Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝ^N

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

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

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

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

where λ > 0.

Let f and h satisfy the following conditions:

(f 1) f is a positive continuous function in ℝ^Nand lim_{|z| → ∞}f(z) = f_∞ > 0.

(f 2) there exist k points a¹, a²,..., a^kin ℝ^Nsuch that

$$f\left(a^i\right)=f_{\text{max}}=\underset{z\in {ℝ}^N}{\text{max}}f\left(z\right)\text{for}\text{1}\le \text{i}\le k,$$

and f_∞ < f_max.

(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

$$\left\{\begin{array}{c}-\Delta u=ch\left(z\right){\left|u\right|}^{q-2}u+{\left|u\right|}^{p-2}u\text{in}\Omega ;\hfill \\ u\in H_0^1\left(\Omega \right),\hfill \end{array}\right.\left(E_c\right)$$

have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2] $h\in C\left(\bar{\Omega }\right)$ 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 ℝ^N. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝ^N, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (E _λ ) admits at least two positive solutions in ℝ^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 ℝ^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 ℝ^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)v^p-1+ λh(z) v^q-1in ℝ^Nfor sufficiently small λ > 0.

By the change of variables

$$\epsilon ={\lambda }^{-\frac{1}{2}}\text{and}u\text{(}z\text{)}\text{ = }{\epsilon }^{\frac{2}{p-2}}v\left(\epsilon z\right),$$

Equation (E _λ ) is transformed to

$$\left\{\begin{array}{c}-\Delta u+u=f\left(\epsilon z\right)u^{p-1}+{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right)u^{q-1}\text{in}{ℝ}^N;\hfill \\ u\in H^1\left({ℝ}^N\right),\hfill \end{array}\right.\left(E_{\epsilon }\right)$$

Associated with Equation (E _ε ), we consider the C¹-functional J _ε , for u ∈ H¹ (ℝ^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 ${∥u∥}_H^2=\underset{{ℝ}^N}{\int }\left({\left|\Delta u\right|}^2+{\left|u\right|}^2\right)dz$ is the norm in H¹ (ℝ^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 ≤ i ≤ k.

Let

$$S=\underset{\underset{{\Vert u\Vert }_H=1}{u\in H^1({ℝ}^N)}}{\sup }{\Vert u\Vert }_{L^p},$$

then

$${∥u∥}_{L^p}\le S{∥u∥}_H\text{for}\text{any}u\in H^1\left({ℝ}^N\right)\backslash \left\{0\right\}.$$

(1.1)

For the semilinear elliptic equations

$$\left\{\begin{array}{c}-\Delta u+u=f\left(\epsilon z\right)u^{p-1}\text{in}{ℝ}^N;\hfill \\ u\in H^1\left({ℝ}^N\right),\hfill \end{array}\right.$$

(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 (ℝ^N) \ {0} | u₊ ≢ 0 and $⟨I_{\epsilon }^{\prime }\left(u\right),u⟩=0$}. Note that

(i) if f ≡ f_∞, 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 (ℝ^N) \ {0} | u₊ ≢ 0 and $⟨I_{\infty }^{\prime }\left(u\right),u⟩=0$};

(ii) if f ≡ f_max, 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 N_max = {u ∈ H¹ (ℝ^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 = ε^2(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(ℝ^N) for some functional J.

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

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

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

$${\mathbf{M}}_{\epsilon }=\left\{u\in H^1\left({ℝ}^N\right)\backslash \left\{0\right\}\left|u_{+}\not\equiv 0\text{and}⟨{J^{\prime }}_{\epsilon }\left(u\right),u⟩=0\right.\right\},$$

(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

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

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

$$\begin{array}{ll}\hfill ⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩& =2{∥u∥}_H^2-p\underset{{ℝ}^N}{\int }f\left(\epsilon z\right)u_{+}^{p}dz-q{\underset{{ℝ}^N}{\int }\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right)u_{+}^{q}dz\\ =\left(p-q\right)\underset{{ℝ}^N}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right)u_{+}^{q}dz-\left(p-2\right){∥u∥}_H^2\end{array}$$

(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

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

Lemma 2.3 Under assumptions (f 1) and (h 1), if 0 < Λ (= ε^{2(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 (ℝ^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

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

(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

$$\begin{array}{ll}\hfill J_{\epsilon }\left(u\right)& =\left(\frac{1}{2}-\frac{1}{p}\right){∥u∥}_H^2-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right)u_{+}^{q}dz\\ \ge \frac{{∥u∥}_H^q}{p}\left[\frac{p-2}{2}{∥u∥}_H^{2-q}-\frac{p-q}{q}\Lambda {∥h∥}_{\#}{S}^q\right]\\ >\frac{1}{p}{\left[\frac{2-q}{\left(p-q\right)f_{\text{max}}{S}^p}\right]}^{\frac{q}{p-2}}\left[\frac{p-2}{2}{\left[\frac{2-q}{\left(p-q\right)f_{\text{max}}{S}^p}\right]}^{\frac{2-q}{p-2}}-\frac{p-q}{q}\Lambda {∥h∥}_{\#}{S}^q\right].\end{array}$$

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∥ _# , f_max).

For u ∈ H¹ (ℝ^N) \ {0} and u ₊ ≢ 0, let

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

Lemma 2.6 For each u ∈ H¹ (ℝ^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 $t^{-}=t^{-}\left(u\right)>\bar{t}$ such that $t^{-}u\in {\mathbf{M}}_{\epsilon }^{-}$ and J _ε (t^-u) = sup_{t ≥ 0}J _ε (tu);

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

$$J_{\epsilon }\left(t^{+}u\right)=\underset{0\le t\le \bar{t}}{\text{inf}}{J}_{\epsilon }\left(tu\right),J_{\epsilon }\left(t^{-}u\right)=\underset{t\ge \bar{t}}{\text{sup}}{J}_{\epsilon }\left(tu\right).$$

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

Applying Lemma 2.3 $\left({\mathbf{M}}_{\epsilon }^0=\varnothing \text{for}0<\Lambda <{\Lambda }_0\right)$, 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

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

Lemma 2.7 (i) If 0 < Λ ( = ε^{2(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∥ _# , f_max).

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

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

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(ℝ^N) for $\beta \in \left(-\infty ,{\gamma }_{\infty }-C_0{\Lambda }^{\frac{2}{2-q}}\right)$, where Λ = ε^{2(p-q)/(p- 2)}and C₀ is defined in the following lemma.

Lemma 3.1 Assume that h satisfies (h 1) and 0 < Λ ( = ε^{2(p-q)/(p- 2)}) < Λ₀. If {u _n } is a (PS) _β -sequence in H¹(ℝ^N) for J _ε with u _n ⇀ u weakly in H¹ (ℝ^N), then $J_{\epsilon }^{\prime }\left(u\right)=0$ in H^-1 (ℝ^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(ℝ^N) for J _ε with u _n ⇀ u weakly in H¹ (ℝ^N), it is easy to check that $J_{\epsilon }^{\prime }\left(u\right)=0$ in H^-1(ℝ^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 $\left(p_1=\frac{2}{q}\text{and}{p}_2=\frac{2}{2-q}\right)$

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

Lemma 3.2 Assume that f and h satisfy (f 1) and (h 1). If 0 < Λ ( = ε^{2(p-q)/(p- 2)}) < Λ₀, then J _ε satisfies the (PS) _β -condition in H¹(ℝ^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(ℝ^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(ℝ^N). Then

$$\begin{array}{ll}\hfill \left|\beta \right|+c_n+\frac{d_n{∥u_n∥}_H}{p}& \ge J_{\epsilon }\left(u_n\right)-\frac{1}{p}⟨{J^{\prime }}_{\epsilon }\left(u_n\right),\left(u_n\right)⟩\\ =\left(\frac{1}{2}-\frac{1}{p}\right){∥u_n∥}_H^2-\left(\frac{1}{q}-\frac{1}{p}\right)\underset{{ℝ}^N}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){\left(u_n\right)}_{+}^{q}dz\\ \ge \frac{p-2}{2p}{∥u_n∥}_H^2-\frac{p-q}{pq}\Lambda {∥h∥}_{\#}{S}^q{∥u_n∥}_H^q,\end{array}$$

where c _n = o _n (1), d _n = o _n (1) as n → ∞. It follows that {u _n } is bounded in H¹(ℝ^N). Hence, there exist a subsequence {u _n } and a nonnegative u ∈ H¹ (ℝ^N) such that $J_{\epsilon }^{\prime }\left(u\right)=0$ in H^-1 (ℝ^N), u _n ⇀ u weakly in H¹ (ℝ^N), u _n ⇀ u a.e. in ℝ^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

$$\underset{{ℝ}^N}{\int }h\left(\epsilon z\right){\left|u_n-u\right|}^{q}dz\to 0\text{as}n\to \infty \text{.}$$

(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

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

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 (ℝ^N). By (3.1)-(3.4), we deduce that

$$\begin{array}{ll}\hfill {∥p_n∥}_H^2& ={∥u_n∥}_H^2-{∥u∥}_H^2+o_n\left(1\right)\\ =\underset{{ℝ}^N}{\int }f\left(\epsilon z\right){\left(u_n\right)}_{+}^{p}dz-\underset{{ℝ}^N}{\int }{\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right){\left(u_n\right)}_{+}^{q}dz\\ -\underset{{ℝ}^N}{\int }f\left(\epsilon z\right)u^{p}dz+{\underset{{ℝ}^N}{\int }\epsilon }^{\frac{2\left(p-q\right)}{p-2}}h\left(\epsilon z\right)u^{q}dz+o_n\left(1\right)\\ =\underset{{ℝ}^N}{\int }f\left(\epsilon z\right){\left(u_n-u\right)}_{+}^{p}dz+o_n\left(1\right)=\underset{{ℝ}^N}{\int }{f}_{\infty }{\left(p_n\right)}_{+}^{p}dz+o_n\left(1\right).\end{array}$$

Then

$$\begin{array}{ll}\hfill I_{\infty }\left(p_n\right)& =\frac{1}{2}{∥p_n∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }{f}_{\infty }{\left(p_n\right)}_{+}^{p}dz\\ =\left(\frac{1}{2}-\frac{1}{p}\right){∥p_n∥}_H^2+o_n\left(1\right)>0.\end{array}$$

By Theorem 4.3 in Wang [12], there exists a sequence {s _n } ⊂ ℝ⁺ 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

$$\begin{array}{ll}\hfill {\gamma }_{\infty }& \le I_{\infty }\left(s_n{p}_n\right)=I_{\infty }\left(p_n\right)+o_n\left(1\right)\\ =J_{\epsilon }\left(u_n\right)-J_{\epsilon }\left(u\right)+o_n\left(1\right)\\ =\beta -J_{\epsilon }\left(u\right)+o_n\left(1\right)<{\gamma }_{\infty },\end{array}$$

which is a contradiction. Hence, u _n → u strongly in H¹(ℝ^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 ℝ^N.

Theorem 3.4 Under assumptions (f 1), (h 1), if 0 < Λ ( = ε^{2(p-q)/(p- 2)}) < Λ₀, then there exists at least one positive ground state solution u₀ of Equation (E _ε ) in ℝ^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 (ℝ^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 (ℝ^N) such that u _n → u₀ strongly in H¹ (ℝ^N). It is easy to see that $u_0\gneqq 0$ is a solution of Equation (E _ε ) in ℝ^Nand J _ε (u₀) = α _ε . Next, we claim that $u_0\in {\mathbf{M}}_{\epsilon }^{+}$. On the contrary, assume that $u_0\in {\mathbf{M}}_{\epsilon }^{-}\left({\mathbf{M}}_{\epsilon }^0=\varnothing \text{for}0<\Lambda \left(={\epsilon }^{2\left(p-q\right)/\left(p-2\right)}\right)<{\Lambda }_0\right)$.

We get that

$$\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(u_0\right)}_{+}^{q}dz>0.$$

Otherwise,

$$\begin{array}{ll}\hfill 0& =\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(u_0\right)}_{+}^{q}dz=\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(u_n\right)}_{+}^{q}dz+o_n\left(1\right)\\ ={∥u_n∥}_H^2-\underset{{ℝ}^N}{\int }f\left(\epsilon z\right){\left(u_n\right)}_{+}^{p}dz+o_n\left(1\right).\end{array}$$

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 $t^{+}<\bar{t}<t^{-}=1$ 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 ℝ^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 (ℝ^N) be the unique, radially symmetric, and positive ground state solution of Equation (E 0) in ℝ^Nfor f = f_max. 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 ∈ ℝ^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 ≤ i ≤ k, we define

$$w_{\epsilon }^i\left(z\right)=w\left(z-\frac{a^i}{\epsilon }\right),\text{where}f\left(a^i\right)=f_{\text{max}}.$$

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 ≤ i ≤ k.

We need to prove that

$$\underset{\epsilon \to 0+}{\text{lim}}{J}_{\epsilon }\left({\left(t_{\epsilon }^i\right)}^{-}{w}_{\epsilon }^i\right)\le {\gamma }_{\text{max}}\text{uniformly}\text{in}i.$$

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

$$J_{\epsilon }\left(t{w}_{\epsilon }^i\right)<{\gamma }_{\text{max}}uniformlyini;$$

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

$$J_{\epsilon }\left(t{w}_{\epsilon }^i\right)<0uniformlyini.$$

Proof. (i) Since J _ε is continuous in $H^1\left({ℝ}^N\right),\left\{w_{\epsilon }^i\right\}$ is uniformly bounded in H¹ (ℝ^N) for any ε > 0, and γ_max > 0, there is t₀ > 0 such that for 0 ≤ t ≤ t₀ 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) ≥ f_max/2 for z ∈ B^N(aⁱ; r₀) uniformly in i. Then there exists ε₁ > 0 such that for ε < ε₁

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

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

$$J_{\epsilon }\left(t{w}_{\epsilon }^i\right)<0\text{uniformly}\text{in}i.$$

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

$$\underset{\epsilon \to 0+}{\text{lim}}\underset{t\to 0}{\text{sup}}{J}_{\epsilon }\left(t{w}_{\epsilon }^i\right)\le {\gamma }_{\text{max}}uniformlyini.$$

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

$$\underset{\epsilon \to 0+}{\text{lim}}\underset{t_0\le t\le t_1}{\text{sup}}{J}_{\epsilon }\left(t{w}_{\epsilon }^i\right)\le {\gamma }_{\text{max}}\text{uniformly}\text{in}i.$$

We know that sup_{t ≥0}I_max (tw) = γ_max. For t₀ ≤ t ≤ t₁, we get

$$\begin{array}{ll}\hfill J_{\epsilon }\left(t{w}_{\epsilon }^i\right)& =\frac{1}{2}{∥t{w}_{\epsilon }^i∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }f\left(\epsilon z\right){\left(t{w}_{\epsilon }^i\right)}^{p}dz-\frac{1}{q}\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(t{w}_{\epsilon }^i\right)}^{q}dz\\ =\frac{t^2}{2}\underset{{ℝ}^N}{\int }\left[{\left|\nabla w\left(z-\frac{a^i}{\epsilon }\right)\right|}^2+w\left(z-\frac{a^i}{\epsilon }\right)\right]dz\\ -\frac{t^p}{p}\underset{{ℝ}^N}{\int }f\left(\epsilon z\right)w{\left(z-\frac{a^i}{\epsilon }\right)}^{p}dz-\frac{t^q}{q}\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right)w{\left(z-\frac{a^i}{\epsilon }\right)}^{q}dz\\ =\left\{\frac{t^2}{p}\underset{{ℝ}^N}{\int }\left[{\left|\nabla w\right|}^2+w^2\right]dz-\frac{t^p}{p}\underset{{ℝ}^N}{\int }{f}_{\text{max}}{w}^{p}dz\right\}\\ +\frac{t^p}{p}\underset{{ℝ}^N}{\int }\left(f_{\text{max}}-f\left(\epsilon z\right)\right)w{\left(z-\frac{a^i}{\epsilon }\right)}^{p}dz-\frac{t^q}{q}\Lambda \underset{{ℝ}^N}{\int }h\left(\epsilon z\right)w{\left(z-\frac{a^i}{\epsilon }\right)}^{q}dz\\ \le {\gamma }_{\text{max}}+\frac{t_1^p}{p}\underset{{ℝ}^N}{\int }{\left(f_{\text{max}}-f\left(\epsilon z\right)\right)w\left(z-\frac{a^i}{\epsilon }\right)}^{p}dz-\frac{t_0^q}{q}\Lambda \underset{{ℝ}^N}{\int }h\left(\epsilon z\right)w\left(z-\frac{a^i}{\epsilon }\right)dz.\end{array}$$

Since

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

and

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

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

$$0<d_0\le {\alpha }_{\epsilon }^{-}\le {\gamma }_{\text{max}}+o\left(1\right)\text{as}\epsilon \to 0^{+}.$$

Since γ_max < γ_∞, there exists ε₀ > 0 such that

$${\gamma }_{\text{max}}<{\gamma }_{\infty }-C_0{\Lambda }^{\frac{2}{2-q}}\text{for}\text{any}\epsilon <{\epsilon }_0.$$

(4.1)

Choosing 0 < ρ₀ < 1 such that

$$\overline{B_{\rho 0}^N\left(a^i\right)}\cap \overline{B_{\rho 0}^N\left(a^j\right)}=\varnothing \text{for}i\ne j\text{and}1\le i,j\le k,$$

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) = f_max. Define K = {aⁱ| 1 ≤ i ≤ k} 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 (ℝ^N) \ {0} → ℝ^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 : ℝ^N→ ℝ^N, χ (z) = z for |z| ≤ r₀ and χ (z) = r₀z/|z| for |z| > r₀.

Lemma 4.3 There exists 0 < ε⁰ ≤ ε₀ 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 ≤ i ≤ k.

Proof. Since

$$\begin{array}{ll}\hfill Q_{\epsilon }\left({\left(t_{\epsilon }^i\right)}^{-}{w}_{\epsilon }^i\right)& =\frac{{\int }_{{ℝ}^N}\chi \left(\epsilon z\right){\left|w\left(z-\frac{a^i}{\epsilon }\right)\right|}^{p}dz}{{\int }_{{ℝ}^N}{\left|w\left(z-\frac{a^i}{\epsilon }\right)\right|}^{p}dz}\\ =\frac{{\int }_{{ℝ}^N}\chi \left(\epsilon z+a^i\right){\left|w\left(z\right)\right|}^{p}dz}{{\int }_{{ℝ}^N}{\left|w\left(z\right)\right|}^{p}dz}\\ \to a^{i}as\epsilon \to 0^{+},\end{array}$$

there exists ε⁰ > 0 such that

$$Q_{\epsilon }\left({\left(t_{\epsilon }^i\right)}^{-}{w}_{\epsilon }^i\right)\in {\mathbf{K}}_{{\rho }_0/2}\text{for}\text{any}\epsilon <{\epsilon }^0\text{and}\text{each}1\le i\le k.$$

Lemma 4.4 There exists a number $\bar{\delta }>0$ such that if u ∈ N _ε and $I_{\epsilon }\left(u\right)\le {\gamma }_{\text{max}}+\bar{\delta }$, 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 } ⊂ ℝ⁺ 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 ∈ ℕ. It is easy to check that {u _n } is bounded in H¹ (ℝ^N). Suppose u _n → 0 strongly in L^p(ℝ^N). Since

$${∥u_n∥}_H^2=\underset{{ℝ}^N}{\int }f\left({\epsilon }_{n}z\right){\left(u_n\right)}_{+}^{p}dz\text{for}\text{each}n\in \mathbb{N},$$

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 L^p(ℝ^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 $\left\{\widetilde{z_n}\right\}\subset {ℝ}^N$ such that

$$\underset{B^N\left(\widetilde{z_n};1\right)}{\int }{\left|u_n\left(z\right)\right|}^{2}dz\ge d_0>0.$$

(4.2)

Let $v_n\left(z\right)=u_n\left(z+\widetilde{z_n}\right)$, there are a subsequence {v _n } and v ∈ H¹ (ℝ^N) such that v _n ⇀ v weakly in H¹ (ℝ^N). Using the similar computation in Lemma 2.6, there is a sequence $\left\{s_{\text{max}}^n\right\}\subset {ℝ}^{+}$ such that $\widetilde{v_n}=s_{\text{max}}^n{v}_n\in {\mathbf{N}}_{\text{max}}$ and

$$\begin{array}{ll}\hfill 0& <{\gamma }_{\text{max}}\le I_{\text{max}}\left(\widetilde{v_n}\right)\le I_{{\epsilon }_n}\left(s_{\text{max}}^n{u}_n\right)\\ \le I_{{\epsilon }_n}\left(u_n\right)={\gamma }_{\text{max}}+o_n\left(1\right)\text{as}n\to \infty .\end{array}$$

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 $\left\{\widetilde{v_n}\right\}$ and $ṽ\in H^1\left({ℝ}^N\right)$ such that ${ṽ}_n\rightharpoonup ṽ\left(=s_{0}v\right)$ weakly in H¹ (ℝ^N). By (4.2), then $ṽ\ne 0$. Moreover, we can obtain that $\widetilde{v_n}\to ṽ$ strongly in H¹ (ℝ^N) and $I_{\text{max}}\left(ṽ\right)={\gamma }_{\text{max}}$. Now, we want to show that there exists a subsequence $\left\{z_n\right\}=\left\{{\epsilon }_n\widetilde{z_n}\right\}$ such that z _n → z₀ ∈ K.

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

$$\begin{array}{ll}\hfill {\gamma }_{\text{max}}& =I_{\text{max}}\left(ṽ\right)<I_{\infty }\left(ṽ\right)\\ \le \underset{n\to \infty }{\text{lim inf}}\left[\frac{1}{2}{∥\widetilde{v_n}∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }f\left({\epsilon }_{n}z+z_n\right){\left(\widetilde{v_n}\right)}_{+}^{p}dz\right]\\ =\underset{n\to \infty }{\text{lim inf}}\left[\frac{{\left(s_{\text{max}}^n\right)}^2}{2}{∥u_n∥}_H^2-\frac{{\left(s_{\text{max}}^n\right)}^p}{p}\underset{{ℝ}^N}{\int }f\left({\epsilon }_{n}z\right){\left(u_n\right)}_{+}^{p}dz\right]\\ =\underset{n\to \infty }{\text{lim inf}}{I}_{{\epsilon }_n}\left(s_{\text{max}}^n{u}_n\right)\le \underset{n\to \infty }{\text{lim inf}}{I}_{{\epsilon }_n}\left(u_n\right)={\gamma }_{\text{max}},\end{array}$$

which is a contradiction.

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

$$\begin{array}{ll}\hfill {\gamma }_{\text{max}}& =I_{\text{max}}\left(ṽ\right)<\frac{1}{2}{∥ṽ∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }f\left(z_0\right){\left(ṽ\right)}_{+}^{p}dz\\ \le \underset{n\to \infty }{\text{lim inf}}\left[\frac{1}{2}{∥\widetilde{v_n}∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }f\left({\epsilon }_{n}z+z_n\right){\left(\widetilde{v_n}\right)}_{+}^{p}dz\right]\\ ={\gamma }_{\text{max}},\end{array}$$

which is a contradiction. Since v _n → v ≠ 0 in H¹ (ℝ^N), we have that

$$\begin{array}{ll}\hfill Q_{{\epsilon }_n}\left(u_n\right)& =\frac{\underset{{ℝ}^N}{\int }\chi \left({\epsilon }_{n}z\right){\left|v_n\left(z-\widetilde{z_n}\right)\right|}^{p}dz}{\underset{{ℝ}^N}{\int }{\left|v_n\left(z-\widetilde{z_n}\right)\right|}^{p}dz}\\ =\frac{\underset{{ℝ}^N}{\int }\chi \left({\epsilon }_{n}z+{\epsilon }_n\widetilde{z_n}\right){\left|v_n\right|}^{p}dz}{\underset{{ℝ}^N}{\int }{\left|v_n\right|}^{p}dz}\to z_0\subset {\mathbf{K}}_{{\rho }_0/2}\text{as}n\to \infty ,\end{array}$$

which is a contradiction.

Hence, there exists a number $\bar{\delta }>0$ such that if u ∈ N _ε and $I_{\epsilon }\left(u\right)\le {\gamma }_{\text{max}}+\bar{\delta }$, then $Q_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_0/2}$ for any 0 < ε < ε⁰.

From (4.1), choosing $0<{\delta }_0<\bar{\delta }$ such that

$${\gamma }_{\text{max}}+{\delta }_0<{\gamma }_{\infty }-C_0{\Lambda }^{\frac{2}{2-q}}\text{for}\text{any}0<\epsilon <{\epsilon }^0.$$

(4.3)

For each 1 ≤ i ≤ k, define

$$\begin{array}{ll}\hfill O_{\epsilon }^i& =\left\{u\in {\mathbf{M}}_{\epsilon }^{-}|\left|Q_{\epsilon }\left(u\right)-a^i\right|<{\rho }_0\right\},\\ \hfill \partial O_{\epsilon }^i& =\left\{u\in {\mathbf{M}}_{\epsilon }^{-}|\left|Q_{\epsilon }\left(u\right)-a^i\right|={\rho }_0\right\},\end{array}$$

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

Lemma 4.5 If $u\in {\mathbf{M}}_{\epsilon }^{-}$ and J _ε (u) ≤ γ_max + δ₀/2, then there exists a number $0<\bar{\epsilon }<{\epsilon }^0$ such that $Q_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_0/2}$ for any $0<\epsilon <\bar{\epsilon }$.

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 $s_{\epsilon }^u<c$ 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

$${∥u_n∥}_{L^p}^p=o_n\left(1\right)\text{as}n\to \infty .$$

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, $s_{\epsilon }^u<c$ for some constant c > 0 (independent of u). Now, we get that

$$\begin{array}{ll}\hfill {\gamma }_{\text{max}}+{\delta }_0/2& \ge J_{\epsilon }\left(u\right)=\underset{t\ge 0}{\text{sup}}{J}_{\epsilon }\left(tu\right)\ge J_{\epsilon }\left(s_{\epsilon }^{u}u\right)\\ =\frac{1}{2}{∥s_{\epsilon }^{u}u∥}_H^2-\frac{1}{p}\underset{{ℝ}^N}{\int }f\left(\epsilon z\right){\left(s_{\epsilon }^{u}u\right)}_{+}^{p}dz-\frac{1}{q}\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(s_{\epsilon }^{u}u\right)}_{+}^{q}dz\\ \ge I_{\epsilon }\left(s_{\epsilon }^{u}u\right)-\frac{1}{q}\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(s_{\epsilon }^{u}u\right)}_{+}^{q}dz.\end{array}$$

From the above inequality, we deduce that

$$\begin{array}{ll}\hfill I_{\epsilon }\left(s_{\epsilon }^{u}u\right)& \le {\gamma }_{\text{max}}+{\delta }_0/2+\frac{1}{q}\underset{{ℝ}^N}{\int }\Lambda h\left(\epsilon z\right){\left(s_{\epsilon }^{u}u\right)}_{+}^{q}dz\\ \le {\gamma }_{\text{max}}+{\delta }_0/2+\Lambda {∥h∥}_{\text{\# }}{S}^q{∥s_{\epsilon }^{u}u∥}_H^q\\ <{\gamma }_{\text{max}}+{\delta }_0/2+\Lambda c^q{\left(c_1\right)}^{q/2}{∥h∥}_{\text{\# }}{S}^q,\text{where}\Lambda ={\epsilon }^{2\left(p-q\right)/\left(p-2\right)}.\end{array}$$

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

$$I_{\epsilon }\left(s_{\epsilon }^{u}u\right)\le {\gamma }_{\text{max}}+{\delta }_0,\text{where}{s}_{\epsilon }^{u}u\in {\mathbf{N}}_{\epsilon }.$$

By Lemma 4.4, we obtain

$$Q_{\epsilon }\left(s_{\epsilon }^{u}u\right)=\frac{\underset{{ℝ}^N}{\int }\chi \left(\epsilon z\right){\left|s_{\epsilon }^{u}u\left(z\right)\right|}^{p}dz}{\underset{{ℝ}^N}{\int }{\left|s_{\epsilon }^{u}u\left(z\right)\right|}^{p}dz}\in {\mathbf{K}}_{{\rho }_0/2}\text{for}\text{any}0<\epsilon <\bar{\epsilon },$$

or $Q_{\epsilon }\left(u\right)\in {\mathbf{K}}_{{\rho }_0/2}$ for any $0<\epsilon <\bar{\epsilon }$.

Applying the above lemma, we get that

$${\tilde{\beta }}_{\epsilon }^i\ge {\gamma }_{\text{max}}+{\delta }_0/2\text{for}\text{any}0<\epsilon <\bar{\epsilon }.$$

(4.4)

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

$${\beta }_{\epsilon }^i\le J_{\epsilon }\left({\left(t_{\epsilon }^i\right)}^{-}{w}_{\epsilon }^i\right)\le {\gamma }_{\text{max}}+{\delta }_0/3<{\gamma }_{\infty }-C_0{\Lambda }^{\frac{2}{2-q}}\text{for}\text{any}0<\epsilon <{\epsilon }^{*}.$$

(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(ℝ^N) → ℝ⁺ such that $l\left(0\right)=1,l\left(v\right)\left(u-v\right)\in O_{\epsilon }^i$ for any v ∈ B(0;η) and

$$⟨l^{\prime }\left(v\right),\varphi ⟩{|}_{\left(l,v\right)=\left(1,0\right)}=\frac{⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),\varphi ⟩}{⟨{{\psi }^{\prime }}_{\epsilon }\left(u\right),u⟩}forany\varphi \in C_c^{\infty }\left({ℝ}^N\right),$$

(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 ≤ i ≤ k, there is a ${\left(PS\right)}_{{\beta }_{\epsilon }^i}$-sequence $\left\{u_n\right\}\subset O_{\epsilon }^i$ in H¹(ℝ^N) for J _ε .

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

$${\beta }_{\epsilon }^i<{\tilde{\beta }}_{\epsilon }^i\text{for}\text{any}0<\epsilon <{\epsilon }^{*}.$$

(4.7)

Then

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

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

$$J_{\epsilon }\left(u_n^i\right)\le J_{\epsilon }\left(w\right)+{∥w-u_n^i∥}_H/n\text{for}\text{all}w\in O_{\epsilon }^i\cup \partial O_{\epsilon }^i.$$

(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

$$\begin{array}{ll}\hfill \frac{{∥w_{\sigma }-u_n^i∥}_H}{n}& \ge J_{\epsilon }\left(u_n^i\right)-J_{\epsilon }\left(w_{\sigma }\right)\\ =⟨{J^{\prime }}_{\epsilon }\left(t_0{u}_n^i+\left(1-t_0\right)w_{\sigma }\right),u_n^i-w_{\sigma }⟩\text{where}{t}_0\in \left(0,1\right)\\ =⟨{J^{\prime }}_{\epsilon }\left(u_n^i\right),u_n^i-w_{\sigma }⟩+o\left({∥u_n^i-w_{\sigma }∥}_H\right)\left(\because J_{\epsilon }\in C^1\right)\\ =\sigma l_n^i\left(v_{\sigma }\right)⟨{J^{\prime }}_{\epsilon }\left(u_n^i\right),v⟩+\left(1-l_n^i\left(v_{\sigma }\right)\right)⟨{J^{\prime }}_{\epsilon }\left(u_n^i\right),u_n^i⟩+o\left({∥u_n^i-w_{\sigma }∥}_H\right)\\ (\because l_n^i\end{array}$$