The Nehari manifold approach for multiplicity of positive solutions to semilinear elliptic system involving multi-singular inverse square potentials with Sobolev critical exponent

Abstract

Using the Nehari manifold and variational methods, the existence and multiplicity of positive solutions for the multi-singular semilinear elliptic system with critical growth terms in bounded domains are investigated. In addition, under appropriate assumptions, it is shown that the system has at least two positive solutions when the pair of the parameters $(\alpha ,\beta )$ belongs to a certain subset of $\mathbb {R}^{2}$.

Introduction

In this paper, we consider the following semilinear elliptic system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\triangle z-\sigma \frac{z}{|x|^{2}}=\frac{1}{2^{*}}F_{z}(x,z,y)+ \alpha g(x)|z|^{q-2}z&{} \quad \text {in} \, \Omega ,\\ -\triangle y-\sigma \frac{y}{|x|^{2}}=\frac{1}{2^{*}}F_{y}(x,z,y)+ \beta h(x)|y|^{q-2}y&{} \quad \text {in} \, \Omega ,\\ z>0, y>0 &{} \quad \text {in} \,\Omega ,\\ z=y=0 &{} \quad \text {on} \, \partial \Omega . \end{array}\right. } \end{aligned}$$

(1)

where $0\in \Omega \subset \mathbb {R}^{N}(N\ge 3)$ is a bounded domain with smooth boundary $\partial \Omega$, such that $0\le \sigma <\overline{\sigma }:=(\frac{N-2}{2})^{2},$ and $F\in C^{1}(\overline{\Omega }\times (\mathbb {R^{+}})^{2},\mathbb {R^{+}})$ is positively homogeneous of degree $2^{*}$ ($2^{*}:=\frac{2N}{N-2}$ denotes the critical Sobolev exponent), that is, $F(x,tz,ty)=t^{2^{*}}F(x,z,y)(t>0)$ holds for all $(x,z,y)\in \overline{\Omega }\times (\mathbb {R^{+}})^{2},$ with $1\le q<2,\alpha>0 ,\sigma >0$.

We use $H^1_0(\Omega )$ to denote the completion of $C_0^{\infty }(\Omega )$ with respect to the norm:

$$\begin{aligned} ||z||_{H^1_0(\Omega )}=\left( \int _{\Omega }\left( |\nabla z|^2-\sigma \frac{ z^2}{|x|^2}\right) \mathrm{d}x\right) ^{\frac{1}{2}}. \end{aligned}$$

In the Banach space $Q =H^{1}_0(\Omega )\times H^{1}_0(\Omega )$, we introduce the norm:

$$\begin{aligned} ||(z,y)||_{Q}=\left( \int _{\Omega }\left( |\nabla z|^{2}+|\nabla y|^{2}-\sigma \frac{z^{2}+y^{2}}{|x|^{2}}\right) \mathrm{d}x\right) ^{\frac{1}{2}}. \end{aligned}$$

We set $\omega _0=(\frac{q}{2})\omega ,$ where

$$\begin{aligned} \omega =\left( \frac{2-q}{2^*-q}\right) ^{\frac{2}{2^*-2}}\left( \frac{2^*-q}{2^*-2} |\Omega |^{\frac{2^*-q}{2^*}}\right) ^{-\frac{2}{2-q}}S_\sigma ^{\frac{q}{2-q}}I_{F}^{\frac{N}{2}}, \end{aligned}$$

$$\begin{aligned} \Theta _{\xi }=\{(\alpha ,\beta )\in \mathbb {R}^+\times \mathbb {R}^+ | 0< (\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\xi \}. \end{aligned}$$

fore starting our result, we need the following assumptions:

$(F_1)\quad F:\overline{\Omega }\times \mathbb {R}^+ \rightarrow \mathbb {R}^+$ is a $C^1$ function, such that $F(x,tz,ty)=t^{{2^*}}F(x,z,y),$ for $t>0$, $\forall x\in \overline{\Omega },(z,y)\in (\mathbb {R}^+)^2$;

$(F_2)\quad F(x,z,0)=F(x,0,y)=\frac{\partial F}{\partial y}(x,0,y)=\frac{\partial F}{\partial z}(x,z,0),$ where $z,y\in \mathbb {R}^+$;

$(F_3)\quad \frac{\partial F}{\partial z}(x,z,y),\, \frac{\partial F}{\partial y}(x,z,y),$ are strictly increasing functions about z and y for all $z>0,y>0$.

Remark 1.1

We point out some important properties of homogeneous functions. Let $\kappa \ge 1$ and H be a differentiable $\kappa$-homogeneous function, then

• (i) for all $m,n\in \mathbb {R},\,mH_m(m,n)+nH_n(m,n)=\kappa H(m,n);$

• (ii) there exists $M_H>0$, such that $|H(m,n)|\le M_H(|m|^{\kappa }+|n|^{\kappa }),$ for all $m,n\in \mathbb {R},$ where $M_H=\max \{H(m,n): m,n\in \mathbb {R},|m|^{\kappa }+|n|^{\kappa }=1\}$;

• (iii) the maximum $M_H$ is attained for some $(m_0,n_0)\in \mathbb {R}^2$, i.e., $|m_0|^{\kappa }+|n_0|^{\kappa }=1$ and $H(m_0,n_0)=M_H$;

• (iv) $H_m(m,n),H_n(m,n)$ are $(\kappa -1)$ homogeneous.

By $(F_1)$ and the properties of homogeneous functions, we have

$$\begin{aligned} F_z(z,y)z+F_y(z,y)y=2^*F(z,y), \end{aligned}$$

and

$$\begin{aligned} F(z,y)\le (M_F(|z|^2+|y|^2))^{\frac{2^*}{2}}, \end{aligned}$$

(2)

where

$$\begin{aligned} M_F=\max \{(F(z,y))^{\frac{2^*}{2}}: (z,y)\in \mathbb {R}^2,|z|^2+|y|^2=1\}. \end{aligned}$$

Define

$$\begin{aligned} I_{F}:=\inf _{(z,y)\in Q}\left\{ \frac{||(z,y)||^{2}}{\left( \int _{\Omega }F(x,z,y)\mathrm{d}x\right) \frac{2}{2^{*}}} : \int _{\Omega }F(x,z,y)\mathrm{d}x>0\right\} , \end{aligned}$$

(3)

then we have [1]

$$\begin{aligned} I_{F}=M^{-1}_FS_\sigma . \end{aligned}$$

(4)

The main result of this paper is the following theorems.

Theorem 1.2

Assume that parameters $(\alpha ,\beta )\in \Theta _{\omega }$ and $(F_{1})-(F_{3})$ hold. Then, problem (1) has at lest one positive solutions.

Theorem 1.3

Assume that parameters $(\alpha ,\beta )\in \Theta _{\omega ^{*}_{0}}$ and $(F_{1})-(F_{3})$ hold. Then, problem (1) has at lest two positive solutions.

This paper is organized as follows. In Sect. 2, we consider some properties of Nehari manifold. In Sect. 3, we show that $(\mathrm{PS})_c$ condition holds for $E_{(\alpha ,\beta )}$ with c in certain interval, and then, we prove Theorems 1.2 and 1.3.

Nehari manifold

Definition 2.1

A pair of functions $(z,y)\in Q$ is weak solution to (1), if for all $(\phi ,\varphi )\in Q$:

$$\begin{aligned}& \int _{\Omega }(\nabla z\nabla \phi +\nabla y\nabla \varphi )\mathrm{d}x -\int _{\Omega }\frac{\sigma }{|x|^{2}}(z\phi + y\varphi ) \mathrm{d}x -\frac{1}{2^{*}}\int _{\Omega }\frac{\partial F(x,z,y)}{\partial z}\phi \mathrm{d}x\\&\quad -\frac{1}{2^{*}}\int _{\Omega }\frac{\partial F(x,z,y)}{\partial y}\varphi \mathrm{d}x-\alpha \int _{\Omega }g(x)|z|^{q-2}\phi \mathrm{d}x\\ & \quad-\beta \int _{\Omega }h(x)|y|^{q-2}y\varphi \mathrm{d}x=0. \end{aligned}$$

Thus, the corresponding energy functional of problem (1) is defined by

$$\begin{aligned} E_{\alpha ,\beta }(z,y)=\frac{1}{2}||(z,y)||^{2}_{Q}-\frac{1}{2^{*}}\int _{\Omega } F(x,z,y)\mathrm{d}x-\frac{1}{q}D_{\alpha ,\beta }(z,y), \end{aligned}$$

for $(z,y)\in Q$, where $D_{\alpha ,\beta }(z,y)=\alpha \int _{\Omega }g(x)|z|^{q}\mathrm{d}x+\beta \int _{\Omega }h(x)|y|^{q}\mathrm{d}x.$

Throughout of this paper for $0\le \sigma <\overline{\sigma }$, we suppose that $S_{\mu }$ is the best Sobolev embedding constant [5, 6], where $S_{\sigma }$ is independent of $\Omega$ [9].

It is well known that the weak solutions of (1) are the critical points of the energy functional $E_{\alpha ,\beta }$. If $E_{\alpha ,\beta }$ is bounded below and has a minimizer on Q, then this minimizer is a critical point of $E_{\alpha ,\beta }$. Therefore, it is a solution of the corresponding elliptic system. However, the energy function, $E_{\alpha ,\beta }$, is not bounded below on the whole space Q, but is bounded on an appropriate subset, that is called Nehari manifold. Then, we have

$$\begin{aligned} \mathcal {N_{\alpha ,\beta }}=\{(z,y)\in Q : \langle E'_{\alpha ,\beta }(z,y),(z,y)\rangle =0\}. \end{aligned}$$

Thus, $(z,y)\in \mathcal {N_{\alpha ,\beta }}$ if and only if

$$\begin{aligned} \langle E'_{\lambda ,\beta }(z,y),(z,y)\rangle =||(z,y)||^{2}_{Q}-\int _{\Omega } F(x,z,y)\mathrm{d}x-D_{\alpha ,\beta }(z,y)=0. \end{aligned}$$

(5)

Note that $\mathcal {N_{\alpha ,\beta }}$ contains every non-zero solution of Eq. (1). We now recall the following Lemma.

Lemma 2.2

[2, Lemma 2.1] The energy functional $E_{\alpha ,\beta }$ is coercive and bounded below on $\mathcal {N_{\alpha ,\beta }}$.

Now, we split $\mathcal {N}_{\alpha ,\beta }$ into three parts:

$$\begin{aligned} \mathcal {N}^{+}_{\alpha ,\beta }=\{(z,y)\in \mathcal {N_{\alpha ,\beta }} : \langle \psi '_{\alpha ,\beta }(z,y),(z,y)\rangle >0\} \end{aligned}$$

$$\begin{aligned} \mathcal {N}^{0}_{\alpha ,\beta }=\{(z,y)\in \mathcal {N_{\alpha ,\beta }} : \langle \psi '_{\alpha ,\beta }(z,y),(z,y)\rangle =0\} \end{aligned}$$

$$\begin{aligned} \mathcal {N}^{-}_{\alpha ,\beta }=\{(z,y)\in \mathcal {N_{\alpha ,\beta }} : \langle \psi '_{\alpha ,\beta }(z,y),(z,y)\rangle <0\}. \end{aligned}$$

By an argument similar to that [4, Theorem 2.3], we have the following lemma.

Lemma 2.3

Assume that $(z_{0},y_{0})$ is local minimizer for $E_{\alpha ,\beta }$ on $\mathcal {N_{\alpha ,\beta }}$ and $(z_{0},y_{0}) \notin \mathcal {N}^{0}_{\alpha ,\beta }$. Then, $E'_{\alpha ,\beta }(z_{0},y_{0})=0$ in $Q^{-1}$.

Lemma 2.4

There exists a positive number $\omega =\omega (q,N,M_F,S,|\Omega |)$ such that if $(\alpha ,\beta )\in \Theta _{\omega },$ then $\mathcal {N}^{0}_{\alpha ,\beta }=\emptyset$.

Using Lemma 2.4, we have $\mathcal {N}_{\alpha ,\beta }=\mathcal {N}^{+}_{\alpha ,\beta }\cup \mathcal {N}^{-}_{\alpha ,\beta }$. Define

$$\begin{aligned} \vartheta _{\alpha ,\beta }=\inf _{(z,y)\in \mathcal {N}_{\alpha ,\beta }}E_{\alpha ,\beta }(z,y) ;\ \vartheta ^{+}_{\alpha ,\beta }=\inf _{(z,y)\in \mathcal {N}^{+}_{\alpha ,\beta }}E_{\alpha ,\beta }(z,y) ;\ \vartheta ^{-}_{\alpha ,\beta }=\inf _{(z,y)\in \mathcal {N}^{-}_{\alpha ,\beta }}E_{\alpha ,\beta }(z,y). \end{aligned}$$

Now, we recall the following results.

Proposition 2.5

[2, Theorem 2.5]

• (i) If $(\alpha ,\beta )\in \Theta _{\omega },$ then $\vartheta _{\alpha ,\beta }\le \vartheta ^{+}_{\alpha ,\beta }<0$;

• (ii) If $(\alpha ,\beta )\in \Theta _{\omega _0}$, then $\vartheta ^{-}_{\alpha ,\beta }>c_{0}$ for some $c_{0}=c_{0}(\alpha ,\beta ,q,N,S,|\Omega |)>0.$

Lemma 2.6

[2, Lemma 2.6] If $\int _{\Omega }F(x,z,y)\mathrm{d}x>0$, then there are unique $t^{+}$ and $t^{-}$ with $0<t^{+}<t_{\max }<t^{-}$, such that $(t^{+}z,t^{+}y)\in \mathcal {N}^{+}_{\alpha ,\beta },(t^{-}z,t^{-}y)\in \mathcal {N}^{-}_{\alpha ,\beta }\,$(for each $(z,y)\in Q$) and

$$\begin{aligned} E_{\alpha ,\beta }(t^{+}z,t^{+}y)=\inf _{0\le t\le t_{\max }}E_{\alpha ,\beta }(tz,ty) ,\quad E_{\alpha ,\beta }(t^{-}z,t^{-}y)=\sup _{t\ge 0}E_{\alpha ,\beta }(tz,ty). \end{aligned}$$

Proof of Theorems 1.2 and 1.3

This section is devoted to providing of the main results of this paper. In the first, we recall some notions and results. For details, see [7, 8, 11, 13, 14].

Definition 3.1

• (i) For $c\in \mathbb {R}$, a sequence $\{(z_n,y_n)\}$ is a $(\mathrm{PS})_c$ sequence in Q for $E_{\alpha ,\beta }$ if $E_{\alpha ,\beta }(z_n,y_n)=c+o_n(1),$ and $E'_{\alpha ,\beta }(z_n,y_n)=o_n(1)$ strongly in Q,  as $n\rightarrow \infty$.

• (ii) $c\in \mathbb {R}$ is a $(\mathrm{PS})_c$ value in Q for $E_{\alpha ,\beta }$ if there exists a $(\mathrm{PS})_c$ sequence in Q for $E_{\alpha ,\beta }$.

• (iii) $E_{\alpha ,\beta }$ satisfies the $(\mathrm{PS})_c$ condition in Q if any $(\mathrm{PS})_c$ sequence $\{(z_n,y_n)\}$ in Q for $E_{\alpha ,\beta }$ contains a convergent subsequence.

We also need the following theorem due to Tsung-Fang Wu [15].

Lemma 3.2

[15, Proposition 9]

• (i) If $(\alpha ,\beta )\in \Theta _{\omega }$, then there exists a $(\mathrm{PS})_{\vartheta _{\alpha ,\beta }}$ sequence $\{(z_{n},y_{n})\}\subset \mathcal {N}_{\alpha ,\beta }$ in Q for $E_{\alpha ,\beta }$.

• (ii) If $(\alpha ,\beta )\in \Theta _{\omega _0}$, then there exists a $(\mathrm{PS})_{\vartheta ^{-}_{\alpha ,\beta }}$ sequence $\{(z_{n},y_{n})\}\subset \mathcal {N}^{-}_{\alpha ,\beta }$ in Q for $E_{\alpha ,\beta }.$

In the following theorem, we express the existence of a local minimum for $E_{\alpha ,\beta }$ on $\mathcal {N}^{+}_{\alpha ,\beta }.$

Theorem 3.3

Assume that $(\alpha ,\beta )\in \Theta _{\omega },$ and $(F_{1})- (F_{3})$ hold. Then, $E_{\alpha ,\beta }$ has a minimizer $\{(z^{+}_{0},y^{+}_{0})\}\subset \mathcal {N}^{+}_{\alpha ,\beta }$ and

• (i) $E_{\alpha ,\beta }(z^{+}_{0},y^{+}_{0})=\vartheta _{\alpha ,\beta }=\vartheta ^{+}_{\alpha ,\beta }$,

• (ii) $(z^{+}_{0},y^{+}_{0})$ is a positive solution of problem (1).

Proof

Using Lemma 3.2(i), there exists a minimizing sequence $\{(z_n,y_n)\}$ for $E_{\alpha ,\beta }$ on $\mathcal {N}_{\alpha ,\beta }$, such that

$$\begin{aligned} E_{\alpha ,\beta }(z_n,y_n)=\vartheta _{\alpha ,\beta }+o(1)\quad {\text {and}} \quad E'_{\alpha ,\beta }(z_n,y_n)=o(1)\,\,\,(\text {in}\, Q^{-1}). \end{aligned}$$

(6)

Then, by Lemma 2.2 and the compact imbedding theorem, there exist a subsequence $\{(z_n,y_n)\}$ and $\{(z_0^+,y_0^+)\}\in Q$, such that

$$\begin{aligned} {\left\{ \begin{array}{ll} z_n\rightharpoonup z_0^+\quad \text {weakly\, in} \,\, Q_0^{1,p},\\ z_n\rightarrow z_0^+\quad \text {strongly\, in}\,\, L^q(\Omega ),\\ y_n\rightharpoonup y_0^+\quad \text {weakly\, in} \,\, Q_0^{1,p},\\ y_n\rightarrow y_0^+\quad \text {strongly\, in} \,\, L^q(\Omega ).\\ \end{array}\right. } \end{aligned}$$

(7)

This implies that $D_{\alpha ,\beta }(z_n,y_n)\rightarrow D_{\alpha ,\beta }(z_0^+,y_0^+)$ as $n\rightarrow \infty$.

Using (6) and (7), it is not hard to see that $(z_0^+,y_0^+)$ is a weak solution of (1).

Since

$$\begin{aligned} E_{\alpha ,\beta }(z_n,y_n)&=\frac{1}{N}||(z_n,y_n)||^2-\frac{2^*-q}{2^*q}D_{\alpha ,\beta } (z_n,y_n)\\&\ge -\frac{2^*-q}{2^*q}D_{\alpha ,\beta }(z_n,y_n), \end{aligned}$$

and using Lemma 2.5(i)

$$\begin{aligned} E_{\alpha ,\beta }(z_n,y_n)\rightarrow \vartheta ^{+}_{\alpha ,\beta }<0\quad {\text {as}} \quad n\rightarrow \infty . \end{aligned}$$

Letting $n\rightarrow \infty$, we see that $D_{\alpha ,\beta }(z_0^+,y_0^+)>0$. Thus, $(z_0^+,y_0^+)$ is a nontrivial solution of problem (1).

Now, it follows that $z_n\rightarrow z_0^+$ strongly in $H_0^1(\Omega )$, $y_n\rightarrow y_0^+$ strongly in $H_0^1(\Omega )$ and $E_{\alpha ,\beta }(z_0^+,y_0^+)= \vartheta _{\alpha ,\beta }$. As $(z_n,y_n)\in \mathcal {N}_{\alpha ,\beta }$ and applying Fatou’s lemma, we have

$$\begin{aligned} \vartheta _{\alpha ,\beta }&\le E_{\alpha ,\beta }(z_0^+,y_0^+)=\frac{1}{N}||(z_0^+,y_0^+)||^2 -\frac{2^*-q}{2^*q}D_{\alpha ,\beta }(z_0^+,y_0^+)\\&\le \liminf _{n\rightarrow \infty }\left( \frac{1}{N}||(z_n,y_n)||^2-\frac{2^*-q}{2^*q} D_{\alpha ,\beta }(z_n,y_n)\right) \\&\le \liminf _{n\rightarrow \infty }E_{\alpha ,\beta }(z_n,y_n)=\vartheta _{\alpha ,\beta }. \end{aligned}$$

This implies that

$$\begin{aligned} E_{\alpha ,\beta }(z_0^+,y_0^+)=\vartheta _{\alpha ,\beta }\quad {\text {and}} \quad \lim _{n\rightarrow \infty }||(z_n,y_n)||^2=||(z_0^+,y_0^+)||^2. \end{aligned}$$

Let $(\overline{z}_n,\overline{y}_n)=(z_n,y_n)-(z_0^+,y_0^+)$, then Br$\acute{\mathrm{e}}$zis–Lieb lemma [3], implies

$$\begin{aligned} ||(\overline{z}_n,\overline{y}_n)||^2=||(z_n,y_n)||^2-||(z_0^+,y_0^+)||^2. \end{aligned}$$

Therefore, $z_n\rightarrow z_0^+$ strongly in $H_0^1(\Omega )$, $y_n\rightarrow y_0^+$ strongly in $H_0^1(\Omega )$. Moreover, we have $(z_0^+,y_0^+)\in \mathcal {N}_{\alpha ,\beta }^+$. In fact, if $(z_0^+,y_0^+)\in \mathcal {N}_{\alpha ,\beta }^-$, by Lemma 2.6, there are unique $t_0^+$ and $t_0^-$, such that $(t_0^+z_0^+,t_0^+y_0^+)\in \mathcal {N}_{\alpha ,\beta }^+$ and $(t_0^-z_0^+,t_0^-y_0^+)\in \mathcal {N}_{\alpha ,\beta }^-$. In particular, we have $t_0^+<t_0^-=1$. Since

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}E_{\alpha ,\beta }(t_0^+z_0^+,t_0^+y_0^+)=0\quad {\text {and}} \quad \frac{\mathrm{d}^2}{\mathrm{d}t^2}E_{\alpha ,\beta }(t_0^+z_0^+,t_0^+y_0^+)>0, \end{aligned}$$

there exists $t_0^+<\overline{t}\le t_0^-$, such that $E_{\alpha ,\beta }(t_0^+z_0^+, t_0^+y_0^+)<E_{\alpha ,\beta }(\overline{t}z_0^+,\overline{t}y_0^+)$. By Lemma 2.6

$$\begin{aligned} E_{\alpha ,\beta }(t_0^+z_0^+,t_0^+y_0^+)<E_{\alpha ,\beta }(\overline{t}z_0^+,\overline{t}y_0^+) \le E_{\alpha ,\beta }(t_0^-z_0^+,t_0^-y_0^+)=E_{\alpha ,\beta }(z_0^+,y_0^+), \end{aligned}$$

which is a contradiction.

It follows from the maximum principle that $(z_0^+,y_0^+)$ is a positive solution of problem (1). $\square$

Lemma 3.4

[12, Lemma 2.2] Let $\{(z_{n},y_{n})\}\subset Q$ be a $(\mathrm{PS})_c$ sequence for $E_{\alpha ,\beta }$ with $(z_{n},y_{n})\rightharpoonup (z,y)$ in Q, then $E'_{\alpha ,\beta }(z,y)=0$, and there exists a constant $d_{0}=d_0(q,N,S,|\Omega |)$, such that $E_{\alpha ,\beta }(z,y)\ge -d_{0}((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}})$.

Lemma 3.5

[12, Lemma 4.1] Let $\{(z_{n},y_{n})\}\subset Q$ be a $(\mathrm{PS})_{c}$ sequence for $E_{\alpha ,\beta }$, then $(z_{n},y_{n})$ is bounded in Q.

Lemma 3.6

Assume that $(F_1)-(F_3)$ hold. Then, $E_{\alpha ,\beta }$ satisfies the $(\mathrm{PS})_{c}$ condition with c satisfying

$$\begin{aligned} -\infty<c<c_{\infty }=\frac{1}{N}S_{F}^{\frac{N}{2}}-d_0((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+ (\beta |h^+|_{\infty })^{\frac{2}{2-q}}), \end{aligned}$$

where $d_0$ is positive constant given in Lemma 3.4.

Proof

Let $\{(z_{n},y_{n})\}\subset Q$ be a $(\mathrm{PS})_c$ sequence for $E_{\alpha ,\beta }$ with $c\in (-\infty ,c_\infty )$. It follows from Lemma 3.5 that $\{(z_{n},y_{n})\}$ is bounded in Q, and then $(z_{n},y_{n})\rightharpoonup (z,y)$ up to a subsequence, (z, y) is a critical point of $E_{\alpha ,\beta }$. Furthermore, we may assume

$$\begin{aligned} {\left\{ \begin{array}{ll} z_n\rightharpoonup z,\quad y_n\rightharpoonup y\quad {\text {in }}\,\, H_0^1(\Omega ), \\ z_n\rightarrow z,\quad y_n\rightarrow y\quad {\text {in }}\,\, L^s(\Omega ),\quad 1\le s<2^* \\ z_n\rightarrow z,\quad y_n\rightarrow y\quad \text {a.e. on }\,\,\Omega .\\ \end{array}\right. } \end{aligned}$$

(8)

Hence, $E_{\alpha ,\beta }'(z,y)=0,$ and

$$\begin{aligned} D_{\alpha ,\beta }(z_n,y_n)= D_{\alpha ,\beta }(z,y)+o_n(1). \end{aligned}$$

(9)

Let $\overline{z}_n=z_n-z,\,\overline{y}_n=y_n-y$. Using Br$\acute{\mathrm{e}}$zie–Lieb lemma [3], we obtain

$$\begin{aligned} ||(\overline{z}_n,\overline{y}_n)||^2=||(z_n,y_n)||^2-||(z,y)||^2+o_n(1). \end{aligned}$$

(10)

In addition, using an argument similar to that [10, Lemma 8]

$$\begin{aligned} \int _{\Omega }F(x,\overline{z}_n,\overline{y}_n)\mathrm{d}x=\int _{\Omega }F(x,z_n,y_n)\mathrm{d}x- \int _{\Omega }F(x,z,y)\mathrm{d}x+o_n(1). \end{aligned}$$

(11)

Since $E_{\alpha ,\beta }(z_n,y_n)=c+o(1),\,E_{\alpha ,\beta }'(z,y)=o_n(1)$ and (9)–(11), we can deduce that

$$\begin{aligned} \frac{1}{2}||(\overline{z}_n,\overline{y}_n)||^2-\frac{1}{2^*}\int _{\Omega } F(x,\overline{z}_n,\overline{y}_n)\mathrm{d}x=c-E_{\alpha ,\beta }(z,y)+o_n(1), \end{aligned}$$

(12)

and

$$\begin{aligned} ||(\overline{z}_n,\overline{y}_n)||^2-\int _{\Omega }F(x,\overline{z}_n,\overline{y}_n)\mathrm{d}x=o_n(1). \end{aligned}$$

(13)

Hence, we may assume that

$$\begin{aligned} ||(\overline{z}_n,\overline{y}_n)||^2\rightarrow l,\quad \int _{\Omega } F(x,\overline{z}_n,\overline{y}_n)\rightarrow l. \end{aligned}$$

(14)

If $l=0$, then the proof is complete. Assume that $l>0$, then from (14), we obtain

$$\begin{aligned} I_{F}l^{\frac{2}{2^*}}&=I_{F}\lim _{n\rightarrow \infty }\left( \int _{\Omega } F(x,\overline{z}_n,\overline{y}_n)\mathrm{d}x\right) ^{\frac{2}{2^*}}\\&\le \lim _{n\rightarrow \infty }||(\overline{z}_n,\overline{y}_n)||^2=l, \end{aligned}$$

which implies that $l\ge I_{F}^{\frac{N}{2}}$. In addition, from Lemma 3.4, relations (12) and (14), we get

$$\begin{aligned} c&=\left( \frac{1}{2}-\frac{1}{2^*}\right) l+E_{\alpha ,\beta }(z,y)\\&\ge \frac{1}{N}I_{F}^{\frac{N}{2}}-d_0((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}), \end{aligned}$$

which contradicts $c<\frac{1}{N}I_{F}^{\frac{N}{2}}-d_0((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}})$. $\square$

Lemma 3.7

Assume that $(F_1)-$ $(F_3)$ hold. Then, there exist a nonnegative function $(z,y)\in Q\backslash \{(0,0)\}$ and $\omega ^{*}>0$, such that for $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\omega ^{*}$, we have

$$\begin{aligned} \sup _{t\ge 0}E_{\alpha ,\beta }(tz,ty)<c_{\infty }. \end{aligned}$$

In particular, $\vartheta ^{-}_{\alpha ,\beta }<c_{\infty }$ for all $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\omega ^{*}.$

Proof

Since $0\in \Omega$, there is $\varrho _0>0$, such that $B_{2\varrho _0}(0)\subset \Omega$. Now, we define a cut-off function $\gamma (x)$ satisfying $\gamma (x)=1$ for $|x|\le \varrho _0,\gamma (x)=0$ for $|x|>2\varrho _0$, $0\le \gamma <1$ and $|\nabla \gamma |\le C$. For $\varepsilon >0$, let

$$\begin{aligned} z_{\varepsilon }(x)=\frac{\varepsilon ^{\frac{N-2}{4}}\gamma (x)}{( \varepsilon |x|^{\tau '/\sqrt{\overline{\sigma }}}+ |x|^{\tau _/\sqrt{\overline{\sigma }}})^{\sqrt{\overline{\sigma }}}} \end{aligned}$$

where $\overline{\sigma }=(\frac{N-2}{2})^{2},\, \tau '=\sqrt{\overline{\sigma }}- \sqrt{\overline{\sigma }-\sigma _{k}}$ and $\tau =\sqrt{\overline{\sigma }}+ \sqrt{\overline{\sigma }-\sigma _{k}}.$ Using the property (iii) of homogeneous functions, there exists $(e_1,e_2)\in \mathbb {R}^2$, such that $e_{1}^{2}+e_{2}^{2}=1$ and $F(x,e_1,e_2)=M_F^{\frac{2^*}{2}}$.

For $t\ge 0$, we consider $z_{0}=e_{1}z_{\varepsilon }, \,y_{0}= e_{2}z_{\varepsilon }$ and $(z_{0},y_{0})\in Q$. We now define the following functions:

$$\begin{aligned} G(t)&:=E_{\alpha ,\beta }(tz_{0},ty_{0})=E_{\alpha ,\beta }(te_{1}z_{\varepsilon },te_{2}z_{\varepsilon })\\&=\frac{t^{2}}{2}\int _{\Omega }\left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_{\varepsilon })^{2}}{|x|^{2}}\right) -\frac{t^{2^{*}}}{2^{*}} \int _{\Omega }F(x,e_{1}z_{\varepsilon },e_{2}z_{\varepsilon })\mathrm{d}x\\&-\frac{t^{q}}{q}(\alpha e_{1}^{q}+\beta e_{2}^{q})\int _{\Omega } |z_{\varepsilon }|^{q}, \end{aligned}$$

and

$$\begin{aligned} \overline{G}(t):=\frac{t^{2}}{2}\int _{\Omega }\left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_{\varepsilon })^{2}}{|x|^{2}}\right) -\frac{t^{2^{*}}}{2^{*}} \int _{\Omega }F(x,e_{1}z_{\varepsilon },e_{2}z_{\varepsilon })\mathrm{d}x \end{aligned}$$

Step 1 We need to show that

$$\begin{aligned} \sup _{t\ge 0}\overline{g}(t)\le c_{\infty }. \end{aligned}$$

To do this, we need to established the following estimates (as $\varepsilon \rightarrow 0$):

$$\begin{aligned}&\left( \int _{\Omega }|z_{\varepsilon }|^{2^{*}}\mathrm{d}x\right) ^{\frac{2}{2^{*}}} =\left( \int _{\Omega }|U|^{2^{*}}\mathrm{d}x\right) ^{\frac{2}{2^{*}}}+ o(\varepsilon ^{\frac{N}{2}}), \end{aligned}$$

(15)

$$\begin{aligned} \int _{\Omega }\left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_{\varepsilon })^{2}}{|x|^{2}} \right) \mathrm{d}x = \int _{\Omega }\left( |\nabla U|^{2}-\sigma \frac{U^{2}}{|x|^{2}} \right) \mathrm{d}x+o(\varepsilon ^{\frac{N-2}{2}}), \end{aligned}$$

(16)

$$\begin{aligned}&\frac{\int _{\Omega }\left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_{\varepsilon })^{2}}{|x|^{2}} \right) \mathrm{d}x}{\left( \int _{\Omega }|z_{\varepsilon }|^{2^{*}}\mathrm{d}x\right) ^{\frac{2}{2^{*}}}} =S_{\sigma }+o(\varepsilon ^{\frac{N-2}{2}}), \end{aligned}$$

(17)

where $U(x)=(|x|^{\tau '/\sqrt{\overline{\sigma }}}+ |x|^{\tau /\sqrt{\overline{\sigma }}})^{-\sqrt{\overline{\sigma }}},$ and $\omega _{N}=2\pi ^{N/2}/N\Gamma (\frac{N}{2})$ is the volume of the unit ball B(0, 1) in $\mathbb {R}^{N}$. We only show that the equality (15) is valid. By definition of $z_{\varepsilon }$, we have

$$\begin{aligned} \int _{\mathbb {R}^N}|z_{\varepsilon }|^{2^*}\mathrm{d}x=\int _{\mathbb {R}^N} \frac{\varepsilon ^{\frac{N}{2}}\gamma ^{2^*}(x)}{( \varepsilon |x|^{\tau '_{k}/\sqrt{\overline{\sigma }}}+ |x|^{\tau _{k}/\sqrt{\overline{\sigma }}})^{N}}\mathrm{d}x. \end{aligned}$$

(18)

On the other hand, let $x=\varepsilon ^{\frac{\sqrt{\overline{\sigma }}}{\tau _k-\tau '_k}}y$, we can deduce that

$$\begin{aligned} \left( \int _{\mathbb {R}^N}\frac{1}{( \varepsilon |x|^{\tau '_{k}/\sqrt{\overline{\sigma }}}+ |x|^{\tau _{k}/\sqrt{\overline{\sigma }}})^{N}}\mathrm{d}x \right)&=\varepsilon ^{-\frac{N}{2}}\int _{\mathbb {R}^N}\frac{1}{( |y|^{\tau '_{k}/\sqrt{\overline{\sigma }}}+ |y|^{\tau _{k}/\sqrt{\overline{\sigma }}})^{N}}\mathrm{d}y \nonumber \\&=\varepsilon ^{-\frac{N}{2}}|U|^{2^*}_{L^{2^*}(\mathbb {R}^N)}. \end{aligned}$$

(19)

We have

$$\begin{aligned} 0&\le \varepsilon ^{-\frac{N}{2}}|U|^{2^*}_{L^{2^*}(\mathbb {R}^N)}- \varepsilon ^{-\frac{N}{2}}|z_{\varepsilon }|^{2^*}_{2^*}\nonumber \\&=\int _{\mathbb {R}^N\setminus B(0,\varepsilon _0)}\frac{1}{(\varepsilon |x|^{\tau '_{k}/\sqrt{\overline{\sigma }}}+ |x|^{\tau /\sqrt{\overline{\sigma }}})^{N}}\mathrm{d}x\nonumber \\&\le \int _{\mathbb {R}^N\setminus B(0,\varepsilon _0)}\frac{\mathrm{d}x}{|x|^{2^*\tau }} \nonumber \\&=\frac{N-2}{2}\omega _{N}(\varepsilon _0)^{-\frac{2N}{N-2}\beta _k}\nonumber \\&\le d_1=\mathrm{Constant}. \end{aligned}$$

(20)

Hence

$$\begin{aligned} 0\le 1-|z_{\varepsilon }|^{2^*}_{L^{2^*}(\mathbb {R}^N)} |U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)}\le d_1\varepsilon ^{\frac{N}{2}} |U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)} \end{aligned}$$

(21)

that is

$$\begin{aligned} 1-d_1\varepsilon ^{\frac{N}{2}}|U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)} \le |z_{\varepsilon }|^{2^*}_{L^{2^*}(\mathbb {R}^N)} |U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)}\le 1, \end{aligned}$$

(22)

Now, let $\varepsilon$ be small enough, such that $d_1\varepsilon ^{\frac{N}{2}} |U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)}<1$, then from (21), we can deduce that

$$\begin{aligned} 1-d_1\varepsilon ^{\frac{N}{2}}|U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)}\le (1-d_1\varepsilon ^{\frac{N}{2}}|U|^{-2^*}_{L^{2^*}(\mathbb {R}^N)})^{\frac{2}{2^*}} \le |z_{\varepsilon }|^{2}_{L^{2^*}(\mathbb {R}^N)} |U|^{-2}_{L^{2^*}(\mathbb {R}^N)}\le 1, \end{aligned}$$

(23)

which yield that

$$\begin{aligned} |U|^{2}_{L^{2^*}(\mathbb {R}^N)}-d_1\varepsilon ^{\frac{N}{2}} |U|^{2-2^*}_{L^{2^*}(\mathbb {R}^N)} \le |z_{\varepsilon }|^{2}_{L^{2^*}(\mathbb {R}^N)} \le |U|^{2}_{L^{2^*}(\mathbb {R}^N)}, \end{aligned}$$

(24)

equivalently, the equality (15) is valid.

Combining (15) and (16), we obtain that

$$\begin{aligned} \left( \frac{||z_{\varepsilon }||^{2}_{\sigma }}{|z_{\varepsilon }|^{2} _{L^{2^*}(\mathbb {R}^N)}}-S_{\sigma }\right)&=\left[ \frac{||U||^2_{\sigma }+O\left( \varepsilon ^{ \frac{N-2}{2}}\right) }{|U|^2_{L^{2^*}(\mathbb {R}^N)}+O\left( \varepsilon ^{\frac{N}{2}}\right) }- \frac{||U||^{2}_{\sigma }}{|U|^{2}_{L^{2^*}(\mathbb {R}^N)}}\right] \nonumber \\&=\left[ \frac{|U|^{2}_{L^{2^*}(\mathbb {R}^N)}O\left( \varepsilon ^{\frac{N-2}{2}}\right) - ||U||^{2}_{\sigma }O\left( \varepsilon ^{\frac{N}{2}}\right) }{(|U|^2_{L^{2^*}(\mathbb {R}^N)}+O\left( \varepsilon ^{\frac{N}{2}}\right) ) |U|^{2}_{L^{2^*}(\mathbb {R}^N)}}\right] \nonumber \\&=O\left( \varepsilon ^{\frac{N-2}{2}}\right) . \end{aligned}$$

(25)

Step 2 Using the fact that

$$\begin{aligned} \max _{t\ge 0}\left( \frac{t^{2}}{2}S-\frac{t^{2^{*}}}{2^{*}}T\right) =\frac{1}{N}\left( \frac{S}{T^{\frac{2}{2^{*}}}}\right) ^{\frac{N}{2}}, \quad \, \text {for any}\, S,T>0, \end{aligned}$$

and using $(F_1)$, (3), (4), and (15)–(17), we conclude that

$$\begin{aligned} \sup _{t\ge 0}\overline{G}(t)&=\frac{1}{N}\left( \frac{(e^{2}_{1}+e^{2}_{2})\int _{\Omega } \left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_ {\varepsilon })^{2}}{|x|^2}\right) }{\left( \int _{\Omega }F(x,e_1z_{\varepsilon },e_2 z_{\varepsilon })\mathrm{d}x\right) ^\frac{2}{2^*}}\right) ^\frac{N}{2}\nonumber \\&=\frac{1}{N}\left( \frac{\int _{\Omega } \left( |\nabla z_{\varepsilon }|^{2}-\sigma \frac{(z_ {\varepsilon })^{2}}{|x|^2}\right) }{\left( M_F^{\frac{2^*}{2}}\int _{\Omega }|z_{\varepsilon }|^{2^*}\mathrm{d}x\right) ^\frac{2}{2^*}}\right) ^\frac{N}{2}\nonumber \\&\le \frac{1}{N}\left( \frac{1}{M_F}\right) ^{N/2}\left( S_{\sigma }+ o\left( \varepsilon ^{\frac{N-2}{2}}\right) \right) ^{N/2}\nonumber \\&=\frac{1}{N}S_{F}^{\frac{N}{2}}+o\left( \varepsilon ^{\frac{N-2}{2}}\right) . \end{aligned}$$

(26)

Let $d_0$ be the positive constant given in Lemma 3.4. We can choose $\varepsilon _1>0$, such that for all $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+ (\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\varepsilon _1$, then

$$\begin{aligned} c_\infty =\frac{1}{N}S_{F}^{\frac{2}{N}}-d_0((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+ (\beta |h^+|_{\infty })^{\frac{2}{2-q}}>0. \end{aligned}$$

Using the definitions $E_{\alpha ,\beta }$ and $(z_0,y_0)$, we obtain

$$\begin{aligned} E_{\alpha ,\beta }(tz_0,y_0)\le \frac{t^{2}}{2}||(z_0,y_0)||^2,\quad ( t\ge 0 , \alpha>0,\beta >0), \end{aligned}$$

which implies that there exists $t_0\in (0,1)$ satisfying

$$\begin{aligned} \sup _{0\le t\le t_0}E_{\alpha ,\beta }(tz_0,y_0)<c_\infty ,\quad \text {for\,\, all}\quad 0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+ (\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\varepsilon _1 \end{aligned}$$

Using the definitions of $E_{\alpha ,\beta },(z_0,y_0)$, and Step 1, we have

$$\begin{aligned} {\begin{matrix} \sup _{t\ge t_0}E_{\alpha ,\beta }(tz_0,ty_0)&{}=\sup _{t\ge t_0}\left( \overline{G}(t)-\frac{t^{q}}{q}D_{\alpha ,\beta }(z_0,y_0)\right) \\ &{}\le \frac{1}{N}I_{F}^{\frac{N}{2}}+O\left( \varepsilon ^{\frac{N-2}{2}}\right) -\frac{t_0^q}{q} ((e_1^q|g^+|_{\infty }\alpha )+(e_2^q|h^+|_{\infty }\beta )\int _{B_{\varepsilon _0}(0)}|z_{\varepsilon }|^q \mathrm{d}x\\ &{}\le \frac{1}{N}I_{F}^{\frac{N}{2}}+O\left( \varepsilon ^{\frac{N-2}{2}}\right) -\frac{t_0^q}{q}m(\alpha +\sigma ) \int _{B_{\varepsilon _0}(0)}|z_{\varepsilon }|^q \mathrm{d}x, \end{matrix}} \end{aligned}$$

(27)

where $m=\min \{e_1^q|g^+|_{\infty },e_2^q|h^+|_{\infty }\}$. Let $0<\varepsilon \le \varepsilon _0^{\frac{\tau -\tau '}{\sqrt{\overline{\sigma }}}}$, then

$$\begin{aligned} {\begin{matrix} \int _{B_{\varepsilon _0}(0)}|z_{\varepsilon }|^q \mathrm{d}x&{}=\int _{B_{\varepsilon _0}(0)} \frac{\varepsilon ^{\frac{q(N-2)}{4}}}{(\varepsilon |x|^{\tau '/\sqrt{\overline{\sigma }}}+ |x|^{\tau /\sqrt{\overline{\sigma }}})^{q\sqrt{\overline{\sigma }}}}\\ &{}\ge \int _{B_{\varepsilon _0}(0)}\frac{\varepsilon ^{\frac{q(N-2)}{4}}}{(2\varepsilon _0^{\frac{\tau }{\sqrt{\overline{\sigma }}}})^{q\sqrt{\overline{\sigma }}}}\\ &{}=d_1(N,q,\sigma ,\varepsilon _0)\varepsilon ^{\frac{q(N-2)}{4}}. \end{matrix}} \end{aligned}$$

(28)

Combining relations (21) and (28), for all $\varepsilon =((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty }) ^{\frac{2}{2-q}})^{\frac{2}{N-2}}\in (0,\varrho _0^ {\frac{\tau -\tau '}{\sqrt{\overline{\sigma }}}})$, we have

$$\begin{aligned} \sup _{t\ge t_0}E_{\alpha ,\beta }(tz_0,ty_0)\le \frac{1}{N}I_{F}^{\frac{N}{2}}+O(\alpha |g^+|_{\infty })^{\frac{2}{2-q}} +(\beta |h^+|_{\infty })^{\frac{2}{2-q}})-\frac{t_0^q}{q}md_1(\alpha +\sigma ). \end{aligned}$$

(29)

we can choose $\varepsilon _2>0$, such that for all $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}} +(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\varepsilon _2$, we have

$$\begin{aligned} O((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}})-\frac{t_0^q}{q}md_1(\alpha +\sigma )< -d_0((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}). \end{aligned}$$

If we set $\omega ^*=\min \{\varepsilon _1,\varepsilon _0^{N-2},\varepsilon _2\}$ and $\varepsilon = ((\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}})^{\frac{2}{N-2}},$ then for $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\omega ^*$:

$$\begin{aligned} \sup _{t\ge 0}E_{\alpha ,\beta }(tz_0,ty_0)<c_\infty . \end{aligned}$$

(30)

Finally, we prove that $\vartheta ^-_{\alpha ,\beta }<c_\infty$ for all $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\omega ^*$. Recall that $(z_0,y_0)=(e_1z_{\varepsilon },e_2z_{\varepsilon })$. It is easy to see that

$$\begin{aligned} \int _{\Omega }F(x,z_0,y_0)\mathrm{d}x>0,\quad D_{\alpha ,\beta }(z_0,y_0)>0. \end{aligned}$$

Using Lemma 2.6, and from the definition of $\vartheta ^-_{\alpha ,\beta }$ and (30), there exists $t_0>0$, such that $(t_0z_0,t_0y_0)\in \mathcal {N}^-_{ \alpha ,\beta }$ and

$$\begin{aligned} \vartheta ^-_{\alpha ,\beta }\le E_{\alpha ,\beta }(t_0z_0,t_0y_0)\le \sup _{t\ge 0} E_{\alpha ,\beta }(tz_0,ty_0)<c_\infty . \end{aligned}$$

This completes the proof. $\square$

Now, we establish the existence of a local minimum of $E_{\alpha ,\beta }$ on $\mathcal {N}^-_{\alpha ,\beta }$.

Theorem 3.8

If $0<(\alpha |g^+|_{\infty })^{\frac{2}{2-q}}+(\beta |h^+|_{\infty })^{\frac{2}{2-q}}<\omega ^{*}_{0}$, $(F_{1})-(F_{3})$ hold, then $E_{\alpha ,\beta }$ has a minimizer $(z^{-}_{0},y^{-}_{0})$ in $\mathcal {N}^{-}_{\alpha ,\beta }$ and

• (i) $E_{\alpha ,\beta }(z^{-}_{0},y^{-}_{0})=\vartheta ^{-}_{\alpha ,\beta }$,

• (ii) $(z^{-}_{0},y^{-}_{0})$ is a positive solution of problem (1), where

  $$\begin{aligned} \omega ^{*}_{0}=\min \left\{ \omega ^{*},\left( \frac{q}{2}\right) ^{\frac{2}{2-q}}\omega \right\} . \end{aligned}$$

Proof

By Lemma 3.2(ii), there is a $(\mathrm{PS})_{\vartheta ^{-}_{\alpha ,\beta }}$ sequence $\{(z_n,y_n)\}\subset \mathcal {N}^-_{\alpha ,\beta }$ in Q for $E_{\alpha ,\beta }$ for all $\alpha ,\beta \in \Theta _{\omega _0}$. From Lemmas 3.6, 3.7 and 2.5(ii), for $\alpha ,\beta \in \Theta _{\omega ^*}$, $E_{\alpha ,\beta }$ satisfies $(\mathrm{PS})_{\vartheta ^{-}_{\alpha ,\beta }}$ condition and $\vartheta ^{-}_{\alpha ,\beta }>0$. Since $E_{\alpha ,\beta }$ is coercive on $\mathcal {N}^-_{\alpha ,\beta }$, we get that $(z_n,y_n)$ is bounded in Q. Therefore, there exists a subsequence still denoted by $(z_n,y_n)$ and $(z_0^-,y_0^-)\in \mathcal {N}^-_{\alpha ,\beta }$, such that $(z_n,y_n)\rightarrow (z_0^-,y_0^-)$ strongly in Q and $E_{\alpha ,\beta }(z_0^-,y_0^-)={\vartheta ^{-}_{\alpha ,\beta }}>0$ for all $\alpha ,\beta \in \Theta _{\omega _1}$. Finally, by the same arguments as in the proof of Theorem 3.3, for all $\alpha ,\beta \in \Theta _{\omega _1}$, we have that $(z_0^-,y_0^-)$ is a positive solution of problem (1). $\square$

Now, we complete the proof of Theorems 1.2 and 1.3; by Theorem 3.3, we obtain that for all $\alpha ,\beta \in \Theta _{\omega }$, problem (1) has a positive solution $(z_0^+,y_0^+)\in \mathcal {N}^+_{\alpha ,\beta }$. On the other hand, from Theorem 3.8, we get the second positive solution $(z_0^-,y_0^-)\in \mathcal {N}^-_{\alpha ,\beta }$ for all $\alpha ,\beta \in \Theta _{\omega _1}$. Since $\mathcal {N}^+_{\alpha ,\beta }\cap \mathcal {N}^-_{\alpha ,\beta }=\emptyset$, this implies that $(z_0^+,y_0^+)$ and $(z_0^-,y_0^-)$ are distinct. This completes the proof of Theorems 1.2 and 1.3.