In this paper, we consider the following equation

$ \begin{equation} \left\{ \begin{aligned} &- \rm{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)-\lambda\frac{|u|^{p-2}u}{|x|^{p(a+1)}} = f_\mu(x)|u|^{q-2}u+g(x)|u|^{r-2}u,\\ &u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N), \end{aligned} \right. \end{equation} $

(1.1)

where $ N\geqslant3 $, $ 1<p<N $, $ 0\leqslant a<\frac{N-p}{p} $, $ 1\leqslant q<p<r<p^*[a] = \frac{Np}{N-p(a+1)} $, and $ p^*[a] $ is the critical Sobolev-Hardy exponent. The parameter $ \lambda $ satisfies $ 0\leqslant\lambda<\overline{\lambda} = (\frac{N-p}{p}-a)^p $, $ \mu\geqslant0 $, and $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ will be explained later. The weight functions $ f{_\mu}(x) = \mu f_+(x)+f_-(x) $ and $ g(x) = g_1(x)+g_2(x) $ satisfy the following conditions

(A$ _1 $) $ f\in L^{q^*}(\mathbb{R}^N)(q^* = \frac{r}{r-q})\ \text{with} \ f{_\pm}(x) = \pm \max\{\pm f(x),0\}\not\equiv0 $ and there exists a positive constant $ r_f $ such that

$ f_-(x)\geqslant-c_f|x|^{-r_f}\quad \text{ for some }c_f>0\text{ and for all }x\in \mathbb{R}^N; $

(A$ _2 $) $ g\in C(\mathbb{R}^N) $ with $ g_0 = \max\limits_{x\in\mathbb{R}^N}g(x) $ and there exists constants $ r_{g_1},r_{g_2} $ with $ 0<r_{g_2}<\min\{r_f-N,r_{g_1}-N\} $ such that

$ 1\geqslant g_1(x)\geqslant 1-c_{g_1}|x|^{-r_{g_1}}\quad \text{for some }c_{g_1}<1\text{ and for all }x\in \mathbb{R}^N $

and

$ g_2(x)\geqslant c_{g_2}|x|^{-r_{g_2}}\quad \text{for some }c_{g_2}>0\text{ and for all }x\in\mathbb{R}^N. $

Such kind of problem arised from various fields of geometry and physics and was widely used in the applied sciences. We refer to [1, 2, 3] for details on the description about the background.

Elliptic problems on bounded domains involving concave-convex nonlinearity were studied extensively since Ambrosetti, Brezis and Cerami [4] considered the following equation

$ \begin{equation} \left\{ \begin{aligned} &-\Delta u = \mu u^{q-1}+u^{p-1} \qquad& \text{in }\Omega,\\ &u>0 & \text{in }\Omega,\\ &u \in H_0^1(\Omega),& \end{aligned} \right. \end{equation} $

(1.2)

where $ 1<q<2<p\leqslant2^*,\;\mu>0 $. They found that there exists $ \mu_0>0 $ such that (1.2) admits at least two positive solutions for $ \mu\in(0,\mu_0) $, a positive solution for $ \mu = \mu_0 $ and no positive solution exists for $ \mu>\mu_0 $ (see also Ambrosetti, Azorero and Peral [5, 6] for more references therein). In recent years, several authors studied semilinear or quasilinear problems with the help of Nehari manifold (see [7-9]). In particular, Lin [9] studied the following critical problem

$ \begin{equation} \left\{ \begin{aligned} &- \rm{div}(|x|^{-2a}\nabla u)-\lambda\frac{u}{|x|^{2(a+1)}} = \frac{|u|^{2^*(a,b)-2}u}{|x|^{b2^*(a,b)}}+\mu|u|^{q-2}u\ &\text{in }\Omega\backslash\{0\},\\ &u = 0\ &\text{on }\partial\Omega, \end{aligned} \right. \end{equation} $

(1.3)

where $ \Omega\subset\mathbb{R}^N(N\geqslant 3) $ is a bounded domain with smooth boundary, $ 0\leqslant a<\frac{N-2}{2} $, $ a\leqslant b<a+1 $, $ 2^*(a,b) = \frac{2N}{N-2(a+1-b)} $, $ 0\leqslant \lambda<\overline{\lambda} = \frac{(N-2(a+1))^2}{4} $, $ \mu>0 $, and $ 1<q<2 $. He found that (1.3) admits at least two positive and one sign-changing solutions.

Actually, Fan and Liu [10] established multiple positive solutions of standard $ p $-Laplacian elliptic equations without Hardy term on a bounded domain $ \Omega $ in $ \mathbb{R}^N $. Some other theorems for $ p $-Laplacian elliptic equations without Hardy term can be found in [11, 12]. Hsu and Lin [13] studied the following critical problem via generalized Mountain Pass Theorem [14]

$ \begin{equation} \left\{ \begin{aligned} &- \rm{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)-\lambda\frac{|u|^{p-2}u}{|x|^{p(a+1)}} = \frac{|u|^{p^*(a,b)-2}u}{|x|^{bp^*(a,b)}}+ \mu\frac{|u|^{q-2}u}{|x|^{dp^*(a,d)}}&\text{in } \Omega,\\ &u = 0 &\text{on } \partial\Omega, \end{aligned} \right. \end{equation} $

(1.4)

where $ a\leqslant b,d<a+1 $, $ p^*(a,b) = \frac{Np}{N-p(a+1-b)} $ is the critical Sobolev-Hardy exponent. They found that (1.4) admits at least two positive solutions.

However, little is done on $ \mathbb{R}^N $ for the operator $ - \rm{div}(|x|^{-ap}|\nabla\cdot|^{p-2}\nabla\cdot)-\lambda\frac{|\cdot|^{p-2}\cdot}{|x|^{p(a+1)}} $ involving the concave-convex nonlinearity. Since the embedding is not compact on $ \mathbb{R}^N $ and the weight functions $ f $ and $ g $ are sign-changing, we will discuss the concentration behavior of solutions on the corresponding Nehari manifold to overcome these difficulties. Moreover, we get some improvement on multiplicity of positive solutions via the theory of Lusternik-Schnirelmann category (see [15]).

Throughout our paper, we denote by $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ the completion of $ C_0^\infty(\mathbb{R}^N) $ with respect to the standard norm $ ( \int_{\mathbb{R}^N}|x|^{-ap}|\nabla u|^p dx)^\frac{1}{p} $. The function $ u\in\mathbf{W}_a^{1,p}(\mathbb{R}^N) $ is said to be a solution of problem (1.1) if $ u $ satisfies

$ \begin{equation} \int_{\mathbb{R}^N}(|x|^{-ap}|\nabla u|^{p-2}\nabla u\nabla v-\lambda\frac{|u|^{p-2}uv}{|x|^{p(a+1)}}-f_\mu|u|^{q-2}uv-g|u|^{r-2}uv) dx = 0 \end{equation} $

(1.5)

for all $ v\in\mathbf{W}_a^{1,p}(\mathbb{R}^N) $. It is well known that the nontrivial solution of problem (1.1) is equivalent to the corresponding nonzero critical point of the energy functional

$ \begin{equation} I_\mu(u) = \frac{1}{p} \int_{\mathbb{R}^N}(|x|^{-ap}|\nabla u|^p-\lambda\frac{|u|^p}{|x|^{p(a+1)}}) dx- \frac{1}{q} \int_{\mathbb{R}^N} f_\mu|u|^q dx-\frac{1}{r} \int_{\mathbb{R}^N} g|u|^{r} dx. \end{equation} $

(1.6)

Then $ I_\mu(u) $ is well-defined on $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ and belongs to $ C^1(\mathbf{W}_a^{1,p}(\mathbb{R}^N), \mathbb{R}) $.

Problem (1.1) is related to well-known Caffarelli-Kohn-Nirenberg inequality in [16]

$ \begin{equation} \bigg( \int_{\mathbb{R}^N}\frac{|u|^{p^*(a,b)}}{|x|^{bp^*(a,b)}} dx\bigg)^\frac{p}{p^*(a,b)} \leqslant C \int_{\mathbb{R}^N}|x|^{-ap}|\nabla u|^p dx, \quad\forall u\in C_0^\infty(\mathbb{R}^N). \end{equation} $

(1.7)

If $ b = a+1 $, then $ p^*(a,b) = p $ and the following Hardy inequality holds [17]

$ \begin{equation} \int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p(a+1)}} dx\leqslant \frac{1}{\overline{\lambda}} \int_{\mathbb{R}^N}|x|^{-ap}|\nabla u|^p dx,\quad \forall u\in C_0^\infty(\mathbb{R}^N), \end{equation} $

(1.8)

where $ \overline{\lambda} = (\frac{N-p}{p}-a)^p $ is the best Hardy constant. Consequently, for $ \lambda<\overline{\lambda} $, we endow the space $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ with the following norm

$ \begin{equation} \lVert u \rVert = \lVert u \rVert_{\mathbf{W}_a^{1,p}(\mathbb{R}^N)} = ( \int_{\mathbb{R}^N}(|x|^{-ap}|\nabla u|^p-\lambda\frac{|u|^p}{|x|^{p(a+1)}}) dx)^\frac{1}{p}, \end{equation} $

(1.9)

which is equivalent to the usual norm $ ( \int_{\mathbb{R}^N}|x|^{-ap}|\nabla u|^p dx)^\frac{1}{p} $.

We get our main result as follows.

Theorem 1.1 Suppose that the functions $ f $ and $ g $ satisfy condition (A$ _1) $ and (A$ _2) $. Let

$ \begin{equation} L_2 = \frac{q}{p}\bigg(\frac{r-p}{r-q}\bigg) \bigg(\frac{p-q}{g_0(r-q)}\bigg)^{\frac{p-q}{r-p}} \frac{S_\lambda^\frac{r-q}{r-p}}{\lVert f_+\rVert_{L^{q^*}}}, \end{equation} $

(1.10)

where $ S_\lambda $ is the best Sobolev constant for the embedding of $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ into $ L^{r}(\mathbb{R}^N) $ and defined by

$ \begin{equation} S_\lambda: = \inf\limits_{u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus \{0\}} \frac{\lVert u \rVert^p}{( \int_{\mathbb{R}^N}|u|^{r} dx)^\frac{p}{r}}. \end{equation} $

(1.11)

Then

(ⅰ) for $ \mu\in(0, L_2) $, $ \rm(1.1) $ has at least two positive solutions in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ corresponding to negative least energy;

(ⅱ) there exists $ \mu_0\in(0, L_2) $ such that for $ \mu\in(0, \mu_0) $, $ \rm(1.1) $ has at least three positive solutions in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ including two with positive energy.

The paper is organized as follows: in Sections 2–4, based on some related preliminaries, we develop the description of Palais-Smale condition and the estimate of corresponding energy functional $ I_\mu $; in Section 5, we discuss the concentration behavior of solutions on Nehari manifold; in Section 6, we complete the proof of Theorem 1.1.

Since the energy functional $ I_\mu $ in (1.6) is unbounded below on $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $, we consider the functional on Nehari manifold

$ N{_\mu} = \{u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} \mid \langle I'_\mu(u),u\rangle = 0\}. $

Note that $ N_\mu $ contains all nonzero solutions of (1.1) and $ u\in N_\mu $ if and only if

$ \rVert {u} \rVert^p- \int_{\mathbb{R}^N} f_\mu|u|^q dx- \int_{\mathbb{R}^N} g|u|^{r}dx = 0. $

Lemma 2.1 The energy functional $ I_\mu $ is coercive and bounded below on $ N{_\mu} $.

Proof For $ u\in N{_\mu} $, by the Hölder inequality and Sobolev embedding theorem, we can deduce

$ \begin{aligned} I{_\mu}(u)& = \frac{1}{p}\lVert u \rVert^p-\frac{1}{q} \int_{\mathbb{R}^N} f{_\mu}|u|^q dx-\frac{1}{r} \int_{\mathbb{R}^N}g|u|^{r} dx \\ & = \bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p-\bigg(\frac{1}{q}-\frac{1}{r}\bigg) \int_{\mathbb{R}^N}f_{\mu}|u|^q dx \\ &\geqslant \bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p-\bigg(\frac{1}{q}-\frac{1}{r}\bigg)\mu\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}}\lVert u \rVert^q\\ &\geqslant -C\mu^{\frac{p}{p-q}}, \end{aligned} $

where $ C $ is a positive constant depending on $ N, q, S_\lambda $ and $ \lVert f_+ \rVert_{L^{q^*}} $. This completes the proof.

Define

$ \Psi(u) = \langle I'{_\mu}(u),u \rangle = \lVert u \rVert^p- \int_{\mathbb{R}^N}f_\mu|u|^q dx- \int_{\mathbb{R}^N}g|u|^r dx. $

Then for $ u\in N_\mu $, we have

$ \begin{equation} \begin{aligned} \langle \Psi'(u),u \rangle & = p\lVert u \rVert^p-q \int _{\mathbb{R}^N}f_\mu|u|^q dx- r \int_{\mathbb{R}^N}g|u|^r dx\\ & = (p-q)\lVert u \rVert^p-(r-q) \int_{\mathbb{R}^N}g|u|^r dx\\ & = (p-r)\lVert u \rVert^p-(q-r) \int_{\mathbb{R}^N}f_\mu|u|^q dx. \end{aligned} \end{equation} $

(2.1)

As in [18], we divide $ N_\mu $ into three parts

$ \begin{eqnarray*} &&N_\mu^+ = \{u\in N_\mu\mid \langle \Psi'(u), u \rangle >0\},\\ &&N_\mu^0 = \{u\in N_\mu\mid \langle \Psi'(u), u \rangle = 0\},\\ &&N_\mu^- = \{u\in N_\mu\mid \langle \Psi'(u), u \rangle <0\}. \end{eqnarray*} $

Then we have the following result.

Lemma 2.2 $ \rm{(ⅰ)} $ If $ u\in N_\mu^+ $, then $ \int_{\mathbb{R}^N}f_\mu(x)|u|^q dx>0 $.

$ \rm{(ⅱ)} $ If $ u\in N_\mu^0 $, then $ \int_{\mathbb{R}^N}f_\mu(x)|u|^q dx>0 $ and $ \int_{\mathbb{R}^N}g(x)|u|^r dx>0 $.

$ \rm{(ⅲ)} $ If $ u\in N_\mu^- $, then $ \int_{\mathbb{R}^N}g(x)|u|^r dx>0 $.

Proof By (2.1) we can easily derive these results.

Set $ L_1 = \bigg(\frac{r-p}{r-q}\bigg)\bigg(\frac{p-q}{g_0(r-q)}\bigg)^\frac{p-q}{r-p} \frac{S_\lambda^\frac{r-q}{r-p}}{\lVert f_+ \rVert_{L^{q^*}}} $ and it is easy to see $ L_2 = \frac{q}{p}L_1 $, where $ L_2 $ is defined in (1.10). We define

$ \alpha = \inf\limits_{u\in N_\mu}I_\mu(u),\quad \alpha^+ = \inf\limits_{u\in N_\mu^+} I_\mu(u)\quad \text{and}\quad \alpha^- = \inf\limits_{u\in N_\mu^-} I_\mu(u). $

Then the following lemma is essential for the main result.

Lemma 2.3 $ \rm{(ⅰ)} $ For all $ \mu\in(0, L_1) $, we have $ N_\mu^0 = \emptyset $ and $ \alpha^+<0 $.

$ \rm{(ⅱ)} $ If $ \mu<L_2 $, then we have $ \alpha^->c_0 $ for some $ c_0>0 $. In particular, $ \inf\limits_{u\in N_\mu}I_\mu(u) = \alpha^+ $ for all $ \mu\in (0, L_2) $.

Proof $ \rm{(ⅰ)} $ Suppose the contrary. We may assume that there exists $ \mu_*\in(0, L_1) $ such that $ N_{\mu_*}^0\neq\emptyset $. Thus, for each $ u\in N_{\mu_*}^0 $, by the Hölder and Sobolev inequalities, we can obtain

$ \begin{equation} 0 = \langle \Psi'(u), u\rangle = (p-r)\lVert u \rVert^p-(q-r) \int_{\mathbb{R}^N}f_{\mu_*}|u|^q dx, \end{equation} $

(2.2)

that is,

$ \begin{equation} \lVert u \rVert^p = \frac{r-q}{r-p} \int_{\mathbb{R}^N}f_{\mu_*}|u|^q dx \leqslant \frac{r-q}{r-p}\mu_* \lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}}\lVert u \rVert^q \end{equation} $

(2.3)

and so

$ \begin{equation} \mu_*\geqslant\frac{r-p}{r-q}\lVert u \rVert^{p-q}\frac{S_\lambda^\frac{q}{p}}{\lVert f_+ \rVert_{L^{q^*}}}. \end{equation} $

(2.4)

But (2.1) implies that

$ (p-q)\lVert u \rVert^p = (r-q) \int_{\mathbb{R}^N}g|u|^r dx\leqslant g_0(r-q)\lVert u \rVert^r S_\lambda^{-\frac{r}{p}}, $

which means

$ \begin{equation} \lVert u \rVert^{p-q} \geqslant \bigg(\frac{p-q}{g_0(r-q)}S_\lambda^\frac{r}{p}\bigg)^\frac{p-q}{r-p}. \end{equation} $

(2.5)

Combined (2.4) and (2.5), we have

$ \begin{aligned} \mu_*& \geqslant \bigg(\frac{r-p}{r-q}\bigg) \bigg(\frac{p-q}{g_0(r-q)}\bigg)^\frac{p-q}{r-p}\frac{S_\lambda^{\frac{r}{p}\cdot \frac{p-q}{r-p}+\frac{q}{p}}}{\lVert f_+ \rVert_{L^{q^*}}}\\ & = \bigg(\frac{r-p}{r-q}\bigg) \bigg(\frac{p-q}{g_0(r-q)}\bigg)^\frac{p-q}{r-p} \frac{S_\lambda^\frac{r-q}{r-p}}{\lVert f_+ \rVert_{L^{q^*}}} = L_1. \end{aligned} $

This contradicts to $ \mu_*\in(0, L_1) $. Therefore, $ N_\mu^0 = \emptyset $ and $ N_\mu = N_\mu^+ \cup N_\mu^- $ for $ \mu\in(0,L_1) $. Then for $ u\in N_\mu^+ $, by Lemma 2.2, we get

$ \begin{aligned} I_\mu(u)& = \bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p- \bigg(\frac{1}{q}-\frac{1}{r}\bigg) \int_{\mathbb{R}^N}f_\mu|u|^q dx\\ &<\bigg(\frac{1}{p}-\frac{1}{q}\bigg)\frac{r-q}{r} \int_{\mathbb{R}^N}f_\mu|u|^q dx \end{aligned} $

and so

$ \begin{equation} \alpha^+ = \inf\limits_{u\in N_\mu^+}I_\mu(u)<0. \end{equation} $

(2.6)

$ \rm{(ⅱ)} $ Let $ u\in N_\mu^- $. By (2.1) and the Sobolev inequality, we have

$ (p-q)\lVert u \rVert^p< (r-q) \int_{\mathbb{R}^N}g|u|^r dx\leqslant g_0(r-q)\lVert u \rVert^r S_\lambda^{-\frac{r}{p}} $

or

$ \begin{equation} \lVert u \rVert>\bigg(\frac{p-q}{g_0(r-q)}S_\lambda^\frac{r}{p}\bigg)^\frac{1}{r-p}. \end{equation} $

(2.7)

Then for $ \mu\in (0, L_2) $, we have

$ \begin{aligned} I_\mu(u)& = \bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p- \bigg(\frac{1}{q}-\frac{1}{r}\bigg) \int_{\mathbb{R}^N}f_\mu|u|^q dx\\ & = \bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p- \bigg(\frac{1}{q}-\frac{1}{r}\bigg)\mu\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}} \lVert u \rVert^q+c_0\\ &>\bigg(\frac{1}{p}-\frac{1}{r}\bigg)\lVert u \rVert^p- \bigg(\frac{1}{q}-\frac{1}{r}\bigg)\frac{q}{p}L_1\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}} \lVert u \rVert^q+c_0\\ &>c_0, \end{aligned} $

where

$ c_0 = \bigg(\frac{1}{q}-\frac{1}{r}\bigg)\mu\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}} \lVert u \rVert^q- \bigg(\frac{1}{q}-\frac{1}{r}\bigg) \int_{\mathbb{R}^N}f_\mu|u|^q dx >0. $

This implies, for $ \mu\in (0, L_2) $, $ \alpha^+<0<c_0<\alpha^- $. The proof is completed.

Now we introduce the following function $ m_u:\mathbb{R}^+\rightarrow \mathbb{R} $ in the form

$ m_u(t) = t^{p-q}\lVert u \rVert^p- t^{r-q} \int_{\mathbb{R}^N}g|u|^r dx\ \text{for}\;t>0. $

Clearly, $ tu\in N_\mu $ if and only if $ m_u(t) = \int_{\mathbb{R}^N}f_\mu(x)|u|^q dx $, and

$ \begin{equation} m'_u(t) = (p-q)t^{p-q-1}\lVert u \rVert^p- (r-q)t^{r-q-1} \int_{\mathbb{R}^N}g(x)|u|^r dx. \end{equation} $

(2.8)

It is obvious that if $ tu\in N_\mu $, then $ t^{q+1}m'_u(t) = \langle\Psi'(tu), tu \rangle $. Hence, $ tu\in N_\mu^+ (\text{or}\ N_\mu^-) $ if and only if $ m'_u(t)>0\; (\text{or}\;<0) $.

Suppose $ u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} $. Then by (2.8), $ m_u $ admits a unique critical point at $ t = t_{\max} $, where

$ t_{\max} = \bigg(\frac{(p-q)\lVert u \rVert^p} {(r-q) \int_{\mathbb{R}^N}g|u|^r dx}\bigg)^\frac{1}{r-p}>0, $

and $ m_u $ strictly increases on $ (0,t_{\max}) $ and decreases on $ (t_{\max},\infty) $ with $ \lim\limits_{t\to\infty}m_u(t) = -\infty. $ Furthermore, since $ \mu\in(0,L_1) $, we have

$ \begin{aligned} m_u(t_{\max})& = \bigg(\frac{(p-q)\lVert u \rVert^p} {(r-q) \int_{\mathbb{R}^N}g|u|^r dx}\bigg)^ \frac{p-q}{r-p}\lVert u \rVert^p-\bigg(\frac{(p-q)\lVert u \rVert^p} {(r-q) \int_{\mathbb{R}^N}g|u|^r dx}\bigg)^ \frac{r-q}{r-p} \int_{\mathbb{R}^N}g|u|^r dx\\ & = \bigg(\frac{p-q}{r-q}\bigg)^\frac{p-q}{r-p} \frac{\lVert u \rVert^\frac{p(r-q)}{r-p}} {( \int_{\mathbb{R}^N}g|u|^r dx)^\frac{p-q}{r-p}}- \bigg(\frac{p-q}{r-q}\bigg)^\frac{r-q}{r-p} \frac{\lVert u \rVert^\frac{p(r-q)}{r-p}} {( \int_{\mathbb{R}^N}g|u|^r dx)^\frac{p-q}{r-p}}\\ & = \lVert u \rVert^q\bigg(\frac{r-p}{r-q}\bigg) \bigg(\frac{p-q}{r-q}\bigg)^\frac{p-q}{r-p} \bigg(\frac{\lVert u \rVert^r} { \int_{\mathbb{R}^N}g|u|^r dx}\bigg)^\frac{p-q}{r-p}\\ &\geqslant\frac{1}{\mu}\lVert f_+ \rVert_{L^{q^*}}^{-1} S_\lambda^\frac{r-q}{r-p} \bigg(\frac{p-q}{r-q}\bigg)^\frac{p-q}{r-p}\bigg(\frac{r-p}{r-q}\bigg) \int_{\mathbb{R}^N}f_\mu|u|^q dx\\ &> \int_{\mathbb{R}^N}f_\mu|u|^q dx. \end{aligned} $

Thus, we have the following lemma.

Lemma 2.4 For each $ u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} $, we have

$ \rm{(ⅰ)} $ if $ \int_{\mathbb{R}^N}f_\mu|u|^q dx\leqslant 0 $, then there exists a unique $ t^- = t^-(u)>t_{\max} $ such that $ t^-u\in N_\mu^- $, and

$ \begin{equation} I_\mu(t^-u) = \sup\limits_{t\geqslant 0}I_\mu(tu); \end{equation} $

(2.9)

$ \rm{(ⅱ)} $ if $ \int_{\mathbb{R}^N}f_\mu|u|^q dx>0 $, then there exist unique $ 0<t^+ = t^+(u)<t_{\max}<t^- $ such that $ t^+u\in N_\mu^+ $, $ t^-u\in N_\mu^- $ and

$ \begin{equation} I_\mu(t^+u) = \inf\limits_{0 \leqslant t \leqslant t_{\max}}I_\mu(tu),\ I_\mu(t^-u) = \sup\limits_{t \geqslant t^+}I_\mu(tu); \end{equation} $

(2.10)

$ \rm{(ⅲ)} $ $ t^-(u): \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\}\to\mathbb{R}^+ $ is continuous;

$ \rm{(ⅳ)} $ $ N_\mu^- = \{u\in {\mathbf{W}_a^{1,p}}(\mathbb{R}^N)\setminus \{0\}\mid\frac{1}{\lVert u \rVert}t^-(\frac{u}{\lVert u \rVert}) = 1\} $.

Proof $ \rm{(ⅰ)} $ The equation $ m_u(t) = \int_{\mathbb{R}^N}f_\mu|u|^q dx $ admits a unique solution $ t^->t_{\max} $ and $ m'_u(t^-)<0 $. Thus $ t^-u\in N_\mu^- $, and (2.9) holds by Lemma 2.3.

$ \rm{(ⅱ)} $ The equation $ m_u(t) = \int_{\mathbb{R}^N}f_\mu|u|^q dx $ admits distinctive solutions $ t^+<t_{\max}<t^- $ such that $ m'_u(t^+)>0 $ and $ m'_u(t^-)<0 $, and then we have $ t^+u\in N_\mu^+ $ and $ t^-u\in N_\mu^- $. Thus (2.10) holds by Lemma 2.3 and Lemma 2.4 (ⅰ).

$ \rm{(ⅲ)} $ By the uniqueness and extremal property of $ t^-(u) $, we have $ t^-(u) $ is a continuous function for $ u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} $.

$ \rm{(ⅳ)} $ For $ u\in N_\mu^- $, let $ v = \frac{u}{\lVert u \rVert} $. By $ \rm{(ⅰ)} $ and $ \rm{(ⅱ)} $, there is a unique $ t^-(v)>0 $ such that $ t^-(v)v\in N_\mu^- $ or

$ t^-(\frac{u}{\lVert u \rVert})\frac{1}{\lVert u \rVert}u\in N_\mu^-. $

Since $ \rm{(ⅰ)} $ $ u\in N_\mu^- $, we have $ t^-(\frac{u}{\lVert u \rVert})\frac{1}{\lVert u \rVert} = 1 $, and this implies

$ N_\mu^-\subset\{u\in\mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\}|\frac{1}{\lVert u \rVert}t^-(\frac{u}{\lVert u \rVert}) = 1\}. $

On the other hand, let $ u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} $ such that

$ \frac{1}{\lVert u \rVert }t^-(\frac{u}{\lVert u \rVert}) = 1. $

If $ u\in N_\mu^+ $, then $ t^-(u)>t_{\max}>1 $ and this contradicts $ t_{\max}<1 $ on $ N_\mu^- $. Then

$ t^-(\frac{u}{\lVert u \rVert})\frac{u}{\lVert u \rVert}\in N_\mu^-. $

Thus, the proof is completed.

Remark 2.5 If $ \mu = 0 $, by Lemma 2.4 $ \rm{(ⅰ)} $, $ N_0^+ = \emptyset $ and so $ N_0 = N_0^- $.

Now we consider the limiting problem

$ \begin{equation} \left\{ \begin{aligned} &- \rm{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)-\lambda \frac{|u|^{p-2}u}{|x|^{p(a+1)}} = |u|^{r-2}u, \\ &u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N) \end{aligned} \right. \end{equation} $

(3.1)

and the corresponding energy functional $ I^\infty $ in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ is defined by

$ \begin{equation} I^\infty(u) = \frac{1}{p}\lVert u \rVert^p-\frac{1}{r} \int_{\mathbb{R}^N}|u|^{r} dx. \end{equation} $

(3.2)

Proposition 3.1 For $ 0\leqslant a<\frac{N-p}{p} $, $ 0\leqslant \lambda<\overline{\lambda} $, problem $ \rm(3.1) $ has radially symmetric ground states

$ u_\epsilon(x) = \epsilon^{-(\frac{N-p}{p}-a)}v_\epsilon(\frac{x}{\epsilon}),\qquad \forall\epsilon>0, $

satisfying

$ \int_{\mathbb{R}^N}(|x|^{-ap}|\nabla u_\epsilon(x)|^p-\lambda\frac{|u|^p}{|x|^{p(a+1)}}) dx = \int_{\mathbb{R}^N}|u_\epsilon(x)|^r dx = S_\lambda^\frac{r}{r-p}, $

where $ v_\epsilon(x) = v_\epsilon(|x|) $ is the unique radial solution of $ \rm(3.1) $ up to a dilation. In particular, we have

$ \begin{equation} v_\epsilon(1) = (\frac{r(\overline{\lambda}-\lambda)}{p})^{\frac{1}{r}-p}, \end{equation} $

(3.3)

and $ v_\epsilon $ also has the following properties:

$ \begin{equation} \begin{aligned} \lim\limits_{\xi\to 0}\xi^{a(\lambda)}v_\epsilon(\xi) = c_1>0,\qquad &\lim\limits_{\xi\to 0}\xi^{a(\lambda)+1}v'_\epsilon(\xi) = c_1a(\lambda)\geqslant 0,\\ \lim\limits_{\xi\to +\infty}\xi^{b(\lambda)}v_\epsilon(\xi) = c_2>0,\qquad &\lim\limits_{\xi\to +\infty}\xi^{b(\lambda)+1}v'_\epsilon(\xi) = c_2b(\lambda)>0, \end{aligned} \end{equation} $

(3.4)

where $ c_i\; (i = 1,2) $ are positive constants and $ a(\lambda), b(\lambda) $ are the zeros of the function

$ \phi(t) = (p-1)t^p-\frac{N}{r}t^{p-1}+\lambda,\quad t\geqslant 0,\ 0\leqslant \lambda<\overline{\lambda} $

with $ 0\leqslant a(\lambda)<\frac{N}{r}<b(\lambda)<\frac{Np}{(p-1)r} $.

Furthermore, there exist the positive constants $ c_3,c_4 $ such that

$ \begin{equation} c_3\leqslant v_\epsilon(x)(|x|^{a(\lambda)/\delta}+|x|^{b(\lambda)/\delta})^\delta \leqslant c_4,\quad \delta = \frac{N}{r}. \end{equation} $

(3.5)

Proof As in [19], we can prove that the limiting problem (3.1) has radially symmetric ground states, by which $ S_\lambda $ can be achieved. Let $ u(\xi) $ be a radial solution to (3.1). Then we get that

$ (\xi^{N-1-ap}|u'|^{p-2}u')'+\xi^{N-1}(\lambda\frac{u^{p-1}}{\xi^{p(a+1)}}+u^{r-1}) = 0. $

Set

$ \delta = \frac{N}{r},\quad t = \ln \xi,\quad y(t) = \xi^\delta u(\xi),\quad z(t) = \xi^{(1+\delta)(p-1)}|u'(\xi)|^{p-2}u'(\xi). $

Then we can obtain the following system

$ \left\{ \begin{aligned} \frac{dy}{dt} = &\delta y+|z|^\frac{2-p}{p-1}z,\\ \frac{dz}{dt} = &-\delta z-|y|^{r-2}y-\lambda |y|^{p-2}y. \end{aligned} \right. $

The rest of the proof follows exactly the same lines as that of the limiting problem (3.1) in [19], here we omit it.

By Proposition 3.1, we can easily derive the minimizing problem

$ \begin{equation} \inf\limits_{u\in N^\infty}I^\infty(u) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}, \end{equation} $

(3.6)

where

$ N^\infty = \{u\in {\mathbf{W}_a^{1,p}}(\mathbb{R}^N)\setminus \{0\}|\langle (I^\infty)'(u),u\rangle = 0\}. $

For our purpose, the functional $ I_\mu $ is said to satisfy the (P.S.)$ _c $ condition if any sequence $ \{u_n\}_{n\in\mathbb{N}}\subset \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ such that as $ n\rightarrow \infty $,

$ I_\mu(u_n)\rightarrow c,\quad I'_\mu(u_n)\rightarrow 0 \quad \text{strongly in }(\mathbf{W}_a^{1,p}(\mathbb{R}^N))^* $

contains a convergent subsequence in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $. Then the following proposition develops a precise description for the (P.S.)$ _c $-sequence of $ I_\mu $.

Proposition 3.2 $ \rm{(ⅰ)} $ If $ \mu\in(0,L_1) $, then $ I_\mu $ has a (P.S.)$ _\alpha $-sequence $ \{u_n\}_{n\in\mathbb{N}}\subset N_\mu $.

$ \rm{(ⅱ)} $ If $ \mu\in(0,L_2) $, then $ I_\mu $ has a (P.S.)$ _{\alpha^-} $-sequence $ \{u_n\}_{n\in\mathbb{N}}\subset N_\mu^- $.

Proof The proof is similar to the argument of Proposition 3.3 in [20].

Now, we establish the existence of a local minimizer for $ I_\mu $ on $ N_\mu $.

Proposition 3.3 For $ \mu\in (0,L_1) $, the functional $ I_\mu $ has a minimizer $ u_\mu^+\in N_\mu^+ $ satisfying

$ \rm{(ⅰ)} $ $ I_\mu(u_\mu^+) = \alpha^+ = \alpha $;

$ \rm{(ⅱ)} $ $ u_\mu^+ $ is a positive solution of $ \rm(1.1) $;

$ \rm{(ⅲ)} $ $ \lVert u_\mu^+ \rVert\rightarrow 0\ \text{as}\ \mu\rightarrow 0^+ $.

Proof By Proposition 3.2 $ \rm{(ⅰ)} $, there exists a minimizing sequence $ \{u_n\}_{n\in\mathbb{N}}\subset N_\mu $ such that

$ \begin{equation} I_\mu(u_n) = \alpha+o(1)\text{ and }I'_\mu(u_n) = o(1)\text{ in }(\mathbf{W}_a^{1,p}(\mathbb{R}^N))^{-1}, \end{equation} $

(3.7)

where $ o(1)\to 0 $ as $ n \to \infty $. Since $ I_\mu $ is coercive on $ N_\mu $, we get that $ \{u_n\} $ is bounded in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $. If necessary to a subsequence, there exists $ u_\mu^+\in\mathbf{W}_a^{1,p}(\mathbb{R}^N) $ such that as $ n\to \infty $,

$ \begin{equation} \left\{ \begin{aligned} &u_n\rightharpoonup u_\mu^+\qquad\qquad\text{weakly in }\mathbf{W}_a^{1,p}(\mathbb{R}^N),\\ &u_n\to u_\mu^+\qquad\qquad\text{a.e. in }\mathbb{R}^N,\\ &\nabla u_n\to\nabla u_\mu^+\qquad\ \text{a.e. in }\mathbb{R}^N,\\ &\frac{u_n}{|x|^{a+1}}\rightharpoonup \frac{u_\mu^+}{|x|^{a+1}}\quad\text{a.e. in }L^p(\mathbb{R}^N),\\ & \int_{\mathbb{R}^N}g|u_n|^{r-2}u_nv dx\to \int_{\mathbb{R}^N}g|u_\mu^+|^{r-2}u_\mu^+v dx \quad\text{for all }v\in \mathbf{W}_a^{1,p}(\mathbb{R}^N). \end{aligned} \right. \end{equation} $

(3.8)

Moreover, by the Egorov Theorem and Hölder inequality, we have

$ \int_{\mathbb{R}^N}f_\mu|u_n|^{q-2}u_n v dx = \int_{\mathbb{R}^N}f_\mu|u_\mu^+|^{q-2}u_\mu^+ v dx+o(1). $

Consequently, passing to the limit in $ \langle I'_\mu(u_n),v\rangle $, by (3.7) and (3.8), we have

$ \begin{aligned} & \int_{\mathbb{R}^N}(|x|^{-ap}|\nabla u_\mu^+|^{p-2}\nabla u_\mu^+\nabla v-\lambda \frac{|u_\mu^+|^{p-2}u_\mu^+v}{|x|^{p(a+1)}}) dx\\&- \int_{\mathbb{R}^N} f_\mu|u_\mu^+|^{q-2}u_\mu^+v dx- \int_{\mathbb{R}^N} g|u_\mu^+|^{r-2}u_\mu^+v dx = 0 \end{aligned} $

for all $ v\in \mathbf{W}_a^{1,p}(\mathbb{R}^N) $. That is, $ \langle I'_\mu(u_\mu^+),v\rangle = 0 $. Thus, $ u_\mu^+ $ is a weak solution of (1.1).

Furthermore, since $ u_n\in N_\mu $, we can deduce that

$ \begin{equation} \int_{\mathbb{R}^N} f_\mu|u_n|^q dx = \frac{q(r-p)}{p(r-q)}\lVert u_n \rVert^p-\frac{r\cdot q} {r-q}I_\mu(u_n), \end{equation} $

(3.9)

which implies that

$ \begin{equation} \lim\limits_{n\to\infty} \int_{\mathbb{R}^N}f_\mu|u_n|^q dx = \int_{\mathbb{R}^N} f_\mu|u_\mu^+|^q dx\geqslant -\frac{r\cdot q}{r-q}\alpha>0. \end{equation} $

(3.10)

Thus, $ u_\mu^+\in N_\mu $ is a nontrival solution of (1.1).

Next, we will show, up to a subsequence, that $ u_n\to u_\mu^+ $ strongly in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ and $ I_\mu(u_\mu^+) = \alpha $. In fact, by the Fatou's lemma, it follows that

$ \begin{aligned} \alpha\leqslant I_\mu(u_\mu^+)& = (\frac{1}{p}-\frac{1}{r})\lVert u_\mu^+\rVert^p-(\frac{1}{q}-\frac{1}{r}) \int_{\mathbb{R}^N} f_\mu|u_\mu^+|^q dx\\ &\leqslant \lim\limits_{n\to \infty}\inf((\frac{1}{p}-\frac{1}{r})\lVert u_n \rVert^p-(\frac{1}{q}-\frac{1}{r}) \int_\Omega f_\mu|u_n|^q dx)\\ & = \lim\limits_{n\to \infty}\inf I_\mu(u_n) = \alpha, \end{aligned} $

which implies that $ I_\mu(u_\mu^+) = \alpha $ and $ \lim\limits_{n\to \infty}\lVert u_n \rVert^p = \lVert u_\mu^+ \rVert^p $. Standard argument shows that $ u_n\to u_\mu^+ $ strongly in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $.

Moreover, we have $ u_\mu^+\in N_\mu^+ $. Otherwise, if $ u_\mu^+\in N_\mu^- $, then by Lemma 2.2 and Lemma 2.4, there is a unique $ t^- = \frac{1}{\lVert u_\mu^+\rVert}t^-(\frac{u_\mu^+}{\lVert u_\mu^+\rVert}) $ such that $ t^-u_\mu^+\in N_\mu^- $ and so

$ 0>\alpha^+ = \alpha = I_\mu(u_\mu^+) = I_\mu(t^-u_\mu^+) = \sup\limits_{t\geqslant 0}I_\mu(tu)>\alpha^-, $

which is a contradiction. Since $ I_\mu(u_\mu^+) = I_\mu(|u_\mu^+|) $ and $ |u_\mu^+|\in N_\mu^+ $, we may assume that $ u_\mu^+ $ is a nontrivial nonnegative solution of (1.1). By Harnack inequality, it follows that $ u_\mu^+>0 $ in $ \mathbb{R}^N $.

Finally, by (2.1) and the Hölder inequality, we can obtain

$ \lVert u_\mu^+ \rVert^{p-q}<\mu\frac{r-q}{r-p}\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}}, $

which implies that $ \lVert u_\mu^+ \rVert \to 0 $ as $ \mu\to 0^+ $. This completes the proof.

Let $ u_ l = u_0(x+le) $, for $ l\in \mathbb{R} $ and $ e\in \mathbb{S}^{N-1} $, where $ u_0(x) $ is a radially symmetric positive solution of (3.1) such that $ I^\infty(u_0) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $ and $ \mathbb{S}^{N-1} = \{x\in \mathbb{R}^N||x| = 1\}. $ Then we have the following result.

Lemma 3.4 $ \rm{(ⅰ)} $ $ \lim\limits_{l \rightarrow 0} \lVert u_l \rVert^p = S_\lambda^\frac{r}{r-p} $ uniformly in $ e \in \mathbb{S}^{N-1} $;

$ \rm{(ⅱ)} $ $ \lim\limits_{l \rightarrow 0} \int_{\mathbb{R}^N}|u_l|^r dx = S_\lambda^\frac{r}{r-p} $ uniformly in $ e \in \mathbb{S}^{N-1} $;

$ \rm{(ⅲ)} $ $ \lim\limits_{l \rightarrow 0} I^\infty(u_l) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $ uniformly in $ e \in \mathbb{S}^{N-1} $.

We refer to the argument of Lemma 4.2 in He and Yang (see [21]).

The following statement is paramount to prove our main result.

Proposition 4.1 For $ \mu\in(0,L_2) $, we have $ \alpha^-<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $.

Proof Let $ u_\mu^+\in N_\mu^+ $ be a positive solution of (1.1) in Proposition 3.3. Then we obtain

$ \begin{equation} \begin{aligned} I_\mu(u_\mu^++tu_l)& = \frac{1}{p}\lVert u_\mu^++tu_l \rVert^p-\frac{1}{q} \int_{\mathbb{R}^N}f_\mu|u_\mu^++tu_l|^q dx-\frac{1}{r} \int_{\mathbb{R}^N}g|u_\mu^++tu_l|^r dx\\ & = I_\mu(u_\mu^+)+I^\infty(tu_l)+\frac{1}{p}(\lVert u_\mu^++tu_l \rVert^p-\lVert u_\mu^+ \rVert^p-\lVert tu_l \rVert^p)\\ &\quad -\frac{1}{q} \int_{\mathbb{R}^N}f_\mu(|u_\mu^++tu_l|^q -|u_\mu^+|^q) dx\\ &\quad -\frac{1}{r} \int_{\mathbb{R}^N}g(|u_\mu^++tu_l|^{r} -|u_\mu^+|^{r}) dx +\frac{1}{r} \int_{\mathbb{R}^N}t^r|u_l|^r dx\\ &\leqslant I_\mu(u_\mu^+)+I^\infty(tu_l)+\frac{1}{p}(\lVert u_\mu^++tu_l\rVert^p-\lVert u_\mu^+\rVert^p-\lVert tu_l\rVert^p)\\ &\quad - \int_{\mathbb{R}^N}f_\mu\bigg\{ \int_0^{tu_l}[(u_\mu^++\eta)^{q-1}-(u_\mu^+)^{q-1}] d\eta \bigg\} dx\\ &\quad +\frac{1}{r} \int_{\mathbb{R}^N}(1-g)t^r|u_l|^r dx- \frac{1}{r} \int_{\mathbb{R}^N}g(|u_\mu^++tu_l|^r-|u_\mu^+|^r-t^r|u_l|^r) dx\\ &\leqslant \alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+\frac{1}{p}(\lVert u_\mu^++tu_l\rVert^p-\lVert u_\mu^+\rVert^p-\lVert tu_l\rVert^p)\\ &\quad +\frac{t^q}{q} \int_{\mathbb{R}^N}|f_-||u_l|^q dx+\frac{t^r}{r} \int_{\mathbb{R}^N}(1-g_1)|u_l|^r dx-\frac{t^r}{r} \int_{\mathbb{R}^N}g_2|u_l|^r dx. \end{aligned} \end{equation} $

(4.1)

Since

$ I_\mu(u_\mu^++tu_l)\rightarrow I_\mu(u_\mu^+) = \alpha^+<0\ \text{as}\ t\rightarrow 0 $

and

$ I_\mu(u_\mu^++tu_l)\rightarrow -\infty\ \text{as}\ t\rightarrow +\infty. $

There exist $ 0<t_1<t_2 $ such that

$ \begin{equation} I_\mu(u_\mu^++tu_l)<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}\ \text{for all}\ t\in[0,t_1)\cup(t_2,+\infty). \end{equation} $

(4.2)

Thus we only need to show that there exists $ l_0>0 $ such that for $ l>l_0 $, we have

$ \begin{equation} \sup\limits_{t_1\leqslant t\leqslant t_2}I_\mu(u_\mu^++tu_l)<\alpha^+ +(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}. \end{equation} $

(4.3)

Since $ u_\mu^++tu_l\rightarrow u_\mu^+ $ as $ l\rightarrow \infty $, by Brézis-Lieb lemma, we can find $ l_0>0 $ such that for $ l>l_0 $,

$ \begin{equation} \lVert u_\mu^++tu_l \rVert^p-\lVert u_\mu^+ \rVert^p-\lVert tu_l \rVert^p<\epsilon_l \text{ for }\epsilon_l>0\text{ small enough}. \end{equation} $

(4.4)

For $ u,v>0 $, we can remark that $ (u+v)^r-u^r-v^r\geqslant 0, $ and so

$ \begin{equation} \int_{\mathbb{R}^N}g(|u_\mu^++tu_l|^r-|u_\mu^+|^r-t^r|u_l|^r) dx\geqslant 0. \end{equation} $

(4.5)

From condition (A$ _1) $, (A$ _2) $ and (3.5), we can obtain

$ \begin{eqnarray} \frac{t^q}{q} \int_{\mathbb{R}^N}|f_-||u_l|^q dx &\leqslant& c_f\cdot c_4\frac{t^q}{q} \int_{\mathbb{R}^N}|x+le|^{-r_f}(|x+le|^{a(\lambda)/\delta} +|x+le|^{b(\lambda)/\delta})^{-\delta p}dx\\ &\leqslant& C_f \int_{|x|<l}|x+le|^{-r_f-p\cdot a(\lambda)}dx+C_f \int_{|x|\geqslant l}|x+le|^{-r_f-p\cdot a(\lambda)}dx\\ &\leqslant& C_f l^N \int_{|x|<1}|x+le|^{-r_f-p\cdot a(\lambda)} dx+C_f \int_{|x|\geqslant l}|x+le|^{-r_f-p\cdot a(\lambda)}dx\\ &\leqslant& C_f(l+1)^{N-r_f-p\cdot a(\lambda)}\quad\text{for all }l\geqslant 1, \end{eqnarray} $

(4.6)

$ \begin{eqnarray} \frac{t^r}{r} \int_{\mathbb{R}^N}(1-g_1)|u_l|^r dx &\leqslant& c_{g_1}\cdot c_4 \int_{\mathbb{R}^N}|x+le|^{-r_{g_1}}(|x+le|^{a(\lambda)/\delta} +|x+le|^{b(\lambda)/\delta})^{-\delta r}dx\\ &\leqslant& C_{g_1} \int_{|x|<l}|x+le|^{-r_{g_1}-p\cdot a(\lambda)}dx+C_{g_1} \int_{|x|\geqslant l}|x+le|^{-r_{g_1}-p\cdot a(\lambda)}dx\\ &\leqslant& C_{g_1} l^N \int_{|x|<1}|x+le|^{-r_{g_1}-p\cdot a(\lambda)} dx+C_{g_1} \int_{|x|\geqslant l}|x+le|^{-r_{g_1}-p\cdot a(\lambda)}dx\\ &\leqslant& C_{g_1}(l+1)^{N-r_{g_1}-p\cdot a(\lambda)}\quad\text{for all }l\geqslant 1 \end{eqnarray} $

(4.7)

and

$ \begin{equation} \begin{aligned} \frac{t^r}{r} \int_{\mathbb{R}^N}g_2|u_l|^r dx & = \frac{t^r}{r} \int_{\mathbb{R}^N}g_2(x-le)|u_o|^r dx\geqslant(\min\limits_{x\in B^N(1)}u_0^r(x)) \int_{B^N(1)}g_2(x-le) dx\\ &\geqslant \bigg(\min\limits_{x\in B^N(1)}u_0^r(x)\bigg)c_{g_2} \int_{B^N(1)}l^{-r_{g_2}} dx\\ &\geqslant \bigg(\min\limits_{x\in B^N(1)}u_0^r(x)\bigg)c_{g_2}l^{-r_{g_2}}. \end{aligned} \end{equation} $

(4.8)

Since $ 0<r_{g_2}<\min\{r_f-N,r_{g_1}-N\} $ and $ t_1\leqslant t\leqslant t_2 $, by (4.1)–(4.8), we can find $ l_0>0 $ such that

$ \sup\limits_{t\geqslant 0}I_\mu(u_\mu^++tu_l)< \alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}\quad \text{for all }l>\max\{l_0,1\}. $

In order to complete the proof of Proposition 4.1, it remains to show that there exists a positive number $ t_* $ such that $ u_\mu^++t_*u_l\in N_\mu^- $. Let

$ \begin{aligned} &U_1 = \bigg\{u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\}\bigg| \frac{1}{\lVert u \rVert} t^-\bigg(\frac{u}{\lVert u \rVert}\bigg)>1\bigg\}\cup\{0\},\\ &U_2 = \bigg\{u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\}\bigg| \frac{1}{\lVert u \rVert} t^-\bigg(\frac{u}{\lVert u \rVert}\bigg)<1\bigg\}. \end{aligned} $

Then the manifold $ N_\mu^- $ divides $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ into two connected components $ U_1 $ and $ U_2 $, and $ \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus N_\mu^- = U_1 \cup U_2 $. For each $ u\in N_\mu^+ $, we have $ 1<t_{\max}(u)<t^-(u) $. Since $ t^-(u) = \frac{1}{\lVert u \rVert}t^-(\frac{u}{\lVert u \rVert}) $, we can obtain $ N_\mu^+\subset U_1 $ and so $ u_\mu^+\in U_1 $.

Next we claim that there exists $ t_0>0 $ such that $ u_\mu^++t_0 u_l\in U_2 $. In fact, we find a constant $ c>0 $ such that $ 0<t^-(\frac{u_\mu^++tu_l}{\lVert u_\mu^++tu_l \rVert})<c $ for each $ t>0 $. If not, then we may assume that there exists a sequence $ \{t_n\}_{n\in\mathbb{N}} $ such that $ t_n\rightarrow \infty $ and $ t^-(\frac{u_\mu^++t_nu_l}{\lVert u_\mu^++t_nu_l \rVert})\rightarrow \infty $ as $ n\rightarrow \infty $. Let $ v_n = \frac{u_\mu^++t_n u_l}{\lVert u_\mu^++t_n u_l \rVert} $. Since $ t^-(v_n)v_n\in N_\mu^- $ and by the Lebesgue dominated convergence theorem, we can deduce

$ \begin{aligned} \int_{\mathbb{R}^N}g v_n^r dx& = \frac{1} {\lVert u_\mu^++t_n u_l \rVert^r} \int_{\mathbb{R}^N}g (u_\mu^++t_n u_l)^r dx\\ & = \frac{1}{\lVert \frac{u_\mu^+}{t_n}+u_l \rVert^r} \int_{\mathbb{R}^N}g (\frac{u_\mu^+}{t_n}+u_l)^r dx\rightarrow \frac{ \int_{\mathbb{R}^N}g u_l^r dx} {\lVert u_l \rVert^r}\quad \text{as }n\rightarrow \infty. \end{aligned} $

Then we have

$ \begin{aligned} I_\mu(t^-(v_n)v_n)& = \frac{1}{p}(t^-(v_n))^p-\frac{(t^-(v_n))^q}{q} \int_{\mathbb{R}^N}f_\mu v_n^q dx-\frac{(t^-(v_n))^r}{r} \int_{\mathbb{R}^N}g v_n^r dx\\ &\rightarrow -\infty \text{ as } n\to \infty, \end{aligned} $

which contradicts the fact that $ I_\mu $ is bounded below on $ N_\mu $. Let

$ \begin{equation} t_0 = \bigg(\frac{( \int_{\mathbb{R}^N}|u_l|^r dx)^\frac{p}{r}+1}{\lVert u_l\rVert^p}|c^p-\lVert u_\mu^+ \rVert^p|\bigg)^\frac{1}{p}+1. \end{equation} $

(4.9)

By (4.4) and Lemma 3.4, we have, as $ l\rightarrow \infty $,

$ \begin{aligned} \lVert u_\mu^++t_0u_l \rVert^p& = \lVert u_\mu^+\rVert^p+t_0^p \lVert u_l\rVert^p+o(1) > \lVert u_\mu^+\rVert^p+|c^p-\lVert u_\mu^+ \rVert^p|+o(1)\\ &>c^p+o(1)>(t^-(\frac{u_\mu^++t_0u_l}{\lVert u_\mu^++t_0u_l \rVert}))^p+o(1). \end{aligned} $

Thus there exists $ l_0>0 $ such that for $ l>l_0 $, we get

$ \frac{1}{\lVert u_\mu^++t_0u_l \rVert}t^-(\frac{u_\mu^++t_0u_l}{\lVert u_\mu^++t_0u_l \rVert})<1 $

or $ u_\mu^++t_0u_l\in U_2 $. Define a path $ \gamma(s) = u_\mu^++st_0u_l $ for $ s\in [0,1] $, and so

$ \gamma(0) = u_\mu^+\in U_1,\quad \gamma(1) = u_\mu^++t_0u_l\in U_2. $

By Lemma 2.4, we have $ \frac{1}{\lVert u \rVert}t^-(\frac{u}{\lVert u \rVert}) $ is a continuous function for $ u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\setminus\{0\} $ and $ \gamma([0,1]) $ is connected. Then there exists $ s_0\in(0,1) $ such that $ u_\mu^++s_0t_0u_l\in N_\mu^- $. Take $ t_* = s_0t_0 $ and this proof is completed.

Then we have the following result.

Theorem 4.2 For $ \mu\in(0,L_2) $, $ \rm(1.1) $ has a positive solution $ u_\mu^-\in N_\mu^- $ such that $ I_\mu(u_\mu^-) = \alpha^- $.

Proof By Ekeland's variational principle [22], there exists a minimizing sequence $ \{u_n\}_{n\in\mathbb{N}}\subset N_\mu^- $ such that

$ I_\mu(u_n) = \alpha^-+o(1)\quad \text{and }\quad I'_\mu(u_n) = o(1)\quad \text{in }(\mathbf{W}_a^{1,p}(\mathbb{R}^N))^{-1}. $

Since $ \alpha^-<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $, by Lemma 2.3 and Proposition 3.2, there exists a subsequence $ \{u_n\}_{n\in\mathbb{N}} $ and a non-zero solution $ u_\mu^-\in N_\mu^- $ of (1.1) such that as $ n\to\infty $, it holds

$ u_n\rightarrow u_\mu^-\quad \text{in }\mathbf{W}_a^{1,p}(\mathbb{R}^N). $

Since $ I_\mu(u_\mu^-) = I_\mu(|u_\mu^-|) $ and $ |u_\mu^-|\in N_\mu^- $, $ u_\mu^- $ is a positive solution of (1.1). We finish the proof.

In this section, we discuss the concentration behavior of solutions to (1.1) so that we can get the proof of Theorem 1.1 $ \rm(ⅱ) $.

Lemma 5.1 We have

$ \begin{equation} \inf\limits_{u\in N_0}I_0(u) = \inf\limits_{u\in N^\infty}I^\infty(u) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}. \end{equation} $

(5.1)

Furthermore, $ \rm(1.1) $ does not admit any solution $ w_0\in\mathbf{W}_a^{1,p}(\mathbb{R}^N) $ such that $ I_0(w_0) = \inf\limits_{u\in N_0}I_0(u). $

Proof By Lemma 2.4, there exists the unique $ t^-(u_l)>0 $ such that $ t^-(u_l)u_l\in N_0 $ for all $ l>0 $, that is,

$ \begin{equation} \lVert t^-(u_l)u_l \rVert^p = \int_{\mathbb{R}^N}f_-|t^-(u_l)u_l|^q dx+ \int_{\mathbb{R}^N}g|t^-(u_l)u_l|^r dx. \end{equation} $

(5.2)

Since

$ \begin{equation} \lVert u_l \rVert^p = \int_{\mathbb{R}^N}|u_l|^r dx = S_\lambda^\frac{r}{r-p} \quad\text{for all }l\geqslant 0 \end{equation} $

(5.3)

and

$ \begin{equation} \int_{\mathbb{R}^N}f_-|u_l|^q dx\rightarrow 0\;\text{and } \int_{\mathbb{R}^N}(1-g)|u_l|^r dx\rightarrow 0\quad \text{as }l\rightarrow \infty. \end{equation} $

(5.4)

By (5.2)–(5.4), we have $ t^-(u_l)\rightarrow 1 $ as $ l\rightarrow \infty $. Thus

$ \begin{equation} \lim\limits_{l\to \infty}I_0(t^-(u_l)u_l) = \lim\limits_{l\to \infty}I^\infty(t^-(u_l)u_l) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}. \end{equation} $

(5.5)

Then we can obtain

$ \begin{equation} \inf\limits_{u\in N_0}I_0(u)\leqslant (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} = \inf\limits_{u\in N^\infty}I^\infty(u). \end{equation} $

(5.6)

For $ u\in N_0 $, by Lemma 2.4 $ \rm{(ⅰ)} $,

$ \begin{equation} I_0(u) = I_0(t^-(\frac{u}{\lVert u \rVert})\frac{u}{\lVert u \rVert}) = \sup\limits_{t\geqslant 0}I(tu). \end{equation} $

(5.7)

Moreover, there exists a unique $ t^\infty>0 $ such that $ t^\infty u\in N^\infty $. Thus,

$ \begin{equation} I_0(u) = I_0(t^\infty u)\geqslant I^\infty(t^\infty u)\geqslant (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} \end{equation} $

(5.8)

and so $ \inf\limits_{u\in N_0}I_0(u)\geqslant (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $. Then we have

$ \begin{equation} \inf\limits_{u\in N_0}I_0(u) = \inf\limits_{u\in N^\infty}I^\infty(u) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}. \end{equation} $

(5.9)

In order to show that (1.1) does not admit any solution $ w_0 $ such that $ I_0(w_0) = \inf\limits_{u \in N_0}I_0(u) $, we argue by the contrary. By Lemma 2.4 $ \rm{(ⅰ)} $, we have $ I_0(w_0) = \sup\limits_{t\geqslant 0}I_0(tw_0) $. Moreover, there exists a unique $ t_{w_0}>0 $ such that $ t_{w_0}w_0\in N^\infty $. Thus we obtain

$ \begin{equation} (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} = \inf\limits_{u\in N_0}I_0(u) = I_0(w_0) = I^\infty(t_{w_0}w_0)-\frac{1}{q} \int_{\mathbb{R}^N}f_-|t_{w_0}w_0|^q dx, \end{equation} $

(5.10)

and this implies $ \int_{\mathbb{R}^N}f_-|w_0|^q dx = 0 $, that is, $ w_0\equiv 0 $ in $ \{x\in \mathbb{R^N}|f_-(x)\not = 0\} $ from (A$ _1) $. Then we can obtain

$ (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} = \inf\limits_{u\in N^\infty}I^\infty(u) = I^\infty(t_{w_0}w_0). $

By the Lagrange multiplier and the maximum principle, we may assume that $ t_{w_0}w_0 $ is a positive solution of (1.1). This contradiction completes the proof.

Lemma 5.2 Assume that $ \{u_n\} $ is a minimizing sequence in $ N_0 $ for $ I_0 $. Then

$ \rm{(ⅰ)} $ $ \int_{\mathbb{R}^N}f_-|u_n|^q dx = o(1); $

$ \rm{(ⅱ)} $ $ \int_{\mathbb{R}^N}(1-g)|u_n|^r dx = o(1). $

Furthermore, $ \{u_n\} $ is a (P.S.)$ _{(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}} $-sequence in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ for $ I^\infty $.

Proof For each $ n $, there exists a unique $ t_n>0 $ such that $ t_nu_n\in N^\infty $, that is,

$ t_n^p\lVert u_n \rVert^p = t_n^r \int_{\mathbb{R}^N}|u_n|^r dx. $

By Lemma 2.4 $ \rm{(ⅰ)} $, we have

$ \begin{aligned} I_0(u_n)&\geqslant I_0(t_nu_n) = I^\infty(t_nu_n)-\frac{t_n^q}{q} \int_{\mathbb{R}^N}f_-|u_n|^q dx+ \frac{t_n^r}{r} \int_{\mathbb{R}^N}(1-g)|u_n|^r dx\\ &\geqslant(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}-\frac{t_n^q}{q} \int_{\mathbb{R}^N}f_-|u_n|^q dx+\frac{t_n^r}{r} \int_{\mathbb{R}^N}(1-g)|u_n|^r dx. \end{aligned} $

Since $ I_0(u_n) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+o(1) $ from Lemma 5.1, we have, as $ n\to \infty $,

$ \frac{t_n^q}{q} \int_{\mathbb{R}^N}f_-|u_n|^q dx = o(1) $

and

$ \frac{t_n^r}{r} \int_{\mathbb{R}^N}(1-g)|u_n|^r dx = o(1). $

Next, we will show that there exists $ M>0,\;c_0>0 $ such that $ t_n>c_0 $ for $ n>M $. Suppose the contrary. Then we may assume $ t_n\rightarrow 0 $ as $ n\rightarrow \infty $. As in the proof of Lemma 2.3, we know that $ \lVert u_n \rVert $ is uniformly bounded and so $ \lVert t_nu_n \rVert\rightarrow 0 $ or $ I^\infty(t_nu_n)\rightarrow 0 $. This contradicts the fact $ I^\infty(t_nu_n)\geqslant (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}>0 $ from Lemma 5.1. Then we have

$ \int_{\mathbb{R}^N}f_-|u_n|^q dx = o(1) $

and

$ \int_{\mathbb{R}^N}(1-g)|u_n|^r dx = o(1). $

This implies

$ \lVert u_n \rVert^p = \int_{\mathbb{R}^N}|u_n|^r dx+o(1) $

and so

$ I^\infty(u_n) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+o(1), $

that is, $ \{u_n\} $ is a (P.S.)$ _{(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}} $-sequence in $ \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ for $ I^\infty $. This completes the proof.

Let

$ N_\mu^-(d) = \{u\in N_\mu^-|I_\mu(u)\leqslant (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+d\} \quad\text{for }d<0, $

be the filtration of the Nehari manifold $ N_\mu $. Then we have the following lemmas.

Lemma 5.3 There exists $ d_0<0 $ such that for $ u\in N_0(d_0) $, we have

$ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{-ap}|\nabla u|^p-\frac{\lambda}{|x|^{p(a+1)}}u^p) dx\not = 0. $

Proof Suppose the contrary. We may assume that there exists a sequence $ \{u_n\}_{n\in\mathbb{N}}\subset N_0 $ such that $ I_0(u_n) = (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+o(1) $ and $ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{1-ap}|\nabla u_n|^p-\frac{\lambda}{|x|^{p(a+1)}}u_n^p) dx = o(1) $. By Proposition 3.2 and the concentration-compactness principle (see [23, Theorem 4.1]), there exists a sequence $ \{x_n\}_{n\in\mathbb{N}}\subset\mathbb{R}^N $ such that

$ \begin{equation} \lVert u_n(x)-u_0(x-x_n) \rVert\rightarrow 0 \quad \text{as }n\rightarrow \infty. \end{equation} $

(5.11)

Now we will show that $ |x_n|\rightarrow \infty $ as $ n\rightarrow \infty $ by contradiction. We may assume that $ \{x_n\} $ is bounded and $ x_n\rightarrow x_* $ for some $ x_*\in \mathbb{R}^N $. Then by (5.11),

$ \begin{aligned} \int_{\mathbb{R}^N}f_-|u_n|^q dx& = \int_{\mathbb{R}^N}f_-(x)|u_0(x-x_n)|^q dx+o(1)\\ & = \int_{\mathbb{R}^N}f_-(x+x_*)|u_0(x)|^q dx+o(1), \end{aligned} $

this contradicts the result of Lemma 5.2 $ \rm{(ⅰ)} $. Hence we may assume $ \frac{x_n}{|x_n|}\rightarrow e $ as $ n\rightarrow \infty $, where $ e\in\mathbb{S}^{N-1} $. By the Lebesgue dominated convergence theorem, we have

$ \begin{aligned} o(1)& = \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x^{1-ap}||\nabla u_n|^p-\frac{\lambda}{|x|^{-ap}}|u_n|^p) dx\\ & = \int_{\mathbb{R}^N}\frac{x+x_n}{|x+x_n|^{1-ap}}(|x+x_n|^{-ap}|\nabla u_0|^p-\frac{\lambda}{|x+x_n|^{p(a+1)}}|u_0|^p) dx\\ & = \int_{\mathbb{R}^N}e|\nabla u_0|^p dx+o(1). \end{aligned} $

This contradiction completes the proof.

By (2.1) and Lemma 2.4 $ \rm{(ⅰ)} $, for each $ u\in N_\mu^- $, there exists the unique $ t_0^-(u)>0 $ such that $ t_0^-(u)u\in N_0 $ and $ t_0^-(u)>t_{\max}(u)>0 $. Then we have the following result.

Lemma 5.4 Let

$ T = \frac{r-q}{p-q}(1+\frac{r-p}{r-q}). $

For each $ \mu\in(0,L_2) $ and $ u\in N_\mu^-(\alpha^+) $, we have $ t_0^-(u)<T^\frac{1}{r-p} $.

Proof For $ u\in N_\mu^-(\alpha^+) $, we distinguish from the following distinctive cases.

Case ($ \rm{ⅰ} $) $ t_0^-(u)<1 $. Since $ T>1 $, we have $ t_0^-(u)<1<T^\frac{1}{r-p} $.

Case ($ \rm{ⅱ} $) $ t_0^-(u)\geqslant 1 $. Since

$ \begin{aligned} (t_0^-(u))^r \int_{\mathbb{R}^N}g|u|^r dx& = (t_0^-(u))^p\lVert u \rVert^p-(t_0^-(u))^q \int_{\mathbb{R}^N}f_-|u|^q dx\\ &\leqslant (t_0^-(u))^p(\lVert u \rVert^p+ \int_{\mathbb{R}^N}|f_-||u|^q dx) \end{aligned} $

and by Lemma 2.2 $ \rm{(ⅲ)} $, we have

$ \begin{equation} (t_0^-(u))^{r-p}\geqslant \frac{\lVert u \rVert^p+ \int_{\mathbb{R}^N}|f_-||u|^q dx}{ \int_{\mathbb{R}^N}g|u|^r dx}. \end{equation} $

(5.12)

Moreover, from the argument in the proof of Lemma 2.2, we have

$ \begin{eqnarray} && \lVert u \rVert\leqslant \frac{r-q}{r-p} \int_{\mathbb{R}^N}g|u|^r dx, \end{eqnarray} $

(5.13)

$ \begin{eqnarray} &&\lVert u \rVert\geqslant \frac{r-q}{r-p} \int_{\mathbb{R}^N}|f_-||u|^q dx. \end{eqnarray} $

(5.14)

Thus, by (5.12)–(5.14), we have

$ \begin{aligned} (t_0(u))^{r-p}&\leqslant \frac{\lVert u \rVert^p}{ \int_{\mathbb{R}^N}g|u|^r dx}\cdot \frac{1}{\lVert u \rVert^p}(\lVert u \rVert^p+ \int_{\mathbb{R}^N}|f_-||u|^q dx)\\ &\leqslant \frac{r-q}{p-q}(1+\frac{ \int_{\mathbb{R}^N}|f_-||u|^q dx}{\lVert u \rVert^p})\leqslant \frac{r-q}{p-q}(1+\frac{r-p}{r-q}). \end{aligned} $

This completes the proof.

Lemma 5.5 There exists $ \mu_0\in (0,L_2) $ such that for each $ \mu\in(0,\mu_0) $ and $ u\in N_\mu^-(\alpha^+) $,

$ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{-ap}|\nabla u|^p-\lambda\frac{|u|^p}{|x|^{p(a+1)}}) dx\not = 0. $

Proof For $ u\in N_\mu^-(\alpha^+) $, by Lemma 2.4 $ \rm{(ⅰ)} $, there exists $ t_0^-(u)>0 $ such that $ t_0^-(u)u\in N_0 $. Moreover, by Lemma 5.4 and the Hölder inequality and Sobolev embedding theorem, we have

$ \begin{aligned} I_\mu(u) = \sup\limits_{t\geqslant 0}I_\mu(tu)\geqslant I_\mu(t_0^-(u)u) = I_0(t_0^-(u)u)-\frac{\mu(t_0^-(u))^q}{q} \int_{\mathbb{R}^N}f_+|u|^q dx \end{aligned} $

or so

$ \begin{aligned} I_0(t_0^-(u)u)&\leqslant I_\mu(u)+\frac{\mu(t_0^-(u))^q}{q} \int_{\mathbb{R}^N}f_+|u|^q dx\\ &<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+\frac{\mu T^\frac{q}{r-p}}{q}\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}}\lVert u \rVert^q. \end{aligned} $

Since $ I_\mu(u)<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}< (\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} $, by Lemma 2.1, for $ \mu\in(0,L_2) $ and $ u\in N_\mu^-(\alpha^+) $, there exists $ c_* $ independent of $ \mu $ such that $ \lVert u \rVert\leqslant c_* $. Thus,

$ I_0(t_0^-(u)u)<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}+\frac{\mu T^\frac{q}{r-p}}{q}\lVert f_+ \rVert_{L^{q^*}}S_\lambda^{-\frac{q}{p}}c_*^q. $

Then by Lemma 5.3, we have

$ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{-ap}|\nabla(t_0^-(u)u)|^p- \frac{\lambda}{|x|^{p(a+1)}}|t_0^-(u)u|^p) dx\not = 0 $

and this implies

$ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{-ap}|\nabla u|^p-\frac{\lambda}{|x|^{p(a+1)}}|u|^p) dx\not = 0\quad \text{for }u\in N_\mu^-(\alpha^+). $

The proof is completed.

In this section, we will follow an idea in [24] to prove our main result. For $ c\in \mathbb{R}^+ $, we denote

$ [I_\mu\leqslant c] = \{u\in N_\mu^-|u\geqslant 0,\ I_\mu(u)\leqslant c\}. $

Then we try to show that for a sufficiently small $ \sigma>0 $, we have

$ \begin{equation} \hbox{cat}([I_\mu\leqslant \alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}-\sigma])\geqslant 2 \end{equation} $

(6.1)

Here 'cat' means the Lusternik-Schnirelmann category [15]. First, let us recall its definition.

Definition 6.1 A non-empty, closed subset $ Y $ is contractible in a topological space $ \mathbf{X} $ if there exists $ h\in\mathbf{C}([0,1]\times Y,\;\mathbf{X}) $ such that for some $ x_0\in \mathbf{X} $,

$ h(0,x) = x,\quad h(1,x) = x_0. $

Definition 6.2 Let $ Y_1,Y_2,\cdots,Y_k $ be closed subsets of a topological space $ \mathbf{X} $. The category of $ \mathbf{X} $ is the least integer $ k $ such that $ Y_j $ is contractible in $ \mathbf{X} $ for all $ j $ and $ \cup_{j = 1}^k Y_j = \mathbf{X} $, denoted by cat$ (\mathbf{X}) $.

When there do not exist finitely many closed subsets $ Y_1,Y_2,\cdots,Y_ k\subset\mathbf{X} $ such that $ Y_j $ is contractible in $ \mathbf{X} $ for all $ j $ and $ \bigcup\limits_{j = 1}^k Y_j = \mathbf{X} $, we denote cat$ (\mathbf{X}) = \infty $. We need the following lemmas (see Theorem 2.3 in [25] and Lemma 2.5 in [24]).

Lemma 6.3 Let $ \mathbf{X} $ be a Hilbert manifold and $ F\in C^1(\mathbf{X},\mathbb{R}) $. Assume that there are $ c_0\in \mathbb{R} $ and $ k\in \mathbf{N} $ such that

$ \rm{(ⅰ)} $ $ F $ satisfies the Palais-Smale condition for energy level $ c\leqslant c_0 $;

$ \rm{(ⅱ)} $ cat$ (\{x\in \mathbf{X}|F(x)\leqslant c_0\})\geqslant k $.

Then $ F $ has at least $ k $ critical points in $ \{x\in \mathbf{X}|F(x)\leqslant c_0\} $.

Lemma 6.4 Let $ \mathbf{X} $ be a topological space. Assume that there are $ \varphi \in C(\mathbb{S}^{N-1},\mathbf{X}) $ and $ \psi \in C(\mathbf{X},\mathbb{S}^{N-1}) $ such that $ \psi \circ \varphi $ is homotopic to the identity map of $ \mathbb{S}^{N-1} $, that is, there exists $ h\in C([0,1]\times \mathbb{S}^{N-1},\mathbb{S}^{N-1}) $ such that $ h(0,x) = (\psi \circ \varphi)(x),\; \; h(1,x) = x. $ Then cat$ (\mathbf{X})\geqslant 2. $

For $ l>l_0 $, we define a map $ \varphi_\mu: \mathbb{S}^{N-1}\rightarrow \mathbf{W}_a^{1,p}(\mathbb{R}^N) $ by

$ \varphi_\mu(e)(x) = u_\mu^++t_*u_l\quad\text{for }e\in \mathbb{S}^{N-1}, $

where $ u_\mu^++t_*u_l $ is as in the proof of Proposition 4.1. Then we have the following result.

Lemma 6.5 There exists a sequence $ \{\sigma_l\}\subset \mathbb{R}^+ $ with $ \sigma_l\rightarrow 0 $ as $ l\rightarrow \infty $ such that

$ \varphi_\mu(\mathbb{S}^{N-1})\subset [I_\mu\leqslant\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}-\sigma_l]. $

Proof By Proposition 4.1, for $ l>l_0 $, we have $ u_\mu^++t_*u_l\in N_\mu^- $ and

$ \sup\limits_{t\geqslant 0}I_\mu(u_\mu^++tu_l)<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p} \quad \text{uniformly in }e\in \mathbb{S}^{N-1}. $

Since $ \varphi_\mu(\mathbb{S}^{N-1}) $ is compact and $ I_\mu(u_\mu^++t_*u_l)\leqslant \alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}-\sigma_l $, the conclusion holds.

From Lemma 5.5, we define a barycenter map, $ \psi_\mu:[I_\mu<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}]\rightarrow \mathbb{S}^{N-1} $ by

$ \psi_\mu(u) = \frac{ \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x|^{-ap}|\nabla u|^p-\lambda \frac{|u|^p}{|x|^{p(a+1)}}) dx}{| \int_{\mathbb{R}^N}\frac{x}{|x|^{1-ap}}(|x^{-ap}||\nabla u|^p-\lambda\frac{|u|^p}{|x|^{p(a+1)}}) dx|}. $

Then we have the following result.

Lemma 6.6 Let $ \mu_0 $ be as in Lemma 5.5. Then for $ \mu\in(0,\mu_0) $, there exists $ l_*>l_0 $ such that the map

$ \psi_\mu \circ \varphi_\mu:\mathbb{S}^{N-1}\rightarrow \mathbb{S}^{N-1}\quad\text{for }l>l_* $

is homotopic to the identity operator.

Proof Denote

$ \text{supp }\psi_\mu = \{u\in \mathbf{W}_a^{1,p}(\mathbb{R}^N)\backslash\{0\}| \int_{\mathbb{R}^N}\frac{x}{|x|}(|\nabla u|^p-\lambda\frac{|u|^p}{|x|^p}) dx\not = 0\} $

and define $ \widetilde{\psi_\mu}:\text{supp }\psi_\mu\rightarrow \mathbb{S}^{N-1} $ by

$ \widetilde{\psi_\mu}(u) = \frac{ \int_{\mathbb{R}^N}\frac{x}{|x|}(|\nabla u|^p-\lambda\frac{|u|^p}{|x|^p}) dx}{| \int_{\mathbb{R}^N}\frac{x}{|x|}(|\nabla u|^p-\lambda\frac{|u|^p}{|x|^p}) dx|} $

as an extension of $ \psi_\mu $. Since $ u_l\in\text{supp }\psi_\mu $ for all $ e\in \mathbb{S}^{N-1} $ and large enough $ l $, we may assume $ \gamma:[s_1,s_2]\rightarrow \mathbb{S}^{N-1} $ is a regular geodesic between $ \psi_\mu(u_l) $ and $ \widetilde{\psi_\mu}(\varphi_\mu(e)) $ such that

$ \gamma(s_1) = \psi_\mu(u_l),\quad \gamma(s_2) = \widetilde{\psi_\mu}(\varphi_\mu(e)). $

By an argument similar to Lemma 5.3, there exists $ l_*\geqslant l_0 $ such that

$ u_0(x+\frac{l}{2(1-\theta)}e)\in \text{supp }\psi_\mu $

for all $ e\in\mathbb{S}^{N-1},\ l>l_* $ and $ \theta\in[\frac{1}{2},1) $. We define

$ h_l(\theta,e):[0,1]\times \mathbb{S}^{N-1}\rightarrow \mathbb{S}^{N-1} $

by

$ h_l(\theta,e) = \left\{ \begin{aligned} &\gamma(2\theta(s_1-s_2)+s_2)&\text{for }\theta\in[0,\frac{1}{2}),\\ &\widetilde{\psi_\mu}(u_0(x+\frac{l}{2(1-\theta)}e))&\text{for }\theta\in [\frac{1}{2},1),\\ &e&\text{for }\theta = 1. \end{aligned} \right. $

Then $ h_l(0,e) = \widetilde{\psi_\mu}(\varphi_\mu(e)) $ and $ h_l(1,e) = e $. By the standard regularity, we have $ h_l(\theta,e)\in C(\mathbb{R}^N) $.

Next, we will show that $ \lim\limits_{\theta\rightarrow 1^-}h_l(\theta,e) = e $ and $ \lim\limits_{\theta\rightarrow \frac{1}{2}^-}h_l(\theta,e) = \widetilde{\psi_\mu}(u_l). $

$ \rm{(ⅰ)} $ $ \lim\limits_{\theta\rightarrow 1^-}h_l(\theta,e) = e $, since

$ \begin{aligned} &\quad \int_{\mathbb{R}^N}\frac{x}{|x|}(|\nabla u_0(x+\frac{l}{2(1-\theta)}e)|^p- \frac{\lambda}{|x+\frac{l}{2(1-\theta)}e|^p} u_0^p(x+\frac{l}{2(1-\theta)}e)) dx\\ & = \int_{\mathbb{R}^N}\frac{x+\frac{l}{2(1-\theta)}e} {|x+\frac{l}{2(1-\theta)}e|}(|\nabla u_0(x)|^p- \frac{\lambda}{|x+\frac{l}{2(1-\theta)}e|^p}u_0^p(x)) dx\\ & = e \int_{\mathbb{R}^N}|\nabla u_0|^p dx+o(1) \end{aligned} $

as $ \theta\rightarrow 1^- $.

$ \rm{(ⅱ)} $ $ \lim\limits_{\theta\rightarrow \frac{1}{2}^-}h_l(\theta,e) = \widetilde{\psi_\mu}(u_l) $. Since $ \widetilde{\psi_\mu}\in C(\text{supp }\psi_\mu,\mathbb{S}^{N-1}) $, then we have

$ h_l(\theta,e)\in C([0,1]\times \mathbb{S}^{N-1},\mathbb{S}^{N-1}) $

and

$ h_l(0,e) = \psi_\mu(\varphi_\mu(e)),\quad h_l(1,e) = e, $

for all $ e\in \mathbb{S}^{N-1} $ and $ l>l_* $. This completes the proof.

Lemma 6.7 For $ \mu\in (0,\mu_0) $ and $ l>l_* $, the energy functional $ I_\mu $ admits at least two critical points in $ [I_\mu<\alpha^++(\frac{1}{p}-\frac{1}{r})S_\lambda^\frac{r}{r-p}] $.

Proof It is easy to deduce from Lemmas 6.3, 6.4, 6.6 and Proposition 3.2.

Proof of Theorem 1.1 Now we can complete the proof of Theorem 1.1

$ \rm{(ⅰ)\; \; } $by Proposition 3.3 and Theorem 4.2;

$ \rm{(ⅱ)\; \; } $from Proposition 3.3 and Lemma 6.7, (1.1) has at least three positive solutions $ u_\mu^+,u_1^-,u_2^- $, where $ u_\mu^+\in N_\mu^+ $ and $ u_i^-\in N_\mu^- $ for $ i = 1,2 $.