset{u\in {\mathbf{M}}_{\epsilon }^{-}}{\text{inf}}{J}_{\epsilon }\left(u\right).$

Lemma 2.7 (i) If 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0, then ${\alpha }_{\epsilon }\le {\alpha }_{\epsilon }^{+}<0$;

(ii) If 0 < Λ < q Λ0/2, then ${\alpha }_{\epsilon }^{-}\ge {d}_{0}>0$ for some constant d0 = d0 (ε, p, q, S, h # , fmax).

Proof. (i) Let $u\in {\mathbf{M}}_{\epsilon }^{+}$, by (2.2), we get

$\left(p-2\right){∥u∥}_{H}^{2}<\left(p-q\right)\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){u}_{+}^{q}dz.$

Then

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(u\right)& =\left(\frac{1}{2}-\frac{1}{p}\right){∥u∥}_{H}^{2}-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){u}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ <\left[\left(\frac{1}{2}-\frac{1}{p}\right)-\left(\frac{1}{q}-\frac{1}{p}\right)\frac{p-2}{p-q}\right]{∥u∥}_{H}^{2}\phantom{\rule{2em}{0ex}}\\ =-\frac{\left(2-q\right)\left(p-2\right)}{2pq}{∥u∥}_{H}^{2}<0.\phantom{\rule{2em}{0ex}}\end{array}$

By the definitions of α ε and ${\alpha }_{\epsilon }^{+}$, we deduce that ${\alpha }_{\epsilon }\le {\alpha }_{\epsilon }^{+}<0$.

(ii) See the proof of Lemma 2.5 (iv).

Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.

Lemma 2.8 (i) There exists a ${\left(PS\right)}_{{\alpha }_{\epsilon }}$ -sequence {u n } in M ε for J ε ;

(ii) There exists a ${\left(PS\right)}_{{\alpha }_{\epsilon }^{+}}$-sequence {u n } in ${\mathbf{M}}_{\epsilon }^{+}$ for J ε ;

(iii) There exists a ${\left(PS\right)}_{{\alpha }_{\epsilon }^{-}}$-sequence {u n } in ${\mathbf{M}}_{\epsilon }^{-}$ for J ε .

## 3 Existence of a ground state solution

In order to prove the existence of positive solutions, we claim that J ε satisfies the (PS) β -condition in H1(N) for $\beta \in \left(-\infty ,{\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\right)$, where Λ = ε2(p-q)/(p- 2)and C0 is defined in the following lemma.

Lemma 3.1 Assume that h satisfies (h 1) and 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0. If {u n } is a (PS) β -sequence in H1(N) for J ε with u n u weakly in H1 (N), then ${J}_{\epsilon }^{\prime }\left(u\right)=0$ in H-1 (N) and ${J}_{\epsilon }\left(u\right)\ge -{C}_{0}{\Lambda }^{\frac{2}{2-q}}\ge -{C}_{0}^{\prime }$, where

${C}_{0}=\left(2-q\right){\left[\left(p-q\right){∥h∥}_{#}{S}^{q}\right]}^{\frac{2}{2-q}}/\left[2pq{\left(p-2\right)}^{\frac{q}{2-q}}\right],$

and

${C}_{0}^{\prime }=\left[\left(p-2\right){\left(2-q\right)}^{\frac{p}{p-2}}\right]/\left\{2pq{\left[{f}_{\text{max}}\left(p-q\right)\right]}^{\frac{2}{p-2}}{S}^{\frac{2p}{p-2}}\right\}.$

Proof. Since {u n } is a (PS) β -sequence in H1(N) for J ε with u n u weakly in H1 (N), it is easy to check that ${J}_{\epsilon }^{\prime }\left(u\right)=0$ in H-1(N) and u ≥ 0. Then we have $⟨{J}_{\epsilon }^{\prime }\left(u\right),u⟩=0$, that is, ${\int }_{{ℝ}^{N}}f\left(\epsilon z\right){u}^{p}dz={∥u∥}_{H}^{2}-{\int }_{{ℝ}^{N}}\Lambda h\left(\epsilon z\right){u}^{q}dz$. Hence, by the Young inequality $\left({p}_{1}=\frac{2}{q}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}{p}_{2}=\frac{2}{2-q}\right)$

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(u\right)& =\left(\frac{1}{2}-\frac{1}{p}\right){∥u∥}_{H}^{2}-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){u}^{q}dz\phantom{\rule{2em}{0ex}}\\ \ge \frac{p-2}{2p}{∥u∥}_{H}^{2}-\frac{p-q}{pq}\Lambda {∥h∥}_{#}{S}^{q}{∥u∥}_{H}^{q}\phantom{\rule{2em}{0ex}}\\ \ge \frac{p-2}{2p}{∥u∥}_{H}^{2}-\frac{p-2}{pq}\left[\frac{q{∥u∥}_{H}^{2}}{2}+{\left(\frac{p-q}{p-2}\Lambda {∥h∥}_{#}{S}^{q}\right)}^{\frac{2}{2-q}}\frac{2-q}{2}\right]\phantom{\rule{2em}{0ex}}\\ \ge -\frac{\left(p-2\right){\left(2-q\right)}^{\frac{p}{p-2}}}{2pq{\left[{f}_{\text{max}}\left(p-q\right)\right]}^{\frac{2}{p-2}}{S}^{\frac{2p}{p-2}}}.\phantom{\rule{2em}{0ex}}\end{array}$

Lemma 3.2 Assume that f and h satisfy (f 1) and (h 1). If 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0, then J ε satisfies the (PS) β -condition in H1(N) for $\beta \in \left(-\infty ,{\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\right)$.

Proof. Let {u n } be a (PS) β -sequence in H1(N) for J ε such that J ε (u n ) = β + o n (1) and ${J}_{\epsilon }^{\prime }\left({u}_{n}\right)={o}_{n}$ (1) in H-1(N). Then

$\begin{array}{ll}\hfill \left|\beta \right|+{c}_{n}+\frac{{d}_{n}{∥{u}_{n}∥}_{H}}{p}& \ge {J}_{\epsilon }\left({u}_{n}\right)-\frac{1}{p}⟨{{J}^{\prime }}_{\epsilon }\left({u}_{n}\right),\left({u}_{n}\right)⟩\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{2}-\frac{1}{p}\right){∥{u}_{n}∥}_{H}^{2}-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ \ge \frac{p-2}{2p}{∥{u}_{n}∥}_{H}^{2}-\frac{p-q}{pq}\Lambda {∥h∥}_{#}{S}^{q}{∥{u}_{n}∥}_{H}^{q},\phantom{\rule{2em}{0ex}}\end{array}$

where c n = o n (1), d n = o n (1) as n → ∞. It follows that {u n } is bounded in H1(N). Hence, there exist a subsequence {u n } and a nonnegative u H1 (N) such that ${J}_{\epsilon }^{\prime }\left(u\right)=0$ in H-1 (N), u n u weakly in H1 (N), u n u a.e. in N, u n u strongly in ${L}_{loc}^{s}\left({ℝ}^{N}\right)$ for any 1 ≤ s < 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.

$\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}-u\right)}_{+}^{p}dz=\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{p}dz-\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}^{p}dz+{o}_{n}\left(1\right);$
(3.1)
$\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){\left({u}_{n}-u\right)}_{+}^{q}dz=\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{q}dz-\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){u}^{q}dz+{o}_{n}\left(1\right).$
(3.2)

Next, claim that

$\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){\left|{u}_{n}-u\right|}^{q}dz\to 0\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}n\to \infty \text{.}$
(3.3)

For any σ > 0, there exists r > 0 such that ${\int }_{{\left[{B}^{N}\left(0;r\right)\right]}^{c}}h{\left(\epsilon z\right)}^{\frac{p}{p-q}}dz<\sigma$. By the Hölder inequality and the Sobolev embedding theorem, we get

$\begin{array}{ll}\hfill \left|\underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right){\left|{u}_{n}-u\right|}^{q}dz\right|& \le \underset{{B}^{N}\left(0;r\right)}{\int }{h\left(\epsilon z\right)\left|{u}_{n}-u\right|}^{q}dz\phantom{\rule{2em}{0ex}}\\ +\underset{{\left[{B}^{N}\left(0;r\right)\right]}^{c}}{\int }h\left(\epsilon z\right){\left|{u}_{n}-u\right|}^{q}dz\phantom{\rule{2em}{0ex}}\\ \le {∥h∥}_{#}{\left(\underset{{B}^{N}\left(0;r\right)}{\int }{\left|{u}_{n}-u\right|}^{p}dz\right)}^{q/p}\phantom{\rule{2em}{0ex}}\\ +{S}^{q}{\left(\underset{{\left[{B}^{N}\left(0;r\right)\right]}^{c}}{\int }h{\left(\epsilon z\right)}^{\frac{p}{p-q}}dz\right)}^{\frac{p-q}{p}}{∥{u}_{n}-u∥}_{H}^{q}\phantom{\rule{2em}{0ex}}\\ \le {C}^{\prime }\sigma +{o}_{n}\left(1\right).\phantom{\rule{2em}{0ex}}\\ \left(\because \left\{{u}_{n}\right\}\phantom{\rule{2.77695pt}{0ex}}\text{is}\phantom{\rule{2.77695pt}{0ex}}\text{bounded}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{H}^{1}\left({ℝ}^{N}\right)\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}{u}_{n}\to u\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}{L}_{loc}^{p}\left({ℝ}^{N}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}$

Applying (f 1) and u n u in ${L}_{loc}^{p}\left({ℝ}^{N}\right)$, we get that

$\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}-u\right)}_{+}^{p}dz=\underset{{ℝ}^{N}}{\int }{f}_{\infty }{\left({u}_{n}-u\right)}_{+}^{p}dz+{o}_{n}\left(1\right).$
(3.4)

Let p n = u n - u. Suppose p n 0 strongly in H1 (N). By (3.1)-(3.4), we deduce that

$\begin{array}{ll}\hfill {∥{p}_{n}∥}_{H}^{2}& ={∥{u}_{n}∥}_{H}^{2}-{∥u∥}_{H}^{2}+{o}_{n}\left(1\right)\phantom{\rule{2em}{0ex}}\\ =\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{p}dz-\underset{{ℝ}^{N}}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ -\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}^{p}dz+{\underset{{ℝ}^{N}}{\int }\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){u}^{q}dz+{o}_{n}\left(1\right)\phantom{\rule{2em}{0ex}}\\ =\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}-u\right)}_{+}^{p}dz+{o}_{n}\left(1\right)=\underset{{ℝ}^{N}}{\int }{f}_{\infty }{\left({p}_{n}\right)}_{+}^{p}dz+{o}_{n}\left(1\right).\phantom{\rule{2em}{0ex}}\end{array}$

Then

$\begin{array}{ll}\hfill {I}_{\infty }\left({p}_{n}\right)& =\frac{1}{2}{∥{p}_{n}∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }{f}_{\infty }{\left({p}_{n}\right)}_{+}^{p}dz\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{2}-\frac{1}{p}\right){∥{p}_{n}∥}_{H}^{2}+{o}_{n}\left(1\right)>0.\phantom{\rule{2em}{0ex}}\end{array}$

By Theorem 4.3 in Wang [12], there exists a sequence {s n } + such that s n = 1 + o n (1), {s n p n } N and I(s n p n ) = I(p n ) + o n (1). It follows that

$\begin{array}{ll}\hfill {\gamma }_{\infty }& \le {I}_{\infty }\left({s}_{n}{p}_{n}\right)={I}_{\infty }\left({p}_{n}\right)+{o}_{n}\left(1\right)\phantom{\rule{2em}{0ex}}\\ ={J}_{\epsilon }\left({u}_{n}\right)-{J}_{\epsilon }\left(u\right)+{o}_{n}\left(1\right)\phantom{\rule{2em}{0ex}}\\ =\beta -{J}_{\epsilon }\left(u\right)+{o}_{n}\left(1\right)<{\gamma }_{\infty },\phantom{\rule{2em}{0ex}}\end{array}$

which is a contradiction. Hence, u n u strongly in H1(N).

Remark 3.3 By Lemma 1.1, we obtain

${C}_{0}^{\prime }=\frac{2-q}{q}{\left(\frac{2-q}{p-q}\right)}^{\frac{2}{p-2}}{\gamma }_{\text{max}}<{\gamma }_{\text{max}}<{\gamma }_{\infty },$

and ${\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}>0$ for 0 < Λ < Λ0.

By Lemma 2.8 (i), there is a ${\left(\text{PS}\right)}_{{\alpha }_{\epsilon }}$-sequence {u n } in M ε for J ε . Then we prove that Equation (E ε ) admits a positive ground state solution u0 in N.

Theorem 3.4 Under assumptions (f 1), (h 1), if 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0, then there exists at least one positive ground state solution u0 of Equation (E ε ) in N. Moreover, we have that ${u}_{0}\in {\mathbf{M}}_{\epsilon }^{+}$ and

${J}_{\epsilon }\left({u}_{0}\right)={\alpha }_{\epsilon }={\alpha }_{\epsilon }^{+}\ge -{C}_{0}{\Lambda }^{\frac{2}{2-q}}.$
(3.5)

Proof. By Lemma 2.8 (i), there is a minimizing sequence {u n } M ε for J ε such that J ε (u n ) = α ε + o n (1) and ${J}_{\epsilon }^{\prime }\left({u}_{n}\right)={o}_{n}\left(1\right)$ in H-1 (N). Since ${\alpha }_{\epsilon }<0<{\gamma }^{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}$, by Lemma 3.2, there exist a subsequence {u n } and u0 H1 (N) such that u n u0 strongly in H1 (N). It is easy to see that ${u}_{0}\gneqq 0$ is a solution of Equation (E ε ) in Nand J ε (u0) = α ε . Next, we claim that ${u}_{0}\in {\mathbf{M}}_{\epsilon }^{+}$. On the contrary, assume that ${u}_{0}\in {\mathbf{M}}_{\epsilon }^{-}\left({\mathbf{M}}_{\epsilon }^{0}=\varnothing \phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}0<\Lambda \left(={\epsilon }^{2\left(p-q\right)/\left(p-2\right)}\right)<{\Lambda }_{0}\right)$.

We get that

$\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({u}_{0}\right)}_{+}^{q}dz>0.$

Otherwise,

$\begin{array}{ll}\hfill 0& =\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({u}_{0}\right)}_{+}^{q}dz=\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{q}dz+{o}_{n}\left(1\right)\phantom{\rule{2em}{0ex}}\\ ={∥{u}_{n}∥}_{H}^{2}-\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{p}dz+{o}_{n}\left(1\right).\phantom{\rule{2em}{0ex}}\end{array}$

It follows that

${\alpha }_{\epsilon }+{o}_{n}\left(1\right)={J}_{\epsilon }\left({u}_{n}\right)=\left(\frac{1}{2}-\frac{1}{p}\right){∥{u}_{n}∥}_{H}^{2}+{o}_{n}\left(1\right),$

which contradicts to α ε < 0. By Lemma 2.6 (ii), there exist positive numbers ${t}^{+}<\stackrel{̄}{t}<{t}^{-}=1$ such that ${t}^{+}{u}_{0}\in {\mathbf{M}}_{\epsilon }^{+},{t}^{-}{u}_{0}\in {\mathbf{M}}_{\epsilon }^{-}$ and

${J}_{\epsilon }\left({t}^{+}{u}_{0}\right)<{J}_{\epsilon }\left({t}^{-}{u}_{0}\right)={J}_{\epsilon }\left({u}_{0}\right)={\alpha }_{\epsilon },$

which is a contradiction. Hence, ${u}_{0}\in {\mathbf{M}}_{\epsilon }^{+}$ and

$-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\le {J}_{\epsilon }\left({u}_{0}\right)={\alpha }_{\epsilon }={\alpha }_{\epsilon }^{+}.$

By Lemma 2.4 and the maximum principle, then u0 is a positive solution of Equation (E ε ) in N.

## 4 Existence of k+ 1 solutions

From now, we assume that f and h satisfy (f 1)-(f 2) and (h 1). Let w H1 (N) be the unique, radially symmetric, and positive ground state solution of Equation (E 0) in Nfor f = fmax. Recall the facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).

(i) $w\in {L}^{\infty }\left({ℝ}^{N}\right)\cap {C}_{loc}^{2,\theta }\left({ℝ}^{N}\right)$ for some 0 < θ < 1 and $\underset{\left|z\right|\to \infty }{\text{lim}}w\left(z\right)=0$;

(ii) for any ε > 0, there exist positive numbers C1, ${C}_{1},{C}_{2}^{\epsilon }$, and ${C}_{3}^{\epsilon }$ such that for all z N

${C}_{2}^{\epsilon }\text{exp}\left(-\left(1-\epsilon \right)\left|z\right|\right)\le w\left(z\right)\le {C}_{1}\text{exp}\left(-\left|z\right|\right)$

and

$\left|\nabla w\left(z\right)\right|\le {C}_{3}^{\epsilon }\text{exp}\left(-\left(1-\epsilon \right)\left|z\right|.\right)$

For 1 ≤ ik, we define

${w}_{\epsilon }^{i}\left(z\right)=w\left(z-\frac{{a}^{i}}{\epsilon }\right),\phantom{\rule{2.77695pt}{0ex}}\text{where}\phantom{\rule{2.77695pt}{0ex}}f\left({a}^{i}\right)={f}_{\text{max}}.$

Clearly, ${w}_{\epsilon }^{i}\left(z\right)\in {H}^{1}\left({ℝ}^{N}\right)$. By Lemma 2.6 (ii), there is a unique number ${\left({t}_{\epsilon }^{i}\right)}^{-}>0$ such that ${\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\in {\text{M}}_{\epsilon }^{-}\subset {\text{M}}_{\epsilon }$, where 1 ≤ ik.

We need to prove that

$\underset{\epsilon \to 0+}{\text{lim}}{J}_{\epsilon }\left({\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}\phantom{\rule{2.77695pt}{0ex}}\text{uniformly}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}i.$

Lemma 4.1 (i) There exists a number t0 > 0 such that for 0tt0 and any ε > 0, we have that

${J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)<{\gamma }_{\text{max}}\phantom{\rule{2.77695pt}{0ex}}uniformly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}i;$

(ii) There exist positive numbers t1 and ε1 such that for any t > t1 and ε < ε1, we have that

${J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)<0\phantom{\rule{2.77695pt}{0ex}}uniformly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}i.$

Proof. (i) Since J ε is continuous in ${H}^{1}\left({ℝ}^{N}\right),\phantom{\rule{2.77695pt}{0ex}}\left\{{w}_{\epsilon }^{i}\right\}$ is uniformly bounded in H1 (N) for any ε > 0, and γmax > 0, there is t0 > 0 such that for 0 ≤ tt0 and any ε > 0

${J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)<{\gamma }_{\text{max}}.$

(ii) There is an r0 > 0 such that f (z) ≥ fmax/2 for z BN(ai; r0) uniformly in i. Then there exists ε1 > 0 such that for ε < ε1

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)& =\frac{{t}^{2}}{2}{∥{w}_{\epsilon }^{i}∥}_{H}^{2}-\frac{{t}^{p}}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({w}_{\epsilon }^{i}\right)}^{p}dz-\frac{{t}^{q}}{q}\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({w}_{\epsilon }^{i}\right)}^{q}dz\phantom{\rule{2em}{0ex}}\\ \le \frac{{t}^{2}}{2}\underset{{ℝ}^{N}}{\int }\left[{\left|\nabla w\right|}^{2}{w}^{2}\right]dz-\frac{{t}^{p}}{2p}\underset{{B}^{N\left(0;1\right)}}{\int }{f}_{\text{max}}{w}^{p}dz.\phantom{\rule{2em}{0ex}}\end{array}$

Thus, there is t1 >0 such that for any t > t1 and ε < ε1

${J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)<0\phantom{\rule{2.77695pt}{0ex}}\text{uniformly}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}i.$

Lemma 4.2 Under assumptions (f 1), (f 2), and (h 1). If 0 < Λ ( = ε2(p-q)/(p- 2)) < q Λ0/ 2, then

$\underset{\epsilon \to 0+}{\text{lim}}\underset{t\to 0}{\text{sup}}{J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}\phantom{\rule{2.77695pt}{0ex}}uniformly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}i.$

Proof. By Lemma 4.1, we only need to show that

$\underset{\epsilon \to 0+}{\text{lim}}\underset{{t}_{0}\le t\le {t}_{1}}{\text{sup}}{J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}\phantom{\rule{2.77695pt}{0ex}}\text{uniformly}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}i.$

We know that supt ≥0Imax (tw) = γmax. For t0tt1, we get

$\begin{array}{ll}\hfill {J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)& =\frac{1}{2}{∥t{w}_{\epsilon }^{i}∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left(t{w}_{\epsilon }^{i}\right)}^{p}dz-\frac{1}{q}\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left(t{w}_{\epsilon }^{i}\right)}^{q}dz\phantom{\rule{2em}{0ex}}\\ =\frac{{t}^{2}}{2}\underset{{ℝ}^{N}}{\int }\left[{\left|\nabla w\left(z-\frac{{a}^{i}}{\epsilon }\right)\right|}^{2}+w\left(z-\frac{{a}^{i}}{\epsilon }\right)\right]dz\phantom{\rule{2em}{0ex}}\\ -\frac{{t}^{p}}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right)w{\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{p}dz-\frac{{t}^{q}}{q}\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right)w{\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{q}dz\phantom{\rule{2em}{0ex}}\\ =\left\{\frac{{t}^{2}}{p}\underset{{ℝ}^{N}}{\int }\left[{\left|\nabla w\right|}^{2}+{w}^{2}\right]dz-\frac{{t}^{p}}{p}\underset{{ℝ}^{N}}{\int }{f}_{\text{max}}{w}^{p}dz\right\}\phantom{\rule{2em}{0ex}}\\ +\frac{{t}^{p}}{p}\underset{{ℝ}^{N}}{\int }\left({f}_{\text{max}}-f\left(\epsilon z\right)\right)w{\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{p}dz-\frac{{t}^{q}}{q}\Lambda \underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right)w{\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{q}dz\phantom{\rule{2em}{0ex}}\\ \le {\gamma }_{\text{max}}+\frac{{t}_{1}^{p}}{p}\underset{{ℝ}^{N}}{\int }{\left({f}_{\text{max}}-f\left(\epsilon z\right)\right)w\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{p}dz-\frac{{t}_{0}^{q}}{q}\Lambda \underset{{ℝ}^{N}}{\int }h\left(\epsilon z\right)w\left(z-\frac{{a}^{i}}{\epsilon }\right)dz.\phantom{\rule{2em}{0ex}}\end{array}$

Since

$\begin{array}{l}\underset{{ℝ}^{N}}{\int }{\left({f}_{\text{max}}-f\left(\epsilon z\right)\right)w\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{p}dz\phantom{\rule{2em}{0ex}}\\ =\underset{{ℝ}^{N}}{\int }\left[{f}_{\text{max}}-f\left(\epsilon z+{a}^{i}\right)\right]{w}^{p}dz=o\left(1\right)\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\epsilon \to {0}^{+}\text{uniformly}\phantom{\rule{2.77695pt}{0ex}}\text{in}\phantom{\rule{2.77695pt}{0ex}}i,\phantom{\rule{2em}{0ex}}\end{array}$

and

$\Lambda \underset{{ℝ}^{N}}{\int }{h\left(\epsilon z\right)w\left(z-\frac{{a}^{i}}{\epsilon }\right)}^{q}dz\le {\epsilon }^{\frac{2\left(p-q\right)}{p-2}}{∥h∥}_{#}{S}^{q}{∥w∥}_{H}^{q}=o\left(1\right)\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\epsilon \to {0}^{+},$

then ${\text{lim}}_{\epsilon \to 0+}{\text{sup}}_{{t}_{0}\le t\le {t}_{1}}{J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}$, that is, ${\text{lim}}_{\epsilon \to 0+}{\text{sup}}_{t\ge 0}{J}_{\epsilon }\left(t{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}$ uniformly in i.

Applying the results of Lemmas 2.6, 2.7(ii), and 4.2, we can deduce that

$0<{d}_{0}\le {\alpha }_{\epsilon }^{-}\le {\gamma }_{\text{max}}+o\left(1\right)\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\epsilon \to {0}^{+}.$

Since γmax < γ, there exists ε0 > 0 such that

${\gamma }_{\text{max}}<{\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}\epsilon <{\epsilon }_{0}.$
(4.1)

Choosing 0 < ρ0 < 1 such that

$\overline{{B}_{\rho 0}^{N}\left({a}^{i}\right)}\cap \overline{{B}_{\rho 0}^{N}\left({a}^{j}\right)}=\varnothing \phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}i\ne j\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}1\le i,j\le k,$

where $\overline{{B}_{\rho 0}^{N}\left({a}^{i}\right)}=\left\{z\in {ℝ}^{N}|\left|z-{a}^{i}\right|\le \rho 0\right\}$ and f(ai) = fmax. Define K = {ai| 1 ≤ ik} and ${\mathbf{K}}_{{\rho }_{0}/2}={\cup }_{i=1}^{k}\overline{{B}_{{\rho }_{0}/2}^{N}\left({a}^{i}\right)}$. Suppose ${\cup }_{i=1}^{k}\overline{{B}_{{\rho }_{0}}^{N}\left({a}^{i}\right)}\subset {B}_{{r}_{0}}^{N}\left(0\right)$ for some r0 > 0.

Let Q ε : H1 (N) \ {0} → Nbe given by

${Q}_{\epsilon }\left(u\right)=\frac{{\int }_{{ℝ}^{N}}\chi \left(\epsilon z\right){\left|u\right|}^{p}dz}{{\int }_{{ℝ}^{N}}{\left|u\right|}^{p}dz},$

where χ : NN, χ (z) = z for |z| ≤ r0 and χ (z) = r0z/|z| for |z| > r0.

Lemma 4.3 There exists 0 < ε0ε0 such that if ε < ε0, then ${Q}_{\epsilon }\left({\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\right)\in {\mathbf{K}}_{{\rho }_{0}/2}$ for each 1 ≤ ik.

Proof. Since

$\begin{array}{ll}\hfill {Q}_{\epsilon }\left({\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\right)& =\frac{{\int }_{{ℝ}^{N}}\chi \left(\epsilon z\right){\left|w\left(z-\frac{{a}^{i}}{\epsilon }\right)\right|}^{p}dz}{{\int }_{{ℝ}^{N}}{\left|w\left(z-\frac{{a}^{i}}{\epsilon }\right)\right|}^{p}dz}\phantom{\rule{2em}{0ex}}\\ =\frac{{\int }_{{ℝ}^{N}}\chi \left(\epsilon z+{a}^{i}\right){\left|w\left(z\right)\right|}^{p}dz}{{\int }_{{ℝ}^{N}}{\left|w\left(z\right)\right|}^{p}dz}\phantom{\rule{2em}{0ex}}\\ \to {a}^{i}\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}\epsilon \to {0}^{+},\phantom{\rule{2em}{0ex}}\end{array}$

there exists ε0 > 0 such that

${Q}_{\epsilon }\left({\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\right)\in {\mathbf{K}}_{{\rho }_{0}/2}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}\epsilon <{\epsilon }^{0}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\text{each}\phantom{\rule{2.77695pt}{0ex}}1\le i\le k.$

Lemma 4.4 There exists a number $\stackrel{̄}{\delta }>0$ such that if u N ε and ${I}_{\epsilon }\left(u\right)\le {\gamma }_{\text{max}}+\stackrel{̄}{\delta }$, then ${Q}_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_{0}/2}$ for any 0 < ε < ε0.

Proof. On the contrary, there exist the sequences {ε n } + and $\left\{{u}_{n}\right\}\subset {\mathbf{N}}_{{\epsilon }_{n}}$ such that ${\epsilon }_{n}\to 0,{I}_{{\epsilon }_{n}}\left({u}_{n}\right)={\gamma }_{\text{max}}\left(>0\right)+{o}_{n}$ (1) as n → ∞ and ${Q}_{{\epsilon }_{n}}\left({u}_{n}\right)\notin {\mathbf{K}}_{{\rho }_{0}/2}$ for all n . It is easy to check that {u n } is bounded in H1 (N). Suppose u n → 0 strongly in Lp(N). Since

${∥{u}_{n}∥}_{H}^{2}=\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z\right){\left({u}_{n}\right)}_{+}^{p}dz\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{each}\phantom{\rule{2.77695pt}{0ex}}n\in ℕ,$

and

${I}_{{\epsilon }_{n}}\left({u}_{n}\right)=\frac{1}{2}{∥{u}_{n}∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z\right){\left({u}_{n}\right)}_{+}^{p}dz={\gamma }_{\text{max}}+{o}_{n}\left(1\right),$

then

${\gamma }_{\text{max}}+{o}_{n}\left(1\right)={I}_{{\epsilon }_{n}}\left({u}_{n}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z\right){\left({u}_{n}\right)}_{+}^{p}dz={o}_{n}\left(1\right),$

which is a contradiction. Thus, u n 0 strongly in Lp(N). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d0 > 0 and a sequence $\left\{\stackrel{̃}{{z}_{n}}\right\}\subset {ℝ}^{N}$ such that

$\underset{{B}^{N}\left(\stackrel{̃}{{z}_{n}};1\right)}{\int }{\left|{u}_{n}\left(z\right)\right|}^{2}dz\ge {d}_{0}>0.$
(4.2)

Let ${v}_{n}\left(z\right)={u}_{n}\left(z+\stackrel{̃}{{z}_{n}}\right)$, there are a subsequence {v n } and v H1 (N) such that v n v weakly in H1 (N). Using the similar computation in Lemma 2.6, there is a sequence $\left\{{s}_{\text{max}}^{n}\right\}\subset {ℝ}^{+}$ such that $\stackrel{̃}{{v}_{n}}={s}_{\text{max}}^{n}{v}_{n}\in {\mathbf{N}}_{\text{max}}$ and

$\begin{array}{ll}\hfill 0& <{\gamma }_{\text{max}}\le {I}_{\text{max}}\left(\stackrel{̃}{{v}_{n}}\right)\le {I}_{{\epsilon }_{n}}\left({s}_{\text{max}}^{n}{u}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \le {I}_{{\epsilon }_{n}}\left({u}_{n}\right)={\gamma }_{\text{max}}+{o}_{n}\left(1\right)\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}n\to \infty .\phantom{\rule{2em}{0ex}}\end{array}$

We deduce that a convergent subsequence $\left\{{s}_{\text{max}}^{n}\right\}$ satisfies ${s}_{\text{max}}^{n}\to {s}_{0}>0$. Then there are subsequences $\left\{\stackrel{̃}{{v}_{n}}\right\}$ and $ṽ\in {H}^{1}\left({ℝ}^{N}\right)$ such that ${ṽ}_{n}⇀ṽ\left(={s}_{0}v\right)$ weakly in H1 (N). By (4.2), then $ṽ\ne 0$. Moreover, we can obtain that $\stackrel{̃}{{v}_{n}}\to ṽ$ strongly in H1 (N) and ${I}_{\text{max}}\left(ṽ\right)={\gamma }_{\text{max}}$. Now, we want to show that there exists a subsequence $\left\{{z}_{n}\right\}=\left\{{\epsilon }_{n}\stackrel{̃}{{z}_{n}}\right\}$ such that z n z0 K.

(i) Claim that the sequence {z n } is bounded in N. On the contrary, assume that |z n | → ∞, then

$\begin{array}{ll}\hfill {\gamma }_{\text{max}}& ={I}_{\text{max}}\left(ṽ\right)<{I}_{\infty }\left(ṽ\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim inf}}\left[\frac{1}{2}{∥\stackrel{̃}{{v}_{n}}∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z+{z}_{n}\right){\left(\stackrel{̃}{{v}_{n}}\right)}_{+}^{p}dz\right]\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim inf}}\left[\frac{{\left({s}_{\text{max}}^{n}\right)}^{2}}{2}{∥{u}_{n}∥}_{H}^{2}-\frac{{\left({s}_{\text{max}}^{n}\right)}^{p}}{p}\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z\right){\left({u}_{n}\right)}_{+}^{p}dz\right]\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim inf}}{I}_{{\epsilon }_{n}}\left({s}_{\text{max}}^{n}{u}_{n}\right)\le \underset{n\to \infty }{\text{lim inf}}{I}_{{\epsilon }_{n}}\left({u}_{n}\right)={\gamma }_{\text{max}},\phantom{\rule{2em}{0ex}}\end{array}$

(ii) Claim that z0 K. On the contrary, assume that z0 K, that is, f(z0) < fmax. Then using the above argument to obtain that

$\begin{array}{ll}\hfill {\gamma }_{\text{max}}& ={I}_{\text{max}}\left(ṽ\right)<\frac{1}{2}{∥ṽ∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left({z}_{0}\right){\left(ṽ\right)}_{+}^{p}dz\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim inf}}\left[\frac{1}{2}{∥\stackrel{̃}{{v}_{n}}∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left({\epsilon }_{n}z+{z}_{n}\right){\left(\stackrel{̃}{{v}_{n}}\right)}_{+}^{p}dz\right]\phantom{\rule{2em}{0ex}}\\ ={\gamma }_{\text{max}},\phantom{\rule{2em}{0ex}}\end{array}$

which is a contradiction. Since v n v ≠ 0 in H1 (N), we have that

$\begin{array}{ll}\hfill {Q}_{{\epsilon }_{n}}\left({u}_{n}\right)& =\frac{\underset{{ℝ}^{N}}{\int }\chi \left({\epsilon }_{n}z\right){\left|{v}_{n}\left(z-\stackrel{̃}{{z}_{n}}\right)\right|}^{p}dz}{\underset{{ℝ}^{N}}{\int }{\left|{v}_{n}\left(z-\stackrel{̃}{{z}_{n}}\right)\right|}^{p}dz}\phantom{\rule{2em}{0ex}}\\ =\frac{\underset{{ℝ}^{N}}{\int }\chi \left({\epsilon }_{n}z+{\epsilon }_{n}\stackrel{̃}{{z}_{n}}\right){\left|{v}_{n}\right|}^{p}dz}{\underset{{ℝ}^{N}}{\int }{\left|{v}_{n}\right|}^{p}dz}\to {z}_{0}\subset {\mathbf{K}}_{{\rho }_{0}/2}\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}n\to \infty ,\phantom{\rule{2em}{0ex}}\end{array}$

Hence, there exists a number $\stackrel{̄}{\delta }>0$ such that if u N ε and ${I}_{\epsilon }\left(u\right)\le {\gamma }_{\text{max}}+\stackrel{̄}{\delta }$, then ${Q}_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_{0}/2}$ for any 0 < ε < ε0.

From (4.1), choosing $0<{\delta }_{0}<\stackrel{̄}{\delta }$ such that

${\gamma }_{\text{max}}+{\delta }_{0}<{\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <{\epsilon }^{0}.$
(4.3)

For each 1 ≤ ik, define

$\begin{array}{ll}\hfill {O}_{\epsilon }^{i}& =\left\{u\in {\mathbf{M}}_{\epsilon }^{-}|\left|{Q}_{\epsilon }\left(u\right)-{a}^{i}\right|<{\rho }_{0}\right\},\phantom{\rule{2em}{0ex}}\\ \hfill \partial {O}_{\epsilon }^{i}& =\left\{u\in {\mathbf{M}}_{\epsilon }^{-}|\left|{Q}_{\epsilon }\left(u\right)-{a}^{i}\right|={\rho }_{0}\right\},\phantom{\rule{2em}{0ex}}\end{array}$

${\beta }_{\epsilon }^{i}=\underset{u\in {O}_{\epsilon }^{i}}{\text{inf}}{J}_{\epsilon }\left(u\right)$ and ${\stackrel{̃}{\beta }}_{\epsilon }^{i}=\underset{u\in \partial {O}_{\epsilon }^{i}}{\text{inf}}{J}_{\epsilon }\left(u\right)$.

Lemma 4.5 If $u\in {\mathbf{M}}_{\epsilon }^{-}$ and J ε (u) ≤ γmax + δ0/2, then there exists a number $0<\stackrel{̄}{\epsilon }<{\epsilon }^{0}$ such that ${Q}_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_{0}/2}$ for any $0<\epsilon <\stackrel{̄}{\epsilon }$.

Proof. We use the similar computation in Lemma 2.6 to get that there is a unique positive number

${s}_{\epsilon }^{u}={\left(\frac{{∥u∥}_{H}^{2}}{\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){u}_{+}^{p}dz}\right)}^{1/\left(p-2\right)}$

such that ${s}_{\epsilon }^{u}u\in {\mathbf{N}}_{\epsilon }$. We want to show that ${s}_{\epsilon }^{u} for some constant c > 0 (independent of u). First, since $u\in {\mathbf{M}}_{\epsilon }^{-}\subset {\mathbf{M}}_{\epsilon }$,

$0<{d}_{0}\le {\alpha }_{\epsilon }^{-}\le {J}_{\epsilon }\left(u\right)\le {\gamma }_{\text{max}}+{\delta }_{0}/2,$

and J ε is coercive on M ε , then $0<{c}_{2}<{∥u∥}_{H}^{2}<{c}_{1}$ for some constants c1 and c2 (independent of u). Next, we claim that ${∥u∥}_{{L}^{p}}^{p}>{c}_{3}>0$ for some constant c3 > 0 (independent of u). On the contrary, there exists a sequence $\left\{{u}_{n}\right\}\subset {\mathbf{M}}_{\epsilon }^{-}$ such that

${∥{u}_{n}∥}_{{L}^{p}}^{p}={o}_{n}\left(1\right)\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}n\to \infty .$

By (2.3),

$\frac{2-q}{p-q}<\frac{\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({u}_{n}\right)}_{+}^{p}dz}{{∥{u}_{n}∥}_{H}^{2}}\le \frac{{f}_{\text{max}}{∥{u}_{n}∥}_{{L}^{p}}^{p}}{{c}_{2}}={o}_{n}\left(1\right),$

which is a contradiction. Thus, ${s}_{\epsilon }^{u} for some constant c > 0 (independent of u). Now, we get that

$\begin{array}{ll}\hfill {\gamma }_{\text{max}}+{\delta }_{0}/2& \ge {J}_{\epsilon }\left(u\right)=\underset{t\ge 0}{\text{sup}}{J}_{\epsilon }\left(tu\right)\ge {J}_{\epsilon }\left({s}_{\epsilon }^{u}u\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2}{∥{s}_{\epsilon }^{u}u∥}_{H}^{2}-\frac{1}{p}\underset{{ℝ}^{N}}{\int }f\left(\epsilon z\right){\left({s}_{\epsilon }^{u}u\right)}_{+}^{p}dz-\frac{1}{q}\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({s}_{\epsilon }^{u}u\right)}_{+}^{q}dz\phantom{\rule{2em}{0ex}}\\ \ge {I}_{\epsilon }\left({s}_{\epsilon }^{u}u\right)-\frac{1}{q}\underset{{ℝ}^{N}}{\int }\Lambda h\left(\epsilon z\right){\left({s}_{\epsilon }^{u}u\right)}_{+}^{q}dz.\phantom{\rule{2em}{0ex}}\end{array}$

From the above inequality, we deduce that

Hence, there exists $0<\stackrel{̄}{\epsilon }<{\epsilon }^{0}$ such that for $0<\epsilon <\stackrel{̄}{\epsilon }$

${I}_{\epsilon }\left({s}_{\epsilon }^{u}u\right)\le {\gamma }_{\text{max}}+{\delta }_{0},\phantom{\rule{2.77695pt}{0ex}}\text{where}\phantom{\rule{2.77695pt}{0ex}}{s}_{\epsilon }^{u}u\in {\mathbf{N}}_{\epsilon }.$

By Lemma 4.4, we obtain

${Q}_{\epsilon }\left({s}_{\epsilon }^{u}u\right)=\frac{\underset{{ℝ}^{N}}{\int }\chi \left(\epsilon z\right){\left|{s}_{\epsilon }^{u}u\left(z\right)\right|}^{p}dz}{\underset{{ℝ}^{N}}{\int }{\left|{s}_{\epsilon }^{u}u\left(z\right)\right|}^{p}dz}\in {\mathbf{K}}_{{\rho }_{0}/2}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <\stackrel{̄}{\epsilon },$

or ${Q}_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_{0}/2}$ for any $0<\epsilon <\stackrel{̄}{\epsilon }$.

Applying the above lemma, we get that

${\stackrel{̃}{\beta }}_{\epsilon }^{i}\ge {\gamma }_{\text{max}}+{\delta }_{0}/2\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <\stackrel{̄}{\epsilon }.$
(4.4)

By Lemmas 4.2, 4.3, and Equation (4.3), there exists $0<{\epsilon }^{*}\le \stackrel{̄}{\epsilon }$ such that

${\beta }_{\epsilon }^{i}\le {J}_{\epsilon }\left({\left({t}_{\epsilon }^{i}\right)}^{-}{w}_{\epsilon }^{i}\right)\le {\gamma }_{\text{max}}+{\delta }_{0}/3<{\gamma }_{\infty }-{C}_{0}{\Lambda }^{\frac{2}{2-q}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <{\epsilon }^{*}.$
(4.5)

Lemma 4.6 Given $u\in {O}_{\epsilon }^{i}$, then there exist an η > 0 and a differentiable functional l : B(0; η) H1(N) → + such that $l\left(0\right)=1,\phantom{\rule{2.77695pt}{0ex}}l\left(v\right)\left(u-v\right)\in {O}_{\epsilon }^{i}$ for any v B(0;η) and

$⟨{l}^{\prime }\left(v\right),\varphi ⟩{|}_{\left(l,v\right)=\left(1,0\right)}=\frac{⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),\varphi ⟩}{⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}any\phantom{\rule{2.77695pt}{0ex}}\varphi \in {C}_{c}^{\infty }\left({ℝ}^{N}\right),$
(4.6)

where ${\psi }_{\epsilon }\left(u\right)=⟨{J}_{\epsilon }^{\prime }\left(u\right),u⟩$.

Proof. See Cao and Zhou [7].

Lemma 4.7 For each 1 ≤ ik, there is a ${\left(PS\right)}_{{\beta }_{\epsilon }^{i}}$-sequence $\left\{{u}_{n}\right\}\subset {O}_{\epsilon }^{i}$ in H1(N) for J ε .

Proof. For each 1 ≤ ik, by (4.4) and (4.5),

${\beta }_{\epsilon }^{i}<{\stackrel{̃}{\beta }}_{\epsilon }^{i}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <{\epsilon }^{*}.$
(4.7)

Then

${\beta }_{\epsilon }^{i}=\underset{u\in {O}_{\epsilon }^{i}\cup \partial {O}_{\epsilon }^{i}}{\text{inf}}{J}_{\epsilon }\left(u\right)\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{any}\phantom{\rule{2.77695pt}{0ex}}0<\epsilon <{\epsilon }^{*}.$

Let $\left\{{u}_{n}^{i}\right\}\subset {O}_{\epsilon }^{i}\cup \partial {O}_{\epsilon }^{i}$ be a minimizing sequence for ${\beta }_{\epsilon }^{i}$. Applying Ekeland's variational principle, there exists a subsequence $\left\{{u}_{n}^{i}\right\}$ such that ${J}_{\epsilon }\left({u}_{n}^{i}\right)={\beta }_{\epsilon }^{i}+1/n$ and

${J}_{\epsilon }\left({u}_{n}^{i}\right)\le {J}_{\epsilon }\left(w\right)+{∥w-{u}_{n}^{i}∥}_{H}/n\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}w\in {O}_{\epsilon }^{i}\cup \partial {O}_{\epsilon }^{i}.$
(4.8)

Using (4.7), we may assume that ${u}_{n}^{i}\in {O}_{\epsilon }^{i}$ for sufficiently large n. By Lemma 4.6, then there exist an ${\eta }_{n}^{i}>0$ and a differentiable functional ${l}_{n}^{i}:B\left(0;{\eta }_{n}^{i}\right)\subset {H}^{1}\left({ℝ}^{N}\right)\to {ℝ}^{+}$ such that ${l}_{n}^{i}\left(0\right)=1$, and ${l}_{n}^{i}\left(v\right)\left({u}_{n}^{i}-v\right)\in {O}_{\epsilon }^{i}$ for $v\in B\left(0;{\eta }_{n}^{i}\right)$. Let v σ = σv with ║v H = 1 and $0<\sigma <{\eta }_{n}^{i}$. Then ${v}_{\sigma }\in B\left(0,{\eta }_{n}^{i}\right)$ and ${w}_{\sigma }={l}_{n}^{i}\left({v}_{\sigma }\right)\left({u}_{n}^{i}-{v}_{\sigma }\right)\in {O}_{\epsilon }^{i}$. From (4.8) and by the mean value theorem, we get that as σ → 0

$\begin{array}{ll}\hfill \frac{{∥{w}_{\sigma }-{u}_{n}^{i}∥}_{H}}{n}& \ge {J}_{\epsilon }\left({u}_{n}^{i}\right)-{J}_{\epsilon }\left({w}_{\sigma }\right)\phantom{\rule{2em}{0ex}}\\ =⟨{{J}^{\prime }}_{\epsilon }\left({t}_{0}{u}_{n}^{i}+\left(1-{t}_{0}\right){w}_{\sigma }\right),{u}_{n}^{i}-{w}_{\sigma }⟩\phantom{\rule{2.77695pt}{0ex}}\text{where}\phantom{\rule{2.77695pt}{0ex}}{t}_{0}\in \left(0,1\right)\phantom{\rule{2em}{0ex}}\\ =⟨{{J}^{\prime }}_{\epsilon }\left({u}_{n}^{i}\right),{u}_{n}^{i}-{w}_{\sigma }⟩+o\left({∥{u}_{n}^{i}-{w}_{\sigma }∥}_{H}\right)\left(\because {J}_{\epsilon }\in {C}^{1}\right)\phantom{\rule{2em}{0ex}}\\ =\sigma {l}_{n}^{i}\left({v}_{\sigma }\right)⟨{{J}^{\prime }}_{\epsilon }\left({u}_{n}^{i}\right),v⟩+\left(1-{l}_{n}^{i}\left({v}_{\sigma }\right)\right)⟨{{J}^{\prime }}_{\epsilon }\left({u}_{n}^{i}\right),{u}_{n}^{i}⟩+o\left({∥{u}_{n}^{i}-{w}_{\sigma }∥}_{H}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(\because {l}_{n}^{}\end{array}$