In [2], the authors investigated the following semilinear elliptic problem Where $\Omega$ is an open bounded domain in ${\mathbb{R}}^N\left(N\ge 3\right)$, with smooth boundary $\partial \Omega$ and $0\in \Omega ,0\le \mu <\overline{\mu }:={\left(\frac{N-2}{2}\right)}^2,0\le s<2,$ $2^{*}\left(s\right)=2\left(N-s\right)/\left(N-2\right)$ is the Hardy-Sobolev critical exponent. This problem comes from the consideration of standing waves in the anisotropic Schrödinger equation, also it is very important in the field of hydrodynamics, glaciology, quantum field theory and statistical mechanics (see [3,4,5,6]).

Assume that $g\in C\left(\Omega \times \mathbb{R},\mathbb{R}\right)$, $G\left(x,t\right)={\int }_0^{t}g\left(x,s\right)ds$ such that

($g_1$) There exist constants $c_1$, $c_2>0$ and $p\in (2,2^{*})$ (here $2^{*}=2^{*}\left(0\right)=2N/\left(N-2\right)$ is the Sobolev critical exponent) such that

($g_2$) There exists a constant $K>0$ big enough such that

($g_3$) There are constants $\rho >2$ and $\nu >0$ such that

From the proof of Theorem 1, one can conclude that it also need $c_1$ to be sufficiently small (see page 222, line -8 in [2]). So we may wonder how small should $c_1$ be, this problem is one of our objects to be solved in this paper.

Many authors investigated problem (1) when $g(x,u)$ is a special function, such as [1,7,8,9,10,11,12,13,14]. And some authors concentrated on the case when $g(x,u)$ is a general function that satisfies some definite conditions, such as [2,15,16] and so on. For the study on problem (1), the classical method is variational method (see [17,18,19]), we should point out that the classical Mountain Pass Lemma can not be applied directly to (1), because (1) contains the Hardy-Sobolev critical exponent $\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u$. As we all know, the essential reason is the embedding from $H_0^1\left(\Omega \right)$ into $L^{2^{*}}\left(\Omega \right)$ is continuous, but not compact. In order to overcome the difficulty caused by this non-compact embedding, one can use the principle of concentrated compactness proposed by P.L. Lions ([20,21,22,23]), also we could use the Mountain Pass Lemma with ${\left(PS\right)}_c$ conditions proposed by H. Brezis and L. Nirenberg (one can refer to [7,19]), and so on. These theoretical methods has greatly promoted the development of nonlinear analysis, and many excellent results have been obtained, for convenience, we list some which are useful for our study.

The following two theorems describe the existence of positive solution for (1) when the function $g(x,t)$ grows linearly at $t=0.$

(B1) If   $\mu \le \overline{\mu }-1$, then when $\lambda <{\lambda }_1\left(\mu \right),$problem (1) has at least a positive solution in$H_0^1\left(\Omega \right)$, where

(B2) If $\overline{\mu }-1<\mu <\overline{\mu }$,then when ${\lambda }_{*}\left(\mu \right)<\lambda <{\lambda }_1\left(\mu \right)$, problem (1) has at least a positive solution in$H_0^1\left(\Omega \right)$, where

(B3) If $\overline{\mu }-1<\mu <\overline{\mu }$and$\Omega =B(0,R)$ (i.e., the ball centered at$x=0$ with radiusR), then (1) has no positive solution for $\lambda \le {\lambda }_{*}\left(\mu \right).$

One can easily see that when $g(x,u)=\lambda u$, $g(x,u)$ satisfies ($g_1$), but when $\lambda <2K$, ($g_2$) is invalid.

In [15], D.S. Kang and Y.B. Deng obtained the existence of the solution to problem where $a\left(x\right)$ is a nonnegative function and locally bounded in ${\mathbb{R}}^N\backslash \left\{0\right\}$, $a\left(x\right)=O\left({\left|x\right|}^{-s}\right)$ in the bounded neighborhood of the origin, $a\left(x\right)=O\left({\left|x\right|}^{-t}\right)$ as $\left|x\right|\to \infty$, $0\le s<t<2$, $2^{*}\left(t\right)<r<2^{*}\left(s\right)$. They obtained: If $r>max\left\{\frac{N-s}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\frac{N-s-2\sqrt{\overline{\mu }-\mu }}{\sqrt{\overline{\mu }}}\right\},$ then (4) has at least one solution.

On the other hand, when $g(x,u)=\lambda u$ and $s=0$, as E. Jannelli [1] said, “The space dimension N plays a fundamental role when one seeks the positive solutions of (1)". In [24], the authors studied and obtained

Motivated by the above mentioned references, we naturally proposed the following problems:

(P1) In Theorem 1, what would happen when $\overline{\mu }-1\le \mu <\overline{\mu }$.

(P2) Can we further weaken the conditions ($g_1$), ($g_2$) and ($g_3$)?

(P3) For problem (1), if g is a general function, whether the space dimension N still play an important role.

To solve these problems, we tried our best, and obtained

As applications of Theorem 1.1 and Theorem 1.2, we give an example.

where $a\left(r\right)\in C^1\left([0,+\infty ),[0,+\infty )\right)$ and $a^{\prime }\left(r\right)\le 0,$ $N\ge 3,$ $0\le s<2,$ $\Omega =B(0,R),$ by Theorem 1.1 and and Theorem 1.2, we have

(i) If $0\le \mu \le \overline{\mu }-1$, $1<\alpha <\frac{N+2}{N-2},$ then when $\lambda <{\lambda }_1\left(\mu \right),$ problem (6) has at least one positive solution in $H_0^1\left(\Omega \right)$.

(ii) If $\overline{\mu }-1<\mu <\overline{\mu },$ $1<\alpha <\frac{N+2}{N-2},$ then when ${\lambda }_{*}\left(\mu \right)<\lambda <{\lambda }_1\left(\mu \right)$, problem (6) has at least one positive solution in $H_0^1\left(\Omega \right)$.

(iii) If $\overline{\mu }-1<\mu <\overline{\mu }$, $\alpha \ge \frac{N+2}{N-2}$ and $\lambda \le {\lambda }_{*}\left(\mu \right),$ then (6) has no positive solution in $H_0^1\left(\Omega \right).$

where $a\left(r\right)\in C^1\left([0,+\infty ),[0,+\infty )\right)$ and $a^{\prime }\left(r\right)\le 0,$ $N\ge 3,$ $0\le s<2,$ $\Omega =B(0,R).$ If $\overline{\mu }-1<\mu <\overline{\mu }$ and $\lambda \le {\lambda }_{*}\left(\mu \right),$ then (7) has no positive solution in $H_0^1\left(\Omega \right).$ Especially, for any $a\ge 1,$ $\overline{\mu }-1<\mu <\overline{\mu }$ and $\lambda \le {\lambda }_{*}\left(\mu \right),$ the equations has no positive solution in $H_0^1\left(\Omega \right).$

where $a\left(r\right)\in C^1\left([0,+\infty ),[0,+\infty )\right)$ and $a^{\prime }\left(r\right)\le 0,$ $N\ge 4,$ $0\le s<2,$ if $1<\alpha <\frac{N+2}{N-2},$ then when $\lambda <{\lambda }_1\left(0\right),$ problem (9) has at least one positive solution in $H_0^1\left(\Omega \right)$. Especially, for any $a\ge 0,$ $N\ge 4,$ $1<\alpha <\frac{N+2}{N-2}$ and $\lambda <{\lambda }_1\left(0\right),$ the equations has at least one positive solution in $H_0^1\left(\Omega \right).$

We organized the rest paper as follows. In section 2, we give some preliminaries about Hardy inequality, the properties of variational functional corresponding to equation (1) and the properties of extremal functions. In section 3, by using the Mountain Pass Lemma with ${\left(PS\right)}_c$ conditions, we give a detailed proof of Theorem 1.1. In section 4, by establishing Pohozaev-type identity and using the properties of Bessel function, we give a detailed proof of Theorem 1.2.

In this section, we give some lemmas which will be useful for our study, for more details, one can refer to the references and cited therein.

By Lemma 2, we can define equivalent norm and inner product in $H_0^1\left(\Omega \right)$ as following for $0\le \mu <\overline{\mu }$:

Notice that the values of $g(x,t)$ are irrelevant for $t<0$ in Theorem 1.1, so we define

To study the existence of positive solution for (1), we first consider the existence of nontrivial solutions to the problem where

Obviously, the existence of positive solution for (1) is equivalent to the existence of positive solution for (11).

The energy functional $J:H_0^1\left(\Omega \right)\to \mathbb{R}$ corresponding to (11) is given by

$J\left(u\right)$ is well defined with $J\in C^1\left(H_0^1\left(\Omega \right),\mathbb{R}\right)$ and for any $v\in H_0^1\left(\Omega \right),$

For $0\le \mu <\overline{\mu }$, define the best constant (see [13,26])

The following two lemmas could be found in [13].

for all $\epsilon >0$. Moreover, the extremal functions $y_{\epsilon }\left(x\right)$ solve the equation and satisfy

Let and define a cut-off function $\phi \left(x\right)\in C_0^{\infty }\left(\Omega \right)$ such that where $B_{2R}\left(0\right)\subset \Omega ,0\le \phi \left(x\right)\le 1$, for $R<\left|x\right|<2R$ $(R\le R_0,$ $R_0$ will be defined later), set then ${\int }_{\Omega }\frac{{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx=1$.

where

Moreover, let $\xi ={\epsilon }^{\frac{\sqrt{\overline{\mu }}}{2-s}},$ then

And there exists $R_0>0$, when $R\le R_0,$

It follows from $\phi \in C_0^{\infty }\left(\Omega \right)$ and $0\le \phi \left(x\right)\le 1$ that there exists $M>0$, for any $x\in \Omega$,

By $\underset{\epsilon \to 0^{+}}{lim}{U}_{\epsilon }=\frac{1}{{\left|x\right|}^{\beta }}$, ${\int }_{\left|x\right|\ge 2R}\frac{{\left|\nabla \phi \left(x\right)\right|}^2}{{\left|x\right|}^{2\beta }}dx\ge 0$, ${\beta }^2-\mu >0$ and (20), we have

Let $r=\left|x\right|$ and make an N-dimensional spherical coordinate transformation, then where $S_{N-1}$ denotes the N-dimensional unit spherical surface area. Since $2\beta +3-N=1+2\sqrt{\overline{\mu }-\mu }>1$, we have

If $2\beta +2-N=1$, then ${\int }_R^{2R}\frac{1}{r^{2\beta +2-N}}dr=ln2$ and $\underset{R\to 0^{+}}{lim}\frac{{\left(2R\right)}^{N-2-2\beta }}{2\beta +2-N}=+\infty .$ Thus, there exists $R_1>0$, such that $-\left({\beta }^2-\mu \right){\int }_{\left|x\right|\ge 2R}\frac{1}{{\left|x\right|}^{2\beta +2}}dx+2\beta M{\int }_{R\le \left|x\right|\le 2R}\frac{1}{{\left|x\right|}^{2\beta +1}}dx<0,$ for $R\le R_1.$

If $2\beta +2-N<1$, then $\underset{R\to 0^{+}}{lim}{\int }_R^{2R}\frac{1}{r^{2\beta +2-N}}dr=\underset{R\to 0^{+}}{lim}\frac{R^{N-1-2\beta }-{\left(2R\right)}^{N-1-2\beta }}{2\beta +1-N}=0,$ thus there exists a positive constant $R_2>0$, such that $-\left({\beta }^2-\mu \right){\int }_{\left|x\right|\ge 2R}\frac{1}{{\left|x\right|}^{2\beta +2}}dx+2\beta M{\int }_{R\le \left|x\right|\le 2R}\frac{1}{{\left|x\right|}^{2\beta +1}}dx<0,$ for $R\le R_2.$

If $2\beta +2-N>1$, then $\underset{R\to 0^{+}}{lim}{\int }_R^{2R}\frac{1}{r^{2\beta +2-N}}dr=\underset{R\to 0^{+}}{lim}\frac{R^{N-1-2\beta }-{\left(2R\right)}^{N-1-2\beta }}{2\beta +1-N}=+\infty ,$ and $N-2-2\beta <N-1-2\beta <0$ implies that there exists some positive constant $R_3>0,$ such that $-\left({\beta }^2-\mu \right){\int }_{2R}^{+\infty }\frac{1}{r^{2\beta +3-N}}dr+2\beta M{\int }_R^{2R}\frac{1}{r^{2\beta +2-N}}dr<0,$ for $R\le R_3.$

As mentioned above, when $R\le R_0=min\left\{R_1,R_2,R_3\right\}$,

Furthermore, by (14) and (15),

therefore where notice that $\gamma >\frac{pN}{N-q}$ implies that thus

This completes the proof.

In this section, we give the proof of Theorem 1.1. First, we have

J defined as in (12) satisfies the ${\left(PS\right)}_c$ condition.