We consider the solitary wave solutions for quasilinear Schrödinger equations of the form

$ \begin{equation} \begin{array}{ll} i z_{t}=-\Delta z+W(x)z-k(x, z)-\Delta l(|z|^2)l'(|z|^2)z, \end{array} \end{equation} $

(1.1)

where $ z:\, \mathbb{R}\times \mathbb{R}^N\to \mathbb{C} $ is a complex function, $ W:\, \mathbb{R}^N\to \mathbb{R} $ is a given potential function, $ k: \mathbb{R}^N\times \mathbb{R} \rightarrow \mathbb{R} $ and $ l: \mathbb{R}\rightarrow \mathbb{R} $ are suitable functions. Quasilinear equation (1.1) has been derived as models of several physical phenomena, for which we can refer to [1-8] and references therein. For example, for the special case $ l(s)=s $, quasilinear equation(1.1) which has been called the superfluid film equation in fluid mechanics by Kurihara [9], can model the time evolution of the condensate wave function in super-fluid film ([9, 10]). For the special case $ l(s)= (1 + s)^{1/2} $, quasilinear equation(1.1) can model the self-channeling of a high-power ultra short laser in matter. Propagation of a high irradiance laser in a plasma produces an optical refractive index which is nonlinear related to the intensity of the light and gives rise to an interesting new nonlinear wave equation. (see [11-14]).

Set $ z(t, x) = \exp(-iEt)u(x) $, where $ E $ is a real number and $ u(x) $ is a real function, equation (1.1) can be simplified to a quasilinear elliptic equation (see [15])

$ \begin{equation} \begin{array}{ll} -\Delta u +V(x)u- \Delta l(u^2)l'(u^2)u=k(x, u), \quad x\in \mathbb{R}^N. \end{array} \end{equation} $

(1.2)

Thus, in order to find the solitary wave solution of (1.1), only the positive solution of the quasilinear elliptic equation (1.2) is required. If $ l(s)=s $, the superfluid film equation in plasma physics can be obtained as follows:

$ \begin{equation} \begin{array}{ll} -\Delta u +V(x)u-\Delta(u^2)u=k(x, u), \quad x\in \mathbb{R}^N. \end{array} \end{equation} $

(1.3)

Recent studies mainly focus on equation (1.3) with $ k(x, u)= |u|^{q-2}u $ at infinity for $ 4\leq q<22^*, \ N\geq 3 $, where $ 2^*=2N/(N-2) $ is the Sobolev critical exponent. In the spirit of [20], such a nonlinear term is called subcritical growth. In [7] and [16], the existence of a positive ground state solution of the equation (1.3) has been proved. By using a constraint minimization argument, a solution of the equation with unknown Lagrangian multiplier $ \lambda $ in front of the nonlinear term has been given. In [17], the quasilinear equation is transformed into a semilinear equation by a variable transformation. Using an Orlitz space frame as the workspace, a positive solution of equation (1.3) is obtained according to the mountain-Pass lemma(e.g., [18]). This method was later used for subcritical growth in [19]. Along this line of thought, one could also look for a sign-changing solution. For instance, in [20], Liu et al. used Nehari's method to deal with more general quasilinear equations and obtained positive and sign-changing solutions. Recently, the first two authors and Wang obtained infinitely many nodal radial solutions of the equation (1.3) through a construction argument in [21].

As shown in [20], the number $ 22^* $ is similar to the critical exponent of equation (1.3). As a matter of fact, in [20], using the variational identity given by Pucci and Serrin [22], it was proved that equation (1.3) has no positive solutions in $ H^1( \mathbb{R}^N) $ with $ u^2|\nabla u|^2\in L^1( \mathbb{R}^N) $ if $ k(u)=|u|^{p-2}u $, $ p\ge 22^* $ and $ \nabla V(x)\cdot x\ge 0 $ in $ \mathbb{R}^N $. As in [17] Liu et al. pointed out, the critical case for equation (1.3) is as good as a play. In this critical case, Moanemi dealt with the related singularly perturbed equation in reference [23], and obtained a positive radial solution in the case of radial symmetry. Later, in [24] a positive solution was proved to exist according to the mountain pass Lemma. Recently, Liu et al. used a perturbation method to obtain a positive solution of the general quasilinear elliptic equation like (1.3) in [25]. The existence of the nodal solution of the equation (1.3) with critical growth is studied by the variational method in [26].

We note that all of the above results are for the special case $ l(s)=s $. A very natural question is whether there is a general way to study the equation (1.1) for the general function $ l(s) $.

In [27], to handle the general case, Shen and Wang introduce the following new variable substitution

$ g^2(u)=1+\frac{(l(u^2)')^2}{2}. $

Using this transformation, we can reduce (1.2) to quasilinear elliptic equation

$ \begin{equation} \begin{array}{ll} -\text{div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+V(x)u=k(x, u), \quad x\in \mathbb{R}^N. \end{array} \end{equation} $

(1.4)

Setting $ g^2(u)=1+2u^2 $, i.e., $ l(s)=s $, we can get equation (1.3). Setting $ g^2(u)=1+\frac {u^2}{2(1+u^2)} $, i.e., $ l(s)=(1+s)^{\frac 12} $, we can get the quasilinear Schrödinger equation

$ \begin{equation} \begin{array}{ll} -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=k(x, u), \quad x\in \mathbb{R}^N \end{array} \end{equation} $

(1.5)

which models the self-channeling of a high-power ultrashort laser in matter.

According to this line of thought, in [27], Shen and Wang obtained a positive solitary wave solution of (1.2) with a general function $ l(s) $ under some assumptions on $ g $, $ V $, and $ k $ which is a function of subcritical growth. In [21], Deng et al. obtained node solutions of equation (1.4) also with subcritical growth function $ k(x, u) $.

What is the critical exponent of the quasilinear Schrödinger equation (1.2) (or (1.4)) with general function $ l(s) $ (or $ g(s) $)? What about the existence of positive solutions for such an equation with the critical exponent? These questions have recently been addressed by the authors in [28]. To be more precise, the critical exponents $ \alpha 2^* $ of equation (1.4) with the general function $ g(s) $ is obtained if $ g(s) $ satisfies $ \lim\limits_{t \rightarrow +\infty} \frac {g(t)}{t^{\alpha -1}} = \beta >0 $ for some $ \alpha \ge 1 $ (see [28]).

In recent years, we've found that there seems to be little progress on the existence of positive ground state solutions for equation (1.4) with a Hardy-type term.

In the present paper, we assume $ V(x)\equiv 1 $ and $ k(x, t)=h(t)+\frac{g(t)|G(t)|^{2^*(a)-2}G(t)}{|x|^a} $. Whereupon, the quasilinear Schrödinger equation (1.4) can be rewritten as

$ \begin{equation} \begin{array}{ll} -{div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+ u=h(u) +\frac{g(u)|G(u)|^{2^*(a)-2}G(u)}{|x|^a}, \quad x\in \mathbb{R}^N, \end{array} \end{equation} $

(1.6)

where $ N\geq3 $, $ \ 2^*(a)=\frac {2(N-a)}{N-2} $, $ 0\leq a<2 $, $ G(t)=\int_0^t g(\tau)d\tau $ and $ h: \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function.

In order to establish the existence of positive radial ground state solutions for equation (1.6), we need to make some assumptions about $ g(t) $ and $ h(t) $.

$ (g_1) $   $ g\in C^1( \mathbb{R}) $ is a positive even function and $ g'(t)\ge 0 $ for $ \forall t\ge 0 $, $ g(0)=1 $;

$ (h_1) $  $ h(t)\ge 0 $ is differentiable for all $ t\in [0, \ +\infty) $. Moreover, we extend $ h(t)\equiv 0 $ for all $ t\in (-\infty, 0) $;

$ (h_2) $ $ \lim\limits_{t\rightarrow +\infty}\frac{h(t)}{g(t)|G(t)|^{2^*(a)-1}}=0 $ and $ \lim\limits_{t\rightarrow 0^+}\frac{h(t)}{g(t)G(t)}=0 $;

$ (h_3) $  There exists $ \delta \in (0, 2^*(a)-2) $ such that for $ \forall t>0 $, there holds $ (1+\delta) h( t)\le G(t) (\frac {h(t)}{g(t)})' $;

Denoting $ H(u)=\int_0^u h(\tau)d\tau $, we find that the natural variational functional

$ I(u)=\frac{1}{2}\int_{ \mathbb{R}^N}g^2(u)|\nabla u|^2dx+\frac{1}{2}\int_{ \mathbb{R}^N}|u|^2dx-\int_{ \mathbb{R}^N}H(u)dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|G(u)|^{2^*(a)}}{|x|^a} dx $

corresponding to (1.6) may be not well defined in $ H^1( \mathbb{R}^N) $. In order to overcome this difficulty, we need to make a variable substitution which was constructed by Shen and Wang in [27], as

$ v=G(u)=\int_0^u g(t)dt. $

And then we can obtain

$ J(v)=\frac{1}{2}\int_{ \mathbb{R}^N}|\nabla v|^2dx+\frac{1}{2}\int_{ \mathbb{R}^N}|G^{-1}(v)|^2dx-\int_{ \mathbb{R}^N}H(G^{-1}(v))dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|v|^{2^*(a)}}{|x|^a} dx. $

Since $ g(t) $ satisfies the assumption $ (g_{1}) $, we can get $ |G^{-1}(v)|\leq \frac{1}{g(0)}|v|=|v|. $ It thus appears that, the functional $ J(v) $ is well defined in $ H^1( \mathbb{R}^N) $ and $ J\in C^1(H^1( \mathbb{R}^N), \mathbb{R}) $ if the function $ h(t) $ satisfies the assumption $ (h_{2}) $.

If $ u $ is a nontrivial solution of equation (1.6), then it should satisfy

$ \begin{equation} \begin{array}{ll} \int_{ \mathbb{R}^N}[g^{2}(u)\nabla u \nabla \varphi +g(u)'|\nabla u|^{2}\varphi+u\varphi-h(u)\varphi+\frac{g(u)|G(u)|^{2^*(a)-2}G(u)}{|x|^a}\varphi]dx=0 \end{array} \end{equation} $

(1.7)

for $ \forall\varphi \in C_{0}^{\infty}( \mathbb{R}^{N}). $

Let $ \varphi=\frac{1}{g(u)}\psi $, we immediately know (see[27]) that equation (1.7) is equivalent to

$ \begin{equation} \begin{array}{ll} \langle J'(v), \psi \rangle =\int_{ \mathbb{R}^N}[\nabla v \nabla\psi+\frac{G^{-1}(v)}{g(G^{-1}(v))}\psi-\frac{h(G^{-1}(v))}{g(G^{-1}(v))}\psi-\frac{|v|^{2^{*}(a)-2}v\psi}{|x|^{a}}]dx=0 \end{array} \end{equation} $

(1.8)

for $ \forall\psi \in C_{0}^{\infty}( \mathbb{R}^{N}). $

Consequently, to find the nontrivial solutions of (1.6), it is sufficient to investigate the existence of nontrivial solutions to the following equation

$ \begin{equation} \begin{array}{ll} -\Delta v +\frac{G^{-1}(v)}{g(G^{-1}(v))}-\frac{h(G^{-1}(v))}{g(G^{-1}(v))}-\frac{|v|^{2^{*}(a)-2}v}{|x|^{a}}=0. \end{array} \end{equation} $

(1.9)

It is easy to prove that equation (1.6) is equivalent to equation (1.9) and the nontrivial critical point of $ J(v) $ is the nontrivial solution of equation (1.9).

The main results of this paper can be stated by the following theorem:

Theorem 1.1   Assume that $ (g_1) $ and $ (h_1)-(h_3) $ hold, then Equation (1.9) has at least one positive radial ground state solution solution if $ N\geq4. $

Remark 1   Since we want to study the existence of positive solutions to equation (1.9), we rewrite the corresponding variational function $ J(v) $ into the following form:

$ J(v)=\frac{1}{2}\int_{ \mathbb{R}^N}|\nabla v|^2dx+\frac{1}{2}\int_{ \mathbb{R}^N}|G^{-1}(v)|^2dx-\int_{ \mathbb{R}^N}H(G^{-1}(v))dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(v^{+})^{2^*(a)}}{|x|^a} dx, $

where $ v^+(x)= \max \{v(x), 0\} $.

We assert that all nontrivial critical points of the functional $ J $ are positive solutions of equation (1.9). As a matter of fact, let $ v\in H^{1}( \mathbb{R}^N) $ be a nontrivial critical point of the functional $ J $, then $ v $ must be a nontrivial solution of the equation

$ \begin{equation} \begin{array}{ll} -\Delta v +v=v-\frac{G^{-1}(v)}{g(G^{-1}(v))}+\frac{h(G^{-1}(v))}{g(G^{-1}(v))}+\frac{(v^+)^{2^{*}(a)-1}}{|x|^{a}}. \end{array} \end{equation} $

(1.10)

According to standard regularity argument, we know that $ v\in C^2( \mathbb{R}^N) $. Furthermore, by the assumption$ (h_2) $ and Lemma 2.1(2) in section 2, we can obtain that the right side of equation (1.10) is nonnegative since $ G(u)g(u)>G^2(u)/u>u. $ So, by the strong maximum principle, we know that $ v $ is positive.

This paper is organized as follows: In section 2, we will prove some useful lemmas. To be precise, firstly, we give some properties for $ G, G^{-1} $and $ H $. After that, based on these properties, we show that the functional $ J(v) $ satisfies Mountain Pass geometry and that the corresponding $ (PS)_{c} $ sequence is bounded. In Section 3, based on Section 2, we prove the main theorem 1.1 of this paper.

In what follows, we denote as usual the functions $ u^+(x)= \max \{u(x), 0\} $ and $ u^-=\max \{-u(x), 0\} $ by $ u^+ $ and $ u^- $ if $ u\in H^1( \mathbb{R}^N) $. The usual Lebesgue space is denoted by $ L^q( \mathbb{R}^N) $ with norms $ \parallel u\parallel_q =\left(\int_{R^N}|u(x)|^q dx\right)^{\frac{1}{q}}, 1\leq q<\infty $, and the space of radial symmetric functions $ \{u\in H^1( \mathbb{R}^N):u(x)=u(|x|)\} $ is denoted by $ H_r^1( \mathbb{R}^N). $

In this section, firstly, we give some properties for the fuctions $ h, g $ and $ H, G $ which are defined in the introduction. After that, we prove that the functional $ J(v) $ satisfies Mountain Pass geometry and the $ (PS)_{c} $ sequence corresponding to $ J(v) $ is bounded. Finally, we establish a compactness theorem for variational functional $ J(v) $.

Lemma 2.1  (See Lemma 2.1 in [29]) Assume that $ (g_1) $ and $ (h_1)-(h_3) $ hold, then the functions $ h(t), g(t) $ and $ H(t)=\int_0^t h(\tau)d\tau, G(t)=\int_0^t g(\tau)d\tau $ enjoy the following properties:

(1) $ G(t) $ and $ G^{-1}(s) $ are odd functions.

(2) For all $ t\geq0, s\geq0, $ there hold $ G(t)\leq g(t)t, G^{-1}(s)\leq \frac{s}{g(0)}=s. $

(3) For all $ s\geq0 $, the function $ \frac{G^{-1}(s)}{s} $ is nonincreasing and $ \lim\limits_{s\rightarrow 0} \frac {G^{-1}(s)}{s}=1. $ Moreover, if $ g $ is bounded, then $ \lim\limits_{s\rightarrow \infty} \frac {G^{-1}(s)}{s}=\frac{1}{g(\infty)}; $ If $ g $ is unbounded, then $ \lim\limits_{s\rightarrow \infty} \frac {G^{-1}(s)}{s}=0. $

(4) There exists a constant $ \mu \in(2, 2^{*}(a)) $ such that for $ \forall t>0, $ there holds $ h(t)G(t)\geq \mu g(t)H(t). $

(5) $ H(t)\geq\frac{H(M)}{G(M)}(G(t))^{\mu} $ for $ t\geq M. $

(6) Denote $ Q_{1}(s)=\frac{G^{-1}(s)}{g(G^{-1}(s))} $, $ Q_{2}(s)=\frac{h(G^{-1}(s))}{g(G^{-1}(s))} $, then for all $ s\in \mathbb{R} $, there hold

$ s^{2}Q'_{1}(s)\leq Q_{1}(s)s, \quad s^{2}Q'_{2}(s)\geq (1+\delta)Q_{2}(s)s. $

Denote

$ \begin{equation} f(s) = s- \frac {G^{-1}(s)}{g(G^{-1}(s))}+\frac { h (G^{-1}(s))}{g(G^{-1}(s))}, \end{equation} $

(2.1)

$ \begin{equation} F(s) = \int_0^s f(\tau)d\tau =\frac 12 [s^2-(G^{-1}(s))^2] + H(G^{-1}(s)). \end{equation} $

(2.2)

Lemma 2.2  (See Lemma 2.2 in [29]) Assume that $ (g_1) $ and $ (h_1)-(h_3) $ hold, then the function $ f(s), \ F(s) $ enjoy the following properties:

(1) $ f(s)\ge 0 $ for all $ s\ge 0 $.

(2) $ \lim\limits_{s \rightarrow 0^+} \frac {f(s)}{s}= \lim\limits_{s \rightarrow 0^+} \frac {F(s)}{s^2}=0 $.

(3) $ \lim\limits_{s \rightarrow +\infty} \frac {f(s)}{s^{2^*-1}}= \lim\limits_{s \rightarrow +\infty} \frac {F( s)}{s^{2^*}}=0 $.

(4) $ f(s)s \ge 2F(s) $ for $ s\ge 0 $.

Based on the lemmas above, we can prove that the functional $ J(v) $ satisfies the Mountain-Pass geometry.

Lemma 2.3  Under our assumptions, the functional $ J(v) $ enjoy the following properties:

(1) there are $ \alpha\in \mathbb{R}^{+}, \; \rho\in \mathbb{R}^{+}, $ such that $ J(v)\geq \alpha $ for all $ \|v\|=\rho $;

(2) there is $ \omega \in H^{1}( \mathbb{R}^{N}), $ such that $ \|\omega\|>\rho $ and $ J(\omega)<0. $

Proof  By Lemma 2.2 (2), (3) and Sobolev-Hardy inequality, we can obtain, for any $ \varepsilon>0 $, there is a constant $ C_{\varepsilon}>0 $ such that

$ \begin{equation} \begin{array}{ll} J(v)& =\frac{1}{2}\int_{ \mathbb{R}^N}|\nabla v|^2dx+\frac{1}{2}\int_{ \mathbb{R}^N}|G^{-1}(v)|^2dx-\int_{ \mathbb{R}^N}H(G^{-1}(v))dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(v^{+})^{2^*(a)}}{|x|^a} dx\\ & =\frac{1}{2}\int_{ \mathbb{R}^N}(|\nabla v|^2+v^{2})dx-\int_{ \mathbb{R}^N}F(v)dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(v^{+})^{2^*(a)}}{|x|^a} dx\\ & \ge \frac{1}{2}\int_{ \mathbb{R}^N}(|\nabla v|^2+v^{2})dx-\int_{ \mathbb{R}^N}(\varepsilon v^{2}+C_{\varepsilon}|v|^{2^*})dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|v|^{2^*(a)}}{|x|^a} dx \\ & \ge (C-\varepsilon)\int_{ \mathbb{R}^N}(|\nabla v|^2+v^{2})dx-C_{\varepsilon}\int_{ \mathbb{R}^N}|v|^{2^*}dx-C_{1} \Big(\int_{ \mathbb{R}^N}(|\nabla v|^2\Big)^{\frac{2^{*}(a)}{2}}. \end{array} \end{equation} $

(2.3)

Therefore, if we choose $ \varepsilon >0 $ small enough and $ \rho>0 $, then there holds

$ J(v)\geq C\|v\|^{2}-C_{\varepsilon}\|v\|^{2^{*}}-C_{1}\|v\|^{2^{*}(a)}, $

which implies the point (1).

Next, we prove point (2). For a given nontrivial function $ \varphi \in C_{0}^{\infty}( \mathbb{R}^{N}, [0, 1]) $, by Lemma 2.2(1) we can obtain that

$ \begin{equation} \begin{array}{ll} J(t\varphi)& =\frac{1}{2}\int_{ \mathbb{R}^N}(|\nabla t\varphi|^2+(t\varphi)^{2})dx-\int_{ \mathbb{R}^N}F(G^{-1}(t\varphi))dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(t\varphi)^{2^*(a)}}{|x|^a} dx\\[12pt] & \leq \frac{t^{2}}{2}\int_{ \mathbb{R}^N}(|\nabla \varphi|^2+(\varphi)^{2})dx-\frac{t^{2^*(a)}}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(\varphi)^{2^*(a)}}{|x|^a} dx\\[12pt] & \rightarrow -\infty \quad as \quad t\rightarrow +\infty. \end{array} \end{equation} $

(2.4)

Therefore, if we take $ \omega =t\varphi $ with $ t $ large enough, then we can see that point (2) is naturally true.

By Lemma 2.3 and the mountain pass Lemma, we know that for the constant

$ \begin{equation} \begin{array}{ll} c:=\inf\limits_{\gamma\in \Gamma}\sup\limits_{t\in [0, 1]}J(\gamma(t))>0, \end{array} \end{equation} $

(2.5)

where

$ \Gamma=\{\gamma\in C([0, 1], H^1( \mathbb{R}^N)), \gamma(0)=0, \gamma(1)\neq0, J^\infty (\gamma(1))<0\}, $

there is a $ (PS)_{c} $ sequence $ \{v_n\} $ in $ H^1( \mathbb{R}^N) $ at the level $ c , $ that is,

$ \begin{equation} \begin{array}{ll} J(v_n)\rightarrow c \quad\text{and}\quad J'(v_n) \rightarrow 0, \quad\text{as}\ n\rightarrow +\infty. \end{array} \end{equation} $

(2.6)

Lemma 2.4  The $ (PS)_{c} $ sequence $ \{v_n\} $ in (2.6) is bounded in $ H^1( \mathbb{R}^N) $.

Proof  Since $ \{v_n\}\subset H^1(R^N) $ is a $ (PS)_{c} $ sequence, we know that

$ \begin{equation} \begin{array}{ll} J(v_{n})& =\frac{1}{2}\int_{ \mathbb{R}^N}|\nabla v_{n}|^2dx+\frac{1}{2}\int_{ \mathbb{R}^N}|G^{-1}(v_{n})|^2dx \\[12pt] & \quad -\int_{ \mathbb{R}^N}H(G^{-1}(v_{n}))dx-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{(v_{n}^{+})^{2^*(a)}}{|x|^a}dx \rightarrow c \end{array} \end{equation} $

(2.7)

and for $ \forall\psi \in C_{0}^{\infty}( \mathbb{R}^{N}), $

$ \begin{equation} \begin{array}{ll} \langle J'(v_{n}), \psi \rangle & =\int_{ \mathbb{R}^N}[\nabla v_{n} \nabla\psi+\frac{G^{-1}(v_{n})}{g(G^{-1}(v_{n}))}\psi-\frac{h(G^{-1}(v_{n}))}{g(G^{-1}(v_{n}))}\psi-\frac{|v_{n}^{+}|^{2^{*}(a)-2}v_{n}^{+}\psi}{|x|^{a}}]dx\\[10pt] & =o(1)\|\psi\| \end{array} \end{equation} $

(2.8)

as $ n\rightarrow \infty. $ Since $ C_{0}^{\infty}( \mathbb{R}^{N}) $ is dense in $ H^1( \mathbb{R}^N), $ by taking $ \psi=v_{n} $ we can obtain that

$ \begin{equation} \begin{array}{ll} \langle J'(v_{n}), v_{n} \rangle & =\int_{ \mathbb{R}^N}[|\nabla v_{n}|^{2} +\frac{G^{-1}(v_{n})}{g(G^{-1}(v_{n}))}v_{n}-\frac{h(G^{-1}(v_{n}))}{g(G^{-1}(v_{n}))}v_{n}-\frac{(v_{n}^{+})^{2^{*}(a)}}{|x|^{a}}]dx\\[10pt] & =o(1)\|v_{n}\| \end{array} \end{equation} $

(2.9)

as $ n\rightarrow \infty. $ By (2.7), (2.9) and Lemma 2.1(2), (4), we can deduce that

$ \begin{equation} \begin{array}{ll} & \quad \mu c+o(1)-\langle J'(v_{n}), v_{n} \rangle \\[10pt] & =\mu J(v_{n})-\langle J'(v_{n}), v_{n} \rangle\\[12pt] & =\frac{\mu-2}{2}\int_{ \mathbb{R}^{N}}|\nabla v_{n}|^{2}dx+\int_{ \mathbb{R}^{N}}G^{-1}(v_{n})\Big[\frac{1}{2}\mu G^{-1}(v_{n})-\frac{1}{g(G^{-1}(v_{n}))}v_{n}\Big ]dx \\[12pt] & \quad -\int_{ \mathbb{R}^{N}}\Big[\mu H(G^{-1}(v_{n}))-\frac{h(G^{-1}(v_{n}))}{g(G^{-1}(v_{n}))}v_{n}\Big ]dx-\int_{ \mathbb{R}^{N}}\Big [\frac{\mu}{2^{*}(a)}-1\Big ]\frac{(v_{n}^{+})^{2^*(a)}}{|x|^{a}}dx \\[12pt] & \geq\frac{\mu-2}{2}\Big[\int_{ \mathbb{R}^{N}}|\nabla v_{n}|^{2}dx+\int_{ \mathbb{R}^{N}}|G^{-1}(v_{n})|^{2}\Big]. \end{array} \end{equation} $

(2.10)

By Lemma 2.1(4), (5), we can obtain that

$ \begin{equation} \frac{h(t)}{g(t)}G(t)\geq\mu H(t)\geq CG(t)^{\mu}\geq CG(t)^{2} \end{equation} $

(2.11)

for all $ t\geq1. $ Therefore, for the case $ x\in\{x:|G^{-1}(v_{n})|>1\} $, we have that

$ \begin{equation} \begin{array}{ll} \int_{\{x:|G^{-1}(v_{n})|>1\}}|v_{n}|^{2}dx&\leq C\int_{\{x:|G^{-1}(v_{n})|>1\}}H(G^{-1}(v_{n}))dx\\[12pt] & \leq \int_{ \mathbb{R}^{N}}H(G^{-1}(v_{n}))dx+\frac{C}{2^{*}(a)}\int_{ \mathbb{R}^{N}}\frac{|v_{n}^{+}|^{2^{*}(a)}}{|x|^{a}}dx\\[12pt] & \leq C \Big [c+o(1)+\frac{1}{2}\int_{ \mathbb{R}^{N}}|\nabla v_{n}|^{2}dx+\int_{ \mathbb{R}^{N}}|G^{-1}(v_{n})|^{2}\Big ]\\[12pt] & \leq C. \end{array} \end{equation} $

(2.12)

On the other hand, since $ g(t) $ is a nondecreasing function, for the case $ x\in\{x:|G^{-1}(v_{n})|\leq1\} $ we have that

$ \begin{equation} \begin{array}{ll} \frac{1}{g^{2}(1)}\int_{\{x:|G^{-1}(v_{n})|\leq1\}}|v_{n}|^{2}dx\leq C \int_{\{x:|G^{-1}(v_{n})|\leq1\}}|G^{-1}(v_{n})|^{2}dx. \end{array} \end{equation} $

(2.13)

Substiuting (2.12)-(2.13) into (2.10), we immediately know that the $ (PS)_{c} $ sequence $ \{v_{n}\} $ is bounded in $ H^{1}( \mathbb{R}^{N}). $

The following lemma provides the interval in which the $ (PS)_{c} $ condition holds for $ J(v). $

Lemma 2.5  $ J(v) $ defined on $ H_{r}^{1}( \mathbb{R}^{N}) $ satisfies $ (PS)_{c} $ condition if the level value

$ c<\frac{2-a}{2(N-a)}A_{a}^{\frac{N-a}{2-a}} $

where $ A_{a} $ is the best Sobolev-Hardy constant defined as follows:

$ A_{a}:=\inf\limits_{u\in D^{1, 2}( \mathbb{R}^{ \mathbb{N}})-\{0\}} \frac{\int _{ \mathbb{R}^{N}}|\nabla u|^{2}dx}{\Big(\int _{ \mathbb{R}^{N}}\frac{|u|^{2^{*}(a)}}{|x|^{a}}dx\Big ) }. $

Proof  Let $ \{v_{n}\}\subset H_{r}^{1}( \mathbb{R}^{N}) $ be a $ (PS)_{c} $ sequence of $ J(v) $. Similar to the proof of Lemma 2.4, we can easily prove that $ \{v_{n}\} $ is bounded in $ H_{r}^{1}( \mathbb{R}^{N}) $. Then it follows from Strauss Lemma [30] that there is a subsequence of $ \{v_{n}\} $ (still denoted by $ \{v_{n}\}) $ such that

$ \begin{equation} \begin{array}{ll} v_n\rightharpoonup v \quad\text{weakly in}\ H^1_r( \mathbb{R}^N), \\ v_n\rightarrow v\quad\text{strongly in}\ L^p( \mathbb{R}^N), \quad 2<p<2^{*}\\ v_n\rightarrow v \quad\text{a.e. in}\ \mathbb{R}^N, \end{array} \end{equation} $

(2.14)

and $ v $ is a solution of (1.10). It follows from Strauss lemma [30] and (2.14) that

$ \begin{equation} \begin{array}{ll} \lim\limits_{n \rightarrow \infty}\int_{ \mathbb{R}^{N}}F(v_{n})dx=\int_{ \mathbb{R}^{N}}F(v)dx, \quad \lim\limits_{n \rightarrow \infty}\int_{ \mathbb{R}^{N}}f(v_{n})v_{n}dx=\int_{ \mathbb{R}^{N}}f(v)vdx. \end{array} \end{equation} $

(2.15)

Therefore, we can obtain that

$ \begin{equation} \begin{array}{ll} \int_{ \mathbb{R}^{N}}\Big[|\nabla v|^{2}+v^{2}\Big]dx-\int_{ \mathbb{R}^{N}}\frac{(v^{+})^{2^{*}(a)}}{|x|^{a}}dx-\int_{ \mathbb{R}^{N}}f(v)vdx=0 \end{array} \end{equation} $

(2.16)

It is clear that for all $ A, B \in \mathbb{R} $, there holds

$ \begin{equation*} ||A+B|-|A||\leq|B|. \end{equation*} $

Hence, we can obtain that

$ \begin{equation*} \left|\left|\frac{v_n^{2^*(a)}}{|x|^{a}}\right|-\left|\frac{v_n^{2^*(a)}}{|x|^{a}}-\frac{v^{2^*(a)}}{|x|^{a}}\right|- \left|\frac{v^{2^*(a)}}{|x|^{a}}\right|\right|\leq 2 \left|\frac{v^{2^*(a)}}{|x|^{a}}\right|. \end{equation*} $

By the Sobolev-Hardy inequality,

$ \begin{equation*} \int_{ \mathbb{R}^N}\frac{|v|^{2^*(a)}}{|x|^{a}}dx\leq C\left(\int_{ \mathbb{R}^N}|\nabla v|^2dx\right) ^{\frac{2^*(a)}{2}}<\infty. \end{equation*} $

It follows from Lebesgue theorem that

$ \begin{equation} \lim\limits_{n\rightarrow \infty} \int_{ \mathbb{R}^N} \left|\left|\frac{v_n^{2^*(a)}}{|x|^{a}}\right|-\left|\frac{v_n^{2^*(a)}}{|x|^{a}}-\frac{v^{2^*(a)}}{|x|^{a}}\right|- \left|\frac{v^{2^*(a)}}{|x|^{a}}\right|\right|=0. \end{equation} $

(2.17)

Taking $ v'_n=v_n-v, $ by Brezis-Lieb's lemma [31] and (2.15)-(2.17), we can obtain that

$ \begin{equation} \begin{array}{ll} J(v)+\frac{1}{2}\int_{ \mathbb{R}^N}\Big[|\nabla v'_n|^2+|v'_n|^2\Big]-\frac{1}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|v'_n|^{2^*(a)}}{|x|^{a}}=c +o(1), \end{array} \end{equation} $

(2.18)

and

$ \begin{equation*} \begin{array}{ll} \int_{ \mathbb{R}^N}\Big[|\nabla v'_n|^2+|v'_n|^2\Big]-\int_{ \mathbb{R}^N}\frac{|v'_n|^{2^*(a)}}{|x|^{a}}=o(1). \end{array} \end{equation*} $

Suppose that $ v_n $ does not converge to $ v $ in $ H^1( \mathbb{R}^N) $, we may assume that $ \int_{ \mathbb{R}^N}\frac{|v'_n|^{2^*(a)}}{|x|^{a}}=l+o(1) $, where $ l>0 $. Then

$ \int_{ \mathbb{R}^N}(|\nabla v'_n|^2+|v'_n|^2)= l+o(1). $

By the Sobolev-Hardy inequality,

$ \begin{equation*} \begin{array}{ll} o(1)+A_{a}l^{\frac{2}{2^*(a)}}=A_{a}\Big(\int_{ \mathbb{R}^N}\frac{|v'_n|^{2^*(a)}}{|x|^{a}}\Big)^{\frac{2}{2^*(a)}}\leq \int_{ \mathbb{R}^N}|\nabla v'_n|^2\leq\int_{ \mathbb{R}^N}(|\nabla v'_n|^2+|v'_n|^2)=l +o(1). \end{array} \end{equation*} $

Hence,

$ \begin{equation*} \begin{array}{ll} l\geq (A_{a})^{\frac {N-a}{2-a}}. \end{array} \end{equation*} $

By (2.18)we deduce that

$ \begin{equation} \begin{array}{ll} J(v)=c -\Big(\frac 12 -\frac {1}{2^*(a)}\Big)l\le c -\frac{2-a}{2(N-a)}(A_{a})^{\frac {N-a}{2-a}}<0. \end{array} \end{equation} $

(2.19)

On the other hand, by Lemma 2.1, we have that

$ \begin{equation} \begin{array}{ll} \frac{1}{2} f(v)v- F(v)\geq \frac 12 \Big (|G^{-1}(v)|^2 -\frac {G^{-1}(v)v}{g(G^{-1}(v))}\Big ) - H (G^{-1}(v))+ \frac 12 \frac { h(G^{-1}(v))v}{g(G^{-1}(v))}\ge 0. \end{array} \end{equation} $

(2.20)

Thus we can obtain that

$ \begin{array}{ll} J (v)= \int_{ \mathbb{R}^N}\big(\frac{1}{2}-\frac{1}{ 2^*(a)}\big)\frac{|v|^{2^{*}(a)}}{|x|^{a}}dx+\int_{ \mathbb{R}^N}\big(\frac{1}{2} f(v)v- F(v)\big)dx \geq 0. \end{array} $

This is in contradiction with (2.19). Therefore $ l=0. $ By the definition of $ v'_n $ we can conclude that $ J $ satisfies $ (PS)_c $ condition. And then the proof is done.

By Lemmas 2.3-2.5 and the mountain pass Lemma, we can easily obtain the following lemma.

Lemma 2.6  Assume that there is $ v_{0}\in H_{r}^{1}( \mathbb{R}^{N}), v_{0} \neq 0 $ such that

$ \begin{equation} \begin{array}{ll} \sup \limits_{t\geq0 }J(tv_{0})<\frac{2-a}{2(N-a)}A_{a}^{\frac{N-a}{2-a}}. \end{array} \end{equation} $

(2.21)

Then equation (1.9) has at least one positive weak solution.

Now, we will show that the level value $ c $ is in the interval in which the $ (PS)_{c} $ condition holds. For this purpose, we introduce a well-known fact that the function

$ u_\epsilon(x)=\frac{((N-a)(N-2)\epsilon)^{\frac{N-2}{2(2-a)}}}{(\epsilon+|x|^{2-a})^{\frac{N-2}{2-a}}} $

solve the equation

$ -\Delta u=\frac{|u|^{2^{*}(a)-2}}{|x|^{a}}u \quad \text {in}\quad \mathbb{R}^{N}-\{0\} $

and satisfy

$ \int_{ \mathbb{R}^{N}}|\nabla u_\epsilon|^2dx=\int_{ \mathbb{R}^{N}}\frac{|u_\epsilon|^{2^*(a)}}{|x|^{a}}dx=(A_{a})^{\frac{N-a}{2-a}}. $

Let $ \varphi\in C_0^{\infty}( \mathbb{R}^N, [0, 1]) $ be a radial cut-off function such that $ \varphi(|x|)=1 $ for $ |x|\le \rho_\epsilon $, $ \varphi(|x|)\in (0, 1) $ for $ \rho_\epsilon<|x|<2\rho_\epsilon $, and $ \varphi(|x|)=0 $ for $ |x| \geq 2\rho_\epsilon $, where $ \rho_\epsilon=\epsilon^\tau, \ \tau\in(\frac{1}{4}, \frac{1}{2}). $ Taking $ v_\epsilon(x)=\varphi(x) u_\epsilon(x), $ and then we can obtain the following estimations (see [32]):

Lemma 3.1   The function $ v_\epsilon(x) $ satisfies the estimations as follows: as $ \epsilon\rightarrow 0, $

(1) $ \|\nabla v_{\epsilon}\|_{2}^{2}=(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) $.

(2) $ \int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx=(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-a}) $.

(3) $ \| v_{\epsilon}\|_{r}^{r}=\left \{ \begin{array}{ll}O(\epsilon^{N-\frac{r(N-2)}{2}})& \quad for \ \ \ r>\frac{2^{*}}{2}, \\[8pt] O (\epsilon^{\frac{r(N-2)}{2}} |\ln \epsilon|) & \quad for \ \ \ r=\frac{2^{*}}{2}, \\[8pt] O(\epsilon^{\frac{r(N-2)}{2}}) & \quad for \ \ \ r<\frac{2^{*}}{2}.\\ \end{array} \right .\\[8pt] $

(4) $ \| v_{\epsilon}\|_{2}^{2}=\left \{ \begin{array}{ll}O(\epsilon^{2})&\quad for \ \ \ N\ge5, \\ O (\epsilon^{2} |\ln \epsilon|) &\quad for \ \ \ N=4, \\ O(\epsilon^{N-2}) &\quad for \ \ \ N=3.\\ \end{array} \right .\\ $

(5) $ \| v_{\epsilon}\|_{1}=O(\epsilon^{\frac{N-2}{2}}). $

(6) $ \| v_{\epsilon}\|_{2^{*}}^{2^{*}}=D+O(\epsilon^{N}). $

where $ D>0 $ is a constant.

Now we are going to show the following result.

Theorem 3.2   Assume that $ (g_1) $ and $ (h_1)-(h_3) $ hold, then the equation (1.9) has at least one positive radial solution solution if $ N\geq4. $

Proof  By Lemma 2.6, we just need to verify that condition (2.21) is naturally true.

Step 1. We claim that for $ \epsilon>0 $ sufficiently small, there is a constant $ t_{\epsilon}>0 $ such that

$ J(t_{\epsilon}v_{\epsilon})=\max \limits_{t\geq 0}J(tv_{\epsilon}) $

and

$ 0<A_{1}<t_{\epsilon}<A_{2}<+\infty, $

where $ A_{1} $ and $ A_{2} $ are two positive constants that do not depend on $ \epsilon. $

In fact, as a result of $ J(0)=0 $ and $ \lim \limits_{t\rightarrow \infty}J(tv_{\epsilon})=-\infty $, there is a $ t_{\epsilon}>0 $ such that

$ J(t_{\epsilon}v_{\epsilon})=\max \limits_{t\geq 0}J(tv_{\epsilon}) \quad\text{and}\quad \frac{dJ(tv_{\epsilon})}{dt}|_{t=t_{\epsilon}}=0. $

Therefore we can obtain

$ \begin{equation} \frac{\|v_{\epsilon}\|^{2}}{\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx}-t_{\epsilon}^{2^{*}(a)-2} -\frac{\int_{ \mathbb{R}^{N}}f(t_{\epsilon}v_{\epsilon})v_{\epsilon}dx}{t_{\epsilon}\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx}=0. \end{equation} $

(3.1)

By Lemma 2.2 and Lemma 3.1, for any $ \delta>0 $ there exist positive constants $ k_{1}, k_{2}, k_{3} $ such that

$ \begin{array}{ll} & \frac{\int_{ \mathbb{R}^{N}}f(t_{\epsilon}v_{\epsilon})v_{\epsilon}dx}{t_{\epsilon}\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx} \leq k_{1}\frac{\int_{ \mathbb{R}^{N}}f(t_{\epsilon}v_{\epsilon})v_{\epsilon}dx}{t_{\epsilon}} \leq k_{1}\frac{1}{t_{\epsilon}}\int_{ \mathbb{R}^{N}}(\delta t_{\epsilon}^{2^{*}-1}v_{\epsilon}^{2^{*}-1}+k_{2}t_{\epsilon}v_{\epsilon})v_{\epsilon}dx \\[12pt] & =k_{1}\int_{ \mathbb{R}^{N}}(\delta t_{\epsilon}^{2^{*}-2}v_{\epsilon}^{2^{*}}+k_{2}v_{\epsilon}^{2})dx \rightarrow \delta k_{3}t_{\epsilon}^{2^{*}-2} \quad (\epsilon\rightarrow 0). \end{array} $

It follows from (3.1) that

$ 1-t_{\epsilon}^{2^{*}(a)-2}-\delta k_{3}t_{\epsilon}^{2^{*}-2}\leq 0 \quad (\epsilon\rightarrow 0). $

Therefore there exists a constant $ A_{1} >0 $ such that $ t_{\epsilon }> A_{1} $ if $ \epsilon $ is sufficiently small.

On the other hand, by (3.1) we can obtain

$ t_{\epsilon}^{2^{*}(a)-2}=\frac{\|v_{\epsilon}\|^{2}}{\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx} -\frac{\int_{ \mathbb{R}^{N}}f(t_{\epsilon}v_{\epsilon})v_{\epsilon}dx}{t_{\epsilon}\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx} \leq \frac{\|v_{\epsilon}\|^{2}}{\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx}. $

Therefore we have

$ t_{\epsilon}\leq \Big(\frac{\|v_{\epsilon}\|^{2}}{\int_{ \mathbb{R}^{N}}\frac{|v_{\epsilon}|^{2^{*}(a)}}{|x|^{a}}dx}\Big)^{\frac{1}{2^{*}(a)-2}}\leq A_{2}<+\infty\quad\text{ if} \; {\epsilon} \; \text{is sufficiently small}. $

Step 2. We want to estimate $ J(t_{\varepsilon}v_{\epsilon}) $. By Lemma (3.1), we can obtain that

$ \begin{array}{ll} J(t_{\varepsilon}v_{\epsilon})& =\frac{t_{\varepsilon}^{2}}{2}\int_{ \mathbb{R}^N}(|\nabla v_{\epsilon}|^2+v_{\epsilon}^{2})dx-\int_{ \mathbb{R}^N}F(t_{\epsilon}v_{\epsilon})dx-\frac{t_{\epsilon}^{2^*(a)}}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|v_{\epsilon}|^{2^*(a)}}{|x|^a} dx\\[12pt] & =\frac{t_{\varepsilon}^{2}}{2}\int_{ \mathbb{R}^N}|\nabla v_{\epsilon}|^2dx-\frac{t_{\epsilon}^{2^*(a)}}{2^*(a)}\int_{ \mathbb{R}^N}\frac{|v_{\epsilon}|^{2^*(a)}}{|x|^a} dx+\frac{t_{\varepsilon}^{2}}{2}\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx-\int_{ \mathbb{R}^N}F(t_{\epsilon}v_{\epsilon})dx\\[12pt] & \leq \Big(\frac{t_{\varepsilon}^{2}}{2}-\frac{t_{\epsilon}^{2^*(a)}}{2^*(a)}\Big)(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +\frac{t_{\varepsilon}^{2}}{2}\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx-\int_{ \mathbb{R}^N}F(t_{\epsilon}v_{\epsilon})dx. \end{array} $

Since the function $ Q(t)=\frac{t_{\varepsilon}^{2}}{2}-\frac{t_{\epsilon}^{2^*(a)}}{2^*(a)} $ has only maximum at $ t=1, $ we can obtain

$ \begin{array}{ll} J(t_{\varepsilon}v_{\epsilon})& \leq\Big(\frac{1}{2}-\frac{1}{2^*(a)}\Big)(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +\frac{A_{2}^{2}}{2}\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx-\int_{ \mathbb{R}^N}F(t_{\epsilon}v_{\epsilon})dx\\[12pt] & =\frac{2-a}{2(N-a)}(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +C\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx\\[12pt] & \quad -\int_{ \mathbb{R}^N}\Big [\frac{1}{2}\Big(|t_{\epsilon}v_{\epsilon}|^{2}-|G^{-1}(t_{\epsilon}v_{\epsilon})|^{2}\Big )+H(G^{-1}(t_{\epsilon}v_{\epsilon}))\Big ]dx. \end{array} $

It follows from Lemma 2.1(2) that

$ \begin{array}{ll} J(t_{\varepsilon}v_{\epsilon})& \leq\frac{2-a}{2(N-a)}(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +C\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx -\int_{ \mathbb{R}^N}H(G^{-1}(t_{\epsilon}v_{\epsilon}))dx. \end{array} $

By Lemma 2.1(4) we can obtain

$ \mu G'(t) H(t)\leq G(t) H'(t)\Rightarrow H(G^{-1}(s))\geq s^{\mu} \quad \text{for all} s\geq0. $

Therefore

$ \begin{array}{ll} J(t_{\varepsilon}v_{\epsilon})& \leq\frac{2-a}{2(N-a)}(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +C\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx -\int_{ \mathbb{R}^N}|t_{\epsilon}v_{\epsilon}|^{\mu}dx\\[12pt] & \leq\frac{2-a}{2(N-a)}(A_{a})^{\frac{N-a}{2-a}}+O(\epsilon^{N-2}) +C\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx -A_{1}^{\mu}\int_{ \mathbb{R}^N}|v_{\epsilon}|^{\mu}dx.\\[12pt] \end{array} $

Let

$ I=O(\epsilon^{N-2}) +C\int_{ \mathbb{R}^N}|v_{\epsilon}|^{2}dx -A_{1}^{\mu}\int_{ \mathbb{R}^N}|v_{\epsilon}|^{\mu}dx, $

then we only need to prove $ I<0 $ for small $ \epsilon. $ It follows from Lemma (3.1) that

$I=C \epsilon^{N-2}+C\left\{\begin{array}{ll} O\left(\epsilon^2\right) & { for } \quad N \geq 5, \\ O\left(\epsilon^2 \mid \ln \epsilon\right) & { for } \quad N=4, \\ O\left(\epsilon^{N-2}\right) & { for } \quad N=3 . \end{array} -C\left\{\begin{array}{lll} O\left(\epsilon^{N-\frac{\mu(N-2)}{2}}\right) & { for } \quad \mu>\frac{2^*}{2}, \\ O\left(\epsilon^{\frac{\mu(N-2)}{2}}|\ln \epsilon|\right) & { for } \quad \mu=\frac{2^*}{2}, \\ O\left(\epsilon^{\left.\frac{\mu(N-2)}{2}\right)}\right. & { for } \quad \mu<\frac{2^*}{2} . \end{array}\right.\right.$

When $ N \geq 4, \mu > 2, $ we can verify that $ I<0 $ as $ \epsilon>0 $ sufficiently small. Therefore (2.20) is naturally true. And then the proof is done.

Define

$ b=\inf \limits_{v\in \mathcal{N}}J(v), $

where

$ \mathcal{N}=\Big\{u \in H^{1}( \mathbb{R}^{N})\setminus\{0\}|\int_{ \mathbb{R}^{N}}\Big (|\nabla v|^{2}+\frac{G^{-1}(v)}{g(G^{-1}(v))}v-\frac{(v^{+})^{2^{*}(a)}}{|x|^{a}}-\frac{h(G^{-1}(v))}{g(G^{-1}(v))}v \Big)dx=0 \Big \} $

It is clear that $ \mathcal{N}\neq \varnothing $ since the equation (1.9) has at least one positive solution.

Lemma 3.2   Assume that $ (g_{1}) $, and $ (h_{1})-(h_{3}) $ hold, then we have

$ b=c. $

Proof  It is obvious that for all $ u\in \mathbb{N} $, there exists $ t^{*}>0, $ such that

$ J(t^{*}u)= \sup \limits_{t>0} J(tu)\quad \text{and} \quad \frac{dJ(tu)}{dt}|_{t=t^{*}} =0. $

Set

$ k_{1}= \int_{ \mathbb{R}^{N}}|\nabla u|^{2}dx, \quad k_{2}=\int_{ \mathbb{R}^{N}}\frac{(u^{+})^{2^{*}(a)}}{|x|^{a}}dx, $

and then

$ \begin{array}{ll} \frac{dJ(tu)}{dt}& =t \int_{ \mathbb{R}^{N}}|\nabla u|^{2}dx+\int_{ \mathbb{R}^{N}}Q_{1}(tu)udx-t^{2^{*}(a)-1}\int_{ \mathbb{R}^{N}}\frac{(u^{+})^{2^{*}(a)}}{|x|^{a}}dx-\int_{ \mathbb{R}^{N}}Q_{2}(tu)udx\\\\ & =k_{1}t-k_{2}t^{2^{*}(a)-1}+\int_{ \mathbb{R}^{N}}Q_{1}(tu)udx-\int_{ \mathbb{R}^{N}}Q_{2}(tu)udx, \end{array} $

where $ Q_{1}(s), Q_{2}(s) $ are given fuctions by Lemma 2.1(6). Denote

$ \gamma (t)=k_{1}-k_{2}t^{2^{*}(a)-2} +\int_{ \mathbb{R}^{N}}\frac{Q_{1}(tu)u}{t}dx-\int_{ \mathbb{R}^{N}}\frac{Q_{2}(tu)u}{t}dx. $

By Lemma 2.1(6), we can obtain

$ \begin{array}{ll} \gamma' (t)& =-(2^{*}(a)-2)k_{2}t^{2^{*}(a)-3} +\int_{ \mathbb{R}^{N}}\frac{tu Q'_{1}(tu)u-Q_{1}(tu)u}{t^{2}}dx\\[12pt] &\quad -\int_{ \mathbb{R}^{N}}\frac{tuQ'_{2}(tu)u-Q_{2}(tu)u}{t^{2}}dx\\ &<0 \end{array} $

for $ t>0 $. Therefore $ \gamma(t) $ has at most one zero point in $ (0, \infty). $ It follows from $ u \in \mathcal{N} $, $ \left.\frac {{dJ }(tu)}{dt} \right| _{t=t^*} =0 $ and $ \left.\frac{dJ (tu)}{dt} \right| _{t=1} =0 $ that $ t^*=1 $. Thus

$ b =\inf\limits_{u\in \mathbb{N}}J(u)= \inf\limits_{u\in \mathbb{N}} \sup\limits_{t>0} J (tu)\ge c. $

On the other hand, since $ c $ is a critical value of $ J (v) $, we can obtain that $ b\le c $ by the definition of $ b $. Thus

$ b=c. $

And then the proof is done.

By Lemma (3.3), we know that the functional $ J(v) $ can be achieved by a function $ w\in \mathcal{N} $ and $ w $ is a positive ground state solution of (1.9).