本文主要研究拟线性Schrödinger方程

$\begin{eqnarray}-\Delta_pu+|u|^{p-2}u-\Delta_p(|u|^{2\alpha})|u|^{2\alpha-2}u=h(u),x\in {\Bbb R}^N\end{eqnarray}$

(1.1)

的解的存在性,其中$ \alpha>\frac{1}{2}$为参数,$\Delta_pu={\rm div}(|\nabla u|^{p-2}\nabla u)$是$p$-Laplacian算子,$p\in(1,N]$.

当$p=2,\alpha=1$时,方程(1.1)的解为Schrödinger方程

$\begin{eqnarray}{\rm i}z_t=-\Delta z+W(x)z-h_1(|z|^2)z-\kappa\Delta g(|z|^2)g'(|z|^2)z,x\in {\Bbb R}^N \end{eqnarray}$

(1.2)

的驻波解,其中$ z=z(t,x),W(x)$为势函数,常数$\kappa>0$,$h_1,g$为实函数.

众所周知具有$z(t,x)=\exp(-{\rm i}\omega t)u(x)$形式的驻波解满足$g(s)=s$的方程(1.2)当且仅当函数$u(x)$满足椭圆型方程

$ \begin{eqnarray}-\Delta u+V(x)u-\Delta(u^2)u=h(u),x\in{\Bbb R}^N,\end{eqnarray}$

(1.3)

其中$\omega\in {\Bbb R},V(x)=W(x)-\omega$及$h(u)\equiv h_1(|u|^2)u$.

形如(1.2)式的拟线性Schrödinger方程出现在数学物理中,随着非线性项$g$的变化也衍生出了一些关于物理现象的模型.例如Kurihura在文献[13]中把$g(s)=s$应用到等离子物理中的超流体膜方程中.在文献[6]中,采用$g(s)=(1+s)^{1/2}$描绘高功率超短激光在物质上的自沟物理现象.此外该方程也出现在等离子物理和流体力学,力学以及凝聚态理论中,见文献[2, 12, 16].

当$p=2$时出现多种方法研究方程(1.1),如文献[14]采用约束极小化方法来证明方程(1.1)的正基态解的存在;文献[5, 9, 15]中通过变量代换将方程(1.1)转换为半线性方程;而在文献[11]中应用了Nehari流形的方法得出基态解的存在性.特别地,文献[14, 15]研究了具有参数$\alpha(>\frac{1}{2})$的方程

$\text{-}\Delta u+V(x)u-\Delta({{\left| u \right|}^{2\alpha }}){{\left| u \right|}^{2\alpha -2}}u=\lambda {{\left| u \right|}^{p-1}}u,~~x\in {{\mathbb{R}}^{N}}$

(1.4)

的基态解的存在性,其中$\lambda>0$,$2<p+1<\frac{4\alpha N}{N-2}$.文献[1]通过对偶方法得出方程

$ \begin{eqnarray}-\Delta u+\lambda u-\Delta(|u|^{2\alpha})|u|^{2\alpha-2}u=|u|^{p-1}u,~~x\in{\Bbb R}^N \end{eqnarray} $

(1.5)

唯一基态解,其中$\alpha>1$,$p\in(1,\frac{(2\alpha-1)N+2}{N-2})$.

Severo在文献[18]中通过在$W^{1,p}({\Bbb R}^N)$中应用极值方法得出方程

$ \begin{eqnarray}-\Delta_p u+V(x)|u|^{p-2}u-\Delta_p(u^2)u=h(u),x\in{\Bbb R}^N \end{eqnarray}$

(1.6)

在$C^{1,\alpha}_{loc}({\Bbb R}^N)\cap W^{1,p}({\Bbb R}^N)$中存在非平凡解,其中$h$满足次临界增长,$p\in(1,N]$.近期,Wu在文献[19]中通过对偶方法研究了方程(1.4)的解的存在,其中$p$和$V(x)$满足:

$(H_1)$ 参数$\alpha\in(\frac{1}{2},1]$,若$N\ge 3$,$4\alpha<p<\frac{4\alpha N}{N-2}$;若$N=1,2$,$4\alpha<p<\infty$.

$(H_2)$ $V(x)\in C({\Bbb R}^N)$,$V(x)\ge V_{0}>0$,且对$\forall M>0$,meas$(\{x\in{\Bbb R}^N: V(x)\le M\})<\infty$.

显然条件$(H_2)$保证了嵌入$E\hookrightarrow L^s({\Bbb R}^N)$是紧的,其中$2\le s<\frac{2N}{N-2}$,$E$为$W^{1,2}({\Bbb R}^N)$的子空间其范数为

$ \begin{equation}\|u\|_E=\bigg(\int_{{\Bbb R}^N}(|\nabla u|^2+V(x)|u|^2){\rm d}x\bigg)^{1/2}.\end{equation}$

(1.7)

受以上文献启发,本文主要研究方程(1.1)在两种条件$1<p<N$和$p=N$下的正解存在性,由于方程(1.1)中$V(x)\equiv 1$,条件$(H_2)$不再成立.

据作者了解,几乎没有看到当$p=N$时方程(1.1)的解的存在性的相关文献.本文利用文献[4, 8]的思路研究方程(1.1),由于非线性算子$\Delta_p(|u|^{2\alpha})|u|^{2\alpha-2}u$的出现需要考虑合适的空间,因此需要更细致的估测.该文采用与上述文献不同的方法即用Nehari流形和Schwarz对称化的方法来确立方程(1.1)的解的存在性.

本文记$X=W^{1,p}({\Bbb R}^N)$,$1<p<N$和$Y=W^{1,N}({\Bbb R}^N)$并作如下假设：

$(A_1)$ 当$1<p<N$时,$h(t)=\lambda |t|^{m-2}t$,其中$p^*=\frac{pN}{N-p}$,$\lambda>0,2\alpha p<m<2\alpha p*$,$\alpha>1/2$;

$(A_2)$ 当$p=N$时,奇函数$h(t)\in C^1({{\Bbb R}}),h(t)>0,t>0,$且$\exists b_1,\alpha_0,q>0$使得

$|h(t)|\le {{b}_{1}}|t{{|}^{q-1}}[\exp ({{\alpha }_{0}}|t{{|}^{\frac{2\alpha N}{N-1}}})-{{S}_{N-2}}({{\alpha }_{0}},t)],\forall t\in \mathbb{R}=(-\infty ,\infty ),$

(1.8)

其中

$\begin{equation}S_{N-2}(\alpha_0,t)=\sum_{k=0}^{N-2}\frac{\alpha_0^k}{k!}|t|^{\frac{2k\alpha N}{N-1}},\forall t\in {\Bbb R};\end{equation}$

(1.9)

$(A_3)$当$p=N$时,函数$h(t)$还满足

$\begin{equation}(q-1+2\alpha N)\frac{h(t)}{t}\le h'(t),(q+2\alpha N)H(t)\le th(t),\forall t\in {\Bbb R}\backslash\{0\}.\end{equation}$

(1.10)

本文记$H(t)=\int_0^th(s){\rm d}s,t\in {\Bbb R}$.

注1.1 设$\lambda>0,\beta\ge 0$,显然函数$h(t)=\lambda t|t|^{m-2}\exp(\beta|t|^{\frac{2\alpha N}{N-1}}),t\in {\Bbb R},m>2\alpha N$,满足$(A_2)-(A_3)$.

本文主要结论是

定理1.1 假设有$(A_1)$或$(A_2)-(A_3)$,则方程(1.1)在空间$X$或$Y$中至少存在一个正解.

设$\Omega$为${\Bbb R}^N$中一开子集,$L^p(\Omega)(p\ge 1)$和$W^{1,p}(\Omega)(W_0^{1,p}(\Omega))$分别为Lebesgue空间和Sobolev空间,其范数分别为$\|u\|_p\equiv\|u\|_{L^p(\Omega)}=\big(\int_{\Omega}|u|^p{\rm d}x\big)^{1/p}$和

$\begin{equation}\|u\|_{W^{1,p}(\Omega)}=\bigg(\int_{\Omega}(|\nabla u|^p+|u|^p){\rm d}x\bigg)^{1/p}.\end{equation}$

(2.1)

为了方便证明,需要引入以下引理

引理2.1^[10] 设$u\in W_0^{1,N}(\Omega)\cap L^r(\Omega)$,其中$r\ge 1$,$\Omega\subseteq {\Bbb R}^N$为任一区域,则对$q\ge r$,有

$\begin{equation}\|u\|_q\le c(N,r)q^{1-1/N}\|\nabla u\|_N^{1-r/q}\|u\|_r^{r/q}.\end{equation}$

(2.2)

特别的,

$\begin{equation}c(N,N)=\frac{1}{\sqrt{\pi}}\left(\frac{\Gamma(N/2)\Gamma(2N)}{2\Gamma(N)^2}\right)^{1/N}\equiv d_N.\end{equation}$

(2.3)

注2.1 由引理2.1,嵌入$Y\hookrightarrow L^q({\Bbb R}^N),q\ge N$是连续的,且有

$\begin{equation}\|u\|_{L^q(\Omega)}\le d_Nq^{1-1/N}\|u\|_{W_{0}^{1,n}(\Omega)}.\end{equation}$

(2.4)

考虑方程(1.1)对应的能量泛函$I(u$)

$ \begin{equation}I(u)=\frac{1}{p}\int_{{\Bbb R}^N}(1+(2\alpha)^{p-1}|u|^{(2\alpha-1)p})|\nabla u|^p{\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}|u|^p{\rm d}x-\int_{{\Bbb R}^N} H(u){\rm d}x.\end{equation}$

(2.5)

采用文献[9, 18]中的变量代换$u=f(v)$或$v=f^{-1}(u)$,这里$f$为奇函数,其中

$\begin{equation}f'(t)=(1+(2\alpha)^{p-1}|f(t)|^{(2\alpha-1)p})^{-1/p},t\ge0,f(0)=0,f(t)=-f(-t),t\le 0.\end{equation}$

(2.6)

引理2.2^{[5, 9, 18]} (2.6)式所定义的函数$f(t)$满足

$(f_1)$ $f\in C^2$且唯一确定且在${\Bbb R}$上可逆;

$(f_2)$ $0<f'(t)\le 1,\forall t\in {\Bbb R}$;

$(f_3)$ $|f(t)|\le |t|,\forall t\in {\Bbb R}$;

$(f_4)$ 当$t\to 0$时,$\frac{f(t)}{t}\to 1$;

$(f_5)$ $|f(t)|\le(2\alpha)^{1/2\alpha p}|t|^{1/2\alpha},\forall t\in {\Bbb R}$;

$(f_6)$ $\frac{1}{2}f(t)\le \alpha tf'(t)\le \alpha f(t),\forall t\ge 0$及$\alpha f(t)\le \alpha tf'(t)\le \frac{1}{2}f(t),\forall t \le 0$;

$(f_7)$ $\exists a\in(0,(2\alpha)^{1/2\alpha p}]$使得$f(t)t^{-1/2\alpha}\to a$,$t\to+\infty$;

$(f_8)$ $ \exists b_0>0$使得当$|t|\le 1$时有$|f(t)|\ge b_0|t|$,当$|t|\le 1$时有$|f(t)|\ge b_0|t|^{1/2\alpha}$.\\通过变量代换(2.6)后$I(u)$变为

$\begin{eqnarray}J(v)\equiv I(f(v))=\frac{1}{p}\int_{{\Bbb R}^N}|\nabla v|^p{\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}|f(v)|^p{\rm d}x-\int_{{\Bbb R}^N}H(f(v)){\rm d}x,\end{eqnarray}$

(2.7)

易见在假设$(A_1)-(A_3)$下,$J(u)$在$X$和$Y$上有定义.由文献[18]可知若$v\in W^{1,p}({\Bbb R}^N)\cap L_{loc}^{\infty}({\Bbb R}^N)(p>1)$为泛函$J$的临界点,即对所有$\varphi\in W^{1,p}({\Bbb R}^N)$有$J'(v)\varphi=0$,其中

$$\begin{eqnarray}J'(v)\varphi=\int_{{\Bbb R}^N}|\nabla v|^{p-2}\nabla v \nabla\varphi {\rm d}x+\int_{{\Bbb R}^N}|f(v)|^{p-2}f(v)f'(v)\varphi{\rm d}x-\int_{{\Bbb R}^N}h(f(v))f'(v)\varphi {\rm d}x,\end{eqnarray}$$

那么$v$为方程

$\begin{equation}-\Delta_pv=g(v),\qquad x\in {\Bbb R}^N\end{equation}$

(2.9)

的解,其中

$\begin{equation}g(s)=-|f(s)|^{p-2}f(s)f'(s)+h(f(s))f'(s),s\in{\Bbb R},\end{equation}$

(2.10)

由此可知$u=f(v)$为方程$(1.1)$的弱解,下面分别在$X$和$Y$中考虑方程(2.9)解的存在性.

首先,分别构造子空间$X_r(\subset X)$和$Y_r(\subset Y)$,函数$u\in {{L}^{p}}({{\mathbb{R}}^{N}})$称为径向非减函数如果有$u(x)\le u(y)$,$|x|\ge |y|$.记

$\begin{eqnarray}\begin{array}{rl}X_r&=\left\{u\in X: u \mbox{为${\Bbb R}^N$中非负,径向非增函数}\right\},\\[2mm]Y_r&=\left\{u\in Y: u \mbox{为${\Bbb R}^N$中非负,径向非增函数} \right\}.\end{array}\end{eqnarray}$

(2.11)

引理2.3^[3] 若$u\in L^p({\Bbb R}^N)(p\ge 1)$为非负且径向非减函数,则有

$\begin{eqnarray}|u(x)|\le |x|^{-N/p}\omega_N^{-1/p}\|u\|_p,\forall x\in{\Bbb R}^N\backslash\{0\}.\end{eqnarray}$

(2.12)

引理2.4^[7] 以下性质成立:

(i) 若$1<p<N$,$s\in(p,p^*)$,则嵌入$X_r\hookrightarrow L^s({\Bbb R}^N)$为紧的;

(ii) 若$p=N$且$s>N$,则嵌入$Y_r\hookrightarrow L^s({\Bbb R}^N)$为紧的.

设非负函数$u\in L^p({\Bbb R}^N)(p\ge 1)$,记

$ \begin{eqnarray}\Omega(t)=\{x\in {\Bbb R}^N | u(x)>t\};\mu(t)={\rm meas}(\Omega(t)).\end{eqnarray}$

(2.13)

这里meas$(\Omega)$为$\Omega$的Lebesgue测度,因为$u\in L^{p}({\Bbb R}^N)$故$\mu(t)<+\infty$.Schwarz对称化构造径向函数$u^*: {\Bbb R}^N\to {\Bbb R}^{+}$使得

$ \begin{eqnarray}\{x\in {\Bbb R}^N| u^*(x)>t\}=B_{\rho(t)},~~ {meas}(B_{\rho(t)})=\mu(t),\end{eqnarray}$

(2.14)

其中$B_{\rho(t)}$为原点为中心半径为$\rho(t)$的球域,显然$u^*$为径向非减函数,具有如下性质.

引理2.5 ^[4](i)~若$f: {\Bbb R}^+\to {\Bbb R}^+$为连续递增函数且满足$f(0)=0$,则有

$$\int_{{\Bbb R}^N}f(u^*){\rm d}x=\int_{{\Bbb R}^N}f(u){\rm d}x.$$

(ii) 当$ p\ge 1$时.若$u\in W^{1,p}({\Bbb R}^N),u\ge 0$,则有$u^*\in W^{1,p}({\Bbb R}^N)$和

$$\int_{{\Bbb R}^N}|\nabla u^*|^p {\rm d}x \le\int_{{\Bbb R}^N}|\nabla u|^p {\rm d}x.$$

注2.2 引理2.5说明假如$u\in X,u\ge 0$,则$u^*\in X_r$,假如$u\in Y,u\ge 0$,则$u^*\in Y_r$.

为了证明方程(2.9)在$X$和$Y$中非平凡解的存在性,引入Nehari流形

$$\begin{eqnarray*}{\cal N}&=&\left\{v\in X\backslash\{0\}:J'(v)v=0\right\}\\&=&\left\{v\in X\backslash\{0\}:\|\nabla v\|_p^p+\int_{{\Bbb R}^N}[|f(v)|^{p-2}f(v)f'(v)v-h(f(v))f'(v)v]{\rm d}x=0\right\},\\{\cal M}&=&\left\{v\in Y\backslash\{0\}:J'(v)v=0\right\}\\&=&\left\{v\in Y\backslash\{0\}:\|\nabla v\|_N^N+\int_{{\Bbb R}^N}[|f(v)|^{N-2}f(v)f'(v)v-h(f(v))f'(v)v]{\rm d}x=0\right\}\end{eqnarray*} $$

及映射$\phi_v(t)=J(tv)$,$t>0$.显然有$v\in {\cal N}({\cal M})$当且仅当$\phi_v'(1)=0$,更一般的有$tv\in {\cal N}({\cal M})$当且仅当$\phi_v'(t)=0$.由定义可知

$\begin{eqnarray}\begin{array}{l} \phi_v(t)=J(tv)=\frac{1}{p}\|t\nabla v\|_p^p+\frac{1}{p}\int_{{\Bbb R}^N}|f(tv)|^p{\rm d}x-\int_{{\Bbb R}^N}H(f(tv)){\rm d}x,\\[3mm]\phi'_v(t)=t^{p-1}\|\nabla v|_p^p+\int_{{\Bbb R}^N}|f(tv)|^{p-2}f(tv)f'(tv)v{\rm d}x-\int_{{\Bbb R}^N}h(f(tv))f'(tv)v{\rm d}x.\end{array}\end{eqnarray}$

(2.15)

注意到若$v\in {\cal N}$则有

$\begin{eqnarray}J(v)&=&\frac{\lambda}{p}\int_{{\Bbb R}^N}\left[|f(v)|^{m-2}f(v)f'(v)v-\frac{p}{m}|f(v)|^m\right]{\rm d}x\\&&+\frac{1}{p}\int_{{\Bbb R}^N}|f(v)|^{p-2}\left(f^2(v)-f(v)f'(v)v\right){\rm d}x\end{eqnarray}$

(2.16)

以及若$v\in {\cal M}$,有

$ \begin{eqnarray}J(v)&=&\frac{1}{N}\int_{{\Bbb R}^N}(h(f(v))f'(v)v-NH(f(v))){\rm d}x\\&&+\frac{1}{N}\int_{{\Bbb R}^N}|f(v)|^{N-2}(f^2(v)-f(v)f'(v)v){\rm d}x.\end{eqnarray}$

(2.17)

下面在假设$(A_1)-(A_3)$下,讨论${\cal N}$和${\cal M}$的一些性质.

引理2.6 Nehari流形${\cal N}$和${\cal M}$是非空的.

证 选取非负函数$v_0\in C_0^{\infty}({\Bbb R}^N)$使得

$\begin{eqnarray}\int_{{\Bbb R}^N}h(f(v_0))f'(v_0)v_0{\rm d}x>0\end{eqnarray}$

(2.18)

及$\|v_0\|_E\le \rho$对充分小的$\rho>0$成立.对$t\ge 0$,假设

$\begin{eqnarray} \gamma(t)&=&J'(tv_0)tv_0\\ &=&t^p\|\nabla v_0\|_p^p+\int_{{\Bbb R}^N}(f(tv_0))^{p-1}f'(tv_0)tv_0{\rm d}x-\int_{{\Bbb R}^N}h(f(tv_0))f'(tv_0)tv_0{\rm d}x.\end{eqnarray}$

(2.19)

首先假设$(A_1)$取$t>0$足够小使$0\le v_0\le 1$在${\Bbb R}^N$中.由引理2.2,可得

$\begin{equation}\int_{{\Bbb R}^N}(f(tv_0))^{p-1}f'(tv_0)tv_0{\rm d}x\ge\frac{1}{2\alpha}\int_{{\Bbb R}^N}(f(tv_0))^{p}{\rm d}x\ge\frac{1}{2\alpha}b_0^pt^p\|v_0\|_p^p,\end{equation}$

(2.20)

$\begin{equation} \int_{{\Bbb R}^N}h(f(tv_0))f'(tv_0)tv_0{\rm d}x=\lambda\int_{{\Bbb R}^N}|f(tv_0)|^{m-2}f(tv_0)f'(tv_0)tv_0{\rm d}x\le\lambda t^m\|v_0\|_m^m.\end{equation}$

(2.21)

由(2.18)-(2.21)式推出对适当小的$t>0$有$\gamma(t)>0$.又因为$m>2\alpha p$,故可以适当选择$s>p$使$m>2\alpha s$.现在证明当$t\to+\infty$时有$\gamma(t)\to-\infty$.作辅助函数

$\begin{eqnarray}G_1(t)=t^{-s+1}(f(tv_0))^{m-1}f'(tv_0)v_0-(f(v_0))^{m-1}f'(v_0)v_0,t\ge 1.\end{eqnarray}$

(2.22)

由引理2.2,通过计算可得$G_1'(t)\ge 0,t\ge 1$.故$G_1(t)\ge G_1(1)=0$和

$\begin{eqnarray}\gamma(t)\le t^p(\|\nabla v_0\|_p^p+\|v_0\|_p^p)-\lambda t^s\int_{{\Bbb R}^N}(f(v_0))^{m-1}f'(v_0)v_0{\rm d}x\to -\infty,t\to+\infty.\end{eqnarray}$

(2.23)

因此$\exists t_1>1$使得$\gamma(t_1)=0$.显然$t_1v_0\not \equiv 0$,故$t_1v_0\in{\cal N}$和${\cal N}$是非空的.

下面考虑条件$(A_2)-(A_3)$.由引理2.2,当$0\le tv_0\le 1$时,则(2.20)式恒成立.记$\Omega=\{x\in{{\Bbb R}^N}:v_{0}(x)\neq 0\}$,则$\Omega$是${\Bbb R}^N$中有界区域,且$\|v_0\|_{L^{s}(\Omega)}=\|v_0\|_{L^{q}({\Bbb R}^N)}=\|v_0\|_{q},$$\|v_0\|_{W_{0}^{1,n}(\Omega)}\le \|v_0\|_{Y}$.那么由(1.8),(2.4)式可得

$\begin{eqnarray}0&\le& \int_{{\Bbb R}^N}h(f(tv_0))f'(tv_0)tv_0{\rm d}x\\&\le& b_1\int_{{\Bbb R}^N}|f(tv_0)|^q[\exp(\alpha_0|f(tv_0)|^{\frac{2\alpha N}{N-1}})-S_{N-2}(\alpha_0,f(tv_0))]{\rm d}x\\& \le&\sum_{k=N-1}^{\infty}\frac{b_1\alpha_0^k}{k!}\int_{{\Bbb R}^N}|f(tv_0)|^{q+\frac{2\alpha Nk}{N-1}}{\rm d}x\\&\le& \sum_{k=N-1}^{\infty}\frac{b_2\alpha_0^k}{k!}(2\alpha)^{\frac{k}{N-1}}t^{\frac{q}{2\alpha}+\frac{kN}{N-1}}\int_{{\Bbb R}^N}|v_0|^{q_k+q_0}{\rm d}x\\&\le&b_2t^{N+q_0}\sum_{k=N-1}^{\infty}\frac{\alpha_0^k}{k!}(2\alpha)^{\frac{k}{N-1}}d_N^{q_k+q_0}(q_k+q_0)^{(1-\frac{1}{N})(q_k+q_0)}\|v_0\|_Y^{q_k+q_0}\\&\le& b_2t^{q_0+N}d_N^{q_0}\|v_0\|_Y^{q_0}\sum_{k=N-1}^{\infty}a_k,\end{eqnarray}$

(2.24)

其中$d_N$在(2.3)式中给出,以及

$\begin{eqnarray}\begin{array}{l} b_2=b_1(2\alpha)^{\frac{q}{2\alpha N}},~~q_0=\frac{q}{2\alpha},~~\beta=\frac{N+q_0}{N-1},~~q_k=\frac{kN}{N-1},\\[3mm] a_k=\frac{((2\alpha)^{\frac{1}{N-1}}\alpha_0)^k}{k!}d^{q_k}_N\|v_0\|_Y^{q_k}(\beta k)^{k+q_0(1-\frac{1}{N})}.\end{array}\end{eqnarray}$

(2.25)

因为$\rho>0$足够小可使

$\begin{eqnarray}\lim_{k\to\infty}\frac{a_{k+1}}{a_k}=e\alpha_0\beta(2\alpha)^{\frac{1}{N-1}}\|v_0\|_Y^{\frac{N}{N-1}}d_N^{\frac{N}{N-1}}\le e \alpha_0\beta(2\alpha)^{\frac{1}{N-1}}\rho^{\frac{N}{N-1}}d_N^{\frac{N}{N-1}}<1,\end{eqnarray}$

(2.26)

那么正项级数$\sum\limits_{k=N-1}^{\infty}a_k$收敛.因此$\exists C_1>0$使得

$\begin{eqnarray}0\le \int_{{\Bbb R}^N}h(f(tv_0))f'(tv_0)tv_0{\rm d}x \le C_1t^{q_0+N}\|v_0\|_Y^{q_0}.\end{eqnarray}$

(2.27)

由(2.19),(2.20)和(2.27)式可得对充分小的$t>0$有$\gamma(t)>0$.又由于$q>0$,故可以选择$s>N$使得$q>2\alpha(s-N)$.令

$\begin{eqnarray}G_2(t)=t^{-s+1}h(f(tv_0))f'(tv_0)v_0-h(f(v_0))f'(v_0)v_0,t\ge1.\end{eqnarray}$

(2.28)

可知当$t\ge 1$时有$G'_2(t)\ge 0$成立.从而得到

$\begin{eqnarray}h(f(tv_0))f'(tv_0)tv_0\ge t^sh(f(v_0))f'(v_0)v_0,t\ge 1.\end{eqnarray}$

(2.29)

应用(2.19)和(2.29)式那么,当$t\to+\infty$时

$\begin{eqnarray}\gamma(t)\le t^N(\|\nabla v_0\|_N^N+\|v_0\|_N^N)-t^s\int_{{\Bbb R}^N}h(f(v_0))f'(v_0)v_0{\rm d}x\to-\infty.\end{eqnarray}$

(2.30)

故$\exists t_2>1$使得$\gamma(t_2)=0$及$t_2v_0\in{\cal M}$.显然在${\Bbb R}^N$中$t_2v_0\not \equiv 0$,所以${\cal M}$非空.证毕.

引理2.7 假设$(A_1)-(A_3)$,则

$$ d_1=\inf_{v\in{\cal N}}\{\|v\|_X\}>0,~~d_2=\inf_{v\in{\cal N}}\{J(v)\}\ge 0,$$ $$ d_3=\inf_{v\in{\cal M}}\{\|v\|_Y\}>0,~~d_4=\inf_{v\in{\cal M}}\{J(v)\}\ge 0.$$

证 首先假设$(A_1)$成立,设$v\in {\cal N}$,由(2.16)式和$(f_6)$可推出

$\begin{eqnarray}J(v)&=&\frac{1}{p}\int_{{\Bbb R}^N}|f(v)|^{p-2}(f^2(v)-f(v)f'(v)v){\rm d}x\\&&+\frac{\lambda}{p}\int_{{\Bbb R}^N}|f(v)|^{m-2}f(v)f'(v)v{\rm d}x-\frac{\lambda}{m}\int_{{\Bbb R}^N}|f(v)|^{m}{\rm d}x\\&\ge& \lambda\left(\frac{1}{2\alpha p}-\frac{1}{m}\right)\int_{{\Bbb R}^N}|f(v)|^m{\rm d}x>0.\end{eqnarray}$

(2.31)

即$d_2\ge 0$.其次证明$d_1>0$.

若假设矛盾,则$\exists \{v_n\} \subset{\cal N}$使得$0<\|v_n\|_X\le \epsilon_n \to 0$,$n\to\infty$.注意到

$\begin{eqnarray} \|\nabla v_n\|_p^p+\int_{{\Bbb R}^N}|f(v_n)|^{p-2}f(v_n)f'(v_n)v_n{\rm d}x&=&\int_{{\Bbb R}^N}\lambda|f(v_n)|^{m-2}f(v_n)f'(v_n)v_n{\rm d}x\\& \le &\lambda\|v_n\|_m^m\le C_2\|v_n\|_X^m,\end{eqnarray}$

(2.32)

其中$C_0$为(2.2)式中一常数,$C_2=\lambda C_0^m$和

$\begin{eqnarray}\int_{{\Bbb R}^N}|f(v_n)|^{p-2}f(v_n)f'(v_n)v_n{\rm d}x \ge\frac{1}{2\alpha}\int_{{\Bbb R}^N}|f(v_n)|^{p}{\rm d}x\ge\frac{b_0^p}{2\alpha}\int_{{\Bbb R}^N}|v_n|^{p}{\rm d}x.\end{eqnarray}$

(2.33)

取$C_3=\min\{1,b_0^p/2\alpha\}>0$,应用(2.32)-(2.33)式可知$ n\to \infty$时

$\begin{eqnarray}0<C_3\le C_2\|v_n\|_X^{m-p}\le C_2\epsilon_n^{m-p}\to 0.\end{eqnarray}$

(2.34)

矛盾.因此在$(A_1)$条件下有$d_1>0$.

下面考虑$(A_2)-(A_3)$.取$v\in {\cal M}$,由(1.10),(2.17)式,可推出

$\begin{eqnarray}J(v)&=&\frac{1}{N}\int_{{\Bbb R}^N}(h(f(v))f'(v)v-NH(f(v))){\rm d}x+\frac{1}{N}\int_{{\Bbb R}^N}|f(v)|^{N-2}(f^2(v)-f(v)f'(v)v){\rm d}x\\&\ge& \frac{1}{N}\int_{{\Bbb R}^N}h(f(v))f(v)\left(\frac{f'(v)v}{f(v)}-\frac{N}{q+\\alpha N}\right){\rm d}x\\&\ge& \frac{q}{2\alpha N(q+2\alpha N)}\int_{{\Bbb R}^N}h(f(v))f(v){\rm d}x>0.\end{eqnarray}$

(2.35)

故$d_4\ge 0$.最后来说明$d_3>0$.

假设矛盾,则$\exists \{v_n\} \subset{\cal M}$使得$0<\|v_n\|_Y\le \epsilon_n \to 0$,$n\to\infty$.可知

$\begin{eqnarray}&&\|\nabla v|_N^N+\int_{{\Bbb R}^N}|f(v_n)|^{N-2}f(v_n)f'(v_n)v_n{\rm d}x\\&\le& \int_{{\Bbb R}^N}h(f(v_n))f(v_n){\rm d}x\\& \le&\sum_{k=N-1}^{\infty}\frac{b_2\alpha_0^k}{k!}(2\alpha)^{\frac{k}{N-1}}d_N^{q_k+q_0}(q_k+q_0)^{(1-\frac{1}{N})(q_k+q_0)}\|v_n\|_Y^{q_k+q_0}\\&\le&b_2\|v_n\|_Y^{q_0+N}d_N^{q_0}\beta^{\frac{q_0(N-1)}{N}}\sum_{k=N-1}^{\infty}b_k,\end{eqnarray}$

(2.36)

其中$d_N$,$b_2,q_0,\beta,q_k$在前面已知以及

$\begin{eqnarray}b_k=\frac{\alpha_0^k}{k!}\beta^{k}(2\alpha)^{\frac{k}{N-1}}d_N^{q_k}\|v_n\|_Y^{\frac{N(k-N+1)}{N-1}}k^{k+\frac{q_0(N-1)}{N}},k\ge N-1.\end{eqnarray}$

(2.37)

因为$\lim\limits_{k\to\infty}\frac{b_{k+1}}{b_k}\le e \alpha_0\beta(2\alpha)^{\frac{1}{N-1}}d_N^{\frac{N}{N-1}}\epsilon_n^{\frac{N}{N-1}}<1,$所以正项级数$\sum\limits_{k=N-1}^{\infty}b_k$收敛.

记$B_0=\sum\limits_{k=N-1}^{\infty}b_k$.进一步由$(f_6)$,$(f_8)$可得

$\begin{eqnarray}&&\|\nabla v|_N^N+\int_{{\Bbb R}^N}|f(v_n)|^{N-2}f(v_n)f'(v_n)v_n{\rm d}x\\&\ge& \|\nabla v|_N^N+\frac{1}{2\alpha}\int_{{\Bbb R}^N}|f(v_n)|^{N}{\rm d}x\ge \|\nabla v_n\|_N^N+\frac{b_0^N}{2\alpha}\|v_n\|_N^{N}\ge C_3\|v_n\|_Y^N,\end{eqnarray} $

(2.38)

其中$C_3=\min\{1,b_0^N/2\alpha\}>0$.根据(2.36)和(2.38)式获得

$\begin{eqnarray}0<C_3\le b_2\|v_n\|_Y^{q_0}d_N^{q_0}\beta^{\frac{q_0(N-1)}{N}}B_0\le b_2 \epsilon_n^{q_0}d_N^{q_0}\beta^{\frac{q_0(N-1)}{N}}B_0.\end{eqnarray}$

(2.39)

说明若$\epsilon_n$足够小这是矛盾的,因此$d_3>0$.证毕.

引理2.8 当$1<p<N$时,存在非负函数$v_0\in {\cal N}$使得$J(v_0)=d_2>0$.当$p=N$时,存在非负函数$v_0\in {\cal M}$使得$J(v_0)=d_4>0$.

证 首先考虑$p\in(1,N)$的情况,证明在${\cal N}\cap X_r$中存在$d_2$的极小化序列.为此,假设$\{z_n\}\subset{\cal N}$为$d_2$的极小化序列,即当$n \to \infty$时,$J(z_n)\to d_2$和在$X^*$中$J'(z_n)\to 0$.事实上$J$为偶泛函有$J(z_n)=J(|z_n|)$蕴含着$\{|z_n|\}$也为一极小化序列,因此可假设在${\Bbb R}^N$中$z_n\ge 0$几乎处处成立.与文献[17,引理3.1]证明方法类似,可推得$\{z_n\}$在$X$中有界.令$z_n^*\in X_r$为$z_n$的Schwarz对称化.注意到$g_1(t)=|f(t)|^{m-2}f(t)f'(t)t,g_2(t)=|f(t)|^{p-2}f(t)f'(t)t,g_3(t)=|f(t)|^p$和$g_4(t)=|f(t)|^m$皆为${\Bbb R}^+$上的非负递增函数,由引理2.5可知

$\begin{eqnarray}\int_{{\Bbb R}^N}|\nabla z_n^*|^p {\rm d}x\le\int_{{\Bbb R}^N}|\nabla z_n|^p {\rm d}x,~~\int_{{\Bbb R}^N}g_k(z_n^*){\rm d}x=\int_{{\Bbb R}^N}g_k(z_n){\rm d}x,~~ k=1,2,3,4.\end{eqnarray}$

(2.40)

因为$z_n\in {\cal N}$,所以

$\begin{eqnarray}\|\nabla z_n^*\|_p^p+\int_{{\Bbb R}^N}g_2(z_n^*){\rm d}x &\le& \|\nabla z_n\|_p^p+\int_{{\Bbb R}^N}g_2(z_n){\rm d}x\\&=&\lambda\int_{{\Bbb R}^N}g_1(z_n){\rm d}x=\lambda\int_{{\Bbb R}^N}g_1(z_n^*){\rm d}x.\end{eqnarray}$

(2.41)

因此,如果令

$\begin{eqnarray}\gamma_n(t)=J'(tz_n^*)tz_n^*=\|\nabla v_n^*\|_p^p+\int_{{\Bbb R}^N}g_2(tz_n^*){\rm d}x-\lambda\int_{{\Bbb R}^N}g_1(tz_n^*){\rm d}x,t\ge 0,\end{eqnarray}$

(2.42)

则$\gamma_n(1)\le 0$,对充分小的$t>0$有$\gamma_n(t)>0$,则$\exists t_n\in(0,1]$使得$\gamma_n(t_n)=0$和$t_nz_n^*\in {\cal N}$.容易验证函数$G_3(t)=|f(t)|^{m-2}f(t)f'(t)t-\frac{p}{m}|f(t)|^m$,$G_4(t)=f^p(t)-f^{p-1}(t)f'(t)t$在${\Bbb R}^+$上递增.由(2.16)式和引理2.5得

$\begin{eqnarray}d_2\le J(t_nz_n^*)&=&\frac{\lambda}{p}\int_{{\Bbb R}^N}G_3(t_nz_n^*){\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}G_4(t_nz_n^*){\rm d}x\\&\le& \frac{\lambda}{p}\int_{{\Bbb R}^N}G_3(z_n^*){\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}G_4(z_n^*){\rm d}x\\&=&\frac{\lambda}{p}\int_{{\Bbb R}^N}G_3(z_n){\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}G_4(z_n){\rm d}x=J(z_n)\to d_2.\end{eqnarray}$

(2.43)

这表明$\{t_nz_n^*\}$为$d_2$的极小化序列且$t_nz_n^*\in {\cal N}\cap X_r$.记$v_n=t_nz_n^*\ge 0$,不失一般性,可设在$X$中有$v_n\rightharpoonup v_0$.由引理2.4可知$\|V_{n}-V_{0}\|_{L^{s}({\Bbb R}^N)}\to 0$由此得$v_n\to v_0$在${\Bbb R}^N$中几乎处处成立,所以$v_0(x)\ge 0$及$v_0\in X_r$.下面证明$v_0\in {\cal N}$及$J(v_0)=d_2>0.$首先在$X$中$v_0\not=0$,否则$v_n\to 0$,$n\to\infty$.由(2.34)式这是不可能的.于是有$\|v_0\|_{X}>0$.此外由$v_n\in {\cal N}$,弱下半连续和Fatou引理得

$\begin{eqnarray}&&\|\nabla v_0\|_p^p+\int_{{\Bbb R}^N}|f(v_0)|^{p-2}f(v_0)f'(v_0)v_0{\rm d}x\\&\le&\liminf_{n\to\infty}(\|\nabla v_n\|_p^p+\int_{{\Bbb R}^N}|f(v_n)|^{p-2}f(v_n)f'(v_n)v_n){\rm d}x\\&=&\lambda\liminf_{n\to\infty}\int_{{\Bbb R}^N}|f(v_n)|^{m-2}f(v_n)f'(v_n)v_n{\rm d}x.\end{eqnarray}$

(2.44)

接下来会用到如下结论(待证)

$\begin{eqnarray}\begin{array}{l} \lim_{n\to\infty}\int_{{\Bbb R}^N}|f(v_n)|^{m-1}f'(v_n)v_n{\rm d}x=\int_{{\Bbb R}^N}|f(v_0)|^{m-1}f'(v_0)v_0{\rm d}x,\\[3mm]\lim_{n\to\infty}\int_{{\Bbb R}^N}|f(v_n)|^m{\rm d}x=\int_{{\Bbb R}^N}|f(v_0)|^m{\rm d}x.\end{array}\end{eqnarray}$

(2.45)

如果$\|\nabla v_0\|_p^p+\int_{{\Bbb R}^N}|f(v_0)|^{p-2}f(v_0)f'(v_0)v_0{\rm d}x=\lambda\int_{{\Bbb R}^N}|f(v_0)|^{m-2}f(v_0)f'(v_0)v_0{\rm d}x,$则有$v_0\in {\cal N}.$若上式不成立,则有

$\begin{eqnarray}\|\nabla v_0\|_p^p+\int_{{\Bbb R}^N}|f(v_0)|^{p-2}f(v_0)f'(v_0)v_0{\rm d}x<\lambda\int_{{\Bbb R}^N}|f(v_0)|^{m-2}f(v_0)f'(v_0)v_0{\rm d}x.\end{eqnarray}$

(2.46)

令$\gamma(t)=J'(tv_0)tv_0$,显然$\gamma(1)<0$.对充分小的$t>0$,有$\gamma(t)>0$.这表明$\exists t\in(0,1)$使$\gamma(t)=0$和$tv_0\in {\cal N}$成立.由函数$G_3(t)$,$G_4(t)$在${\Bbb R}^+$上递增以及(2.16),(2.45)式和Fatou引理可得

$$\begin{eqnarray*}d_2&\le&J(tv_0)=\frac{\lambda}{p}\int_{{\Bbb R}^N}G_3(tv_0){\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}G_4(tv_0){\rm d}x\\&<& \frac{\lambda}{p}\int_{{\Bbb R}^N}G_3(v_0){\rm d}x+\frac{1}{p}\int_{{\Bbb R}^N}G_4(v_0){\rm d}x\\&\le&\frac{\lambda}{p}\liminf_{n\to\infty}\int_{{\Bbb R}^N}G_3(v_n){\rm d}x+\frac{1}{p}\liminf_{n\to\infty}\int_{{\Bbb R}^N}G_4(v_n){\rm d}x\\&=& \liminf_{n\to\infty}J(v_n)=d_2.\end{eqnarray*}$$

这是矛盾的,因此(2.45)式得证,于是$v_0\in{\cal N}$.又弱下半连续得$J(v_0)\le\liminf\limits_{n\to\infty}J(v_n)=d_2$.另一方面对$\forall v\in {\cal N}$,$J(v)\ge d_2$.于是,$J(v_0)=d_2>0$.

下面来证明(2.45)式.$\{v_n\}$在$X$中有界可假设$\|v_n\|_{X}\le M(n\ge 1)$,$M>0.$则对$\forall r>0$,有$\|v_{n}-v_{0}\|_{L^{s}(B_r)}\to 0,s \in(p,p^*)$,在$B_r$中$v_n(x)\to v_0(x)$几乎处处成立,于是有

$\begin{eqnarray}\begin{array}{l} \lim_{n\to\infty}\int_{B_r}|f(v_n)|^{m-1}f'(v_n)v_n{\rm d}x=\int_{B_r}|f(v_0)|^{m-1}f'(v_0)v_0{\rm d}x,\\[3mm]\lim_{n\to\infty}\int_{B_r}|f(v_n)|^m{\rm d}x=\int_{B_r}|f(v_0)|^m{\rm d}x.\end{array}\end{eqnarray}$

(2.47)

接下来证明,对$\forall \epsilon>0$,$\exists r_0>1$使得$r\ge r_0$和$\forall n\ge 1$,

$ \begin{eqnarray}\begin{array}{l} \int_{B^c_r}|f(v_n)|^{m-1}|f'(v_n)v_n|{\rm d}x<\epsilon,\int_{B^c_r}|f(v_n)|^m{\rm d}x<\epsilon,\\[4mm]\int_{B^c_r}|f(v_0)|^{m-1}|f'(v_0)v_0|{\rm d}x<\epsilon,\int_{B^c_r}|f(v_0)|^m{\rm d}x<\epsilon\end{array}\end{eqnarray}$

(2.48)

成立.又由$(f_2)-(f_3)$及(2.12)式,得到

$ \begin{eqnarray}\int_{B^c_r}|f(v_n)|^{m-1}|f'(v_n)v_n|{\rm d}x&\le&\int_{B^c_r}|v_n(x)|^{m}{\rm d}x \\&\le& M^m\omega_N^{-\frac{m}{p}}\int_{B_r^c}|x|^{-\frac{mN}{p}}{\rm d}x\\&=&C_4r^{\frac{N(p-m)}{p}},\end{eqnarray}$

(2.49)

其中$C_4=pM^m\omega_N^{1-\frac{m}{p}}(m-p)^{-1}N^{-1}.$类似的有

$$\begin{eqnarray*}\int_{B^c_r}|f(v_0)|^{m-1}|f'(v_0)v_0|{\rm d}x,\int_{B^c_r}|f(v_n)|^{m}{\rm d}x,\int_{B^c_r}|f(v_0)|^{m}{\rm d}x\le C_4r^{\frac{N(p-m)}{p}}.\end{eqnarray*}$$

因为$m>2\alpha p>p$,故(2.45)式成立.

其次来考虑$(A_2)-(A_3)$.当$p=N$时,取$z_n\in {\cal M}$满足$J(z_n)\to d_4,n \to \infty$,在$Y^*$中$ J'(z_n)\to 0.$同样令$z_n^*\in Y_r$为$z_n$的Schwarz对称化及$t_n\in(0,1]$使得

$$\gamma_n(t_n)=J'(t_nz_n^*)t_nz_n^*=0,~~ n\ge 1.$$ 注意到函数$G_5(t)=h(f(t))f'(t)t-NH(f(t))$和$G_6(t)=f^N(t)-f^{N-1}(t)f'(t)t$在${\Bbb R}^+$上递增.因此当$n\to\infty$时,有

$\begin{eqnarray}d_4\le J(t_nz_n^*)&=&\frac{1}{N}\int_{{\Bbb R}^N}(G_5(t_nz_n^*)+G_6(t_nz_n^*)){\rm d}x\\&\le&\frac{1}{N}\int_{{\Bbb R}^N}(G_5(z_n^*)+G_6(z_n^*)){\rm d}x\\&=&\frac{1}{N}\int_{{\Bbb R}^N}(G_5(z_n)+G_6(z_n)){\rm d}x=J(z_n)\to d_4.\end{eqnarray}$

(2.50)

即$\{t_nz_n^*\}$也为$d_4$的极小化序列且$t_nz_n^*\in {\cal M}\cap Y_r$.取$v_n=t_nz_n^*\ge 0$.可假设,在$Y$中$v_n\rightharpoonup v_0$.由引理2.4,可得$s>N$时$\|v_{n}-v_{0}\|_{L^{s}({\Bbb R}^N)}\to 0,$在${\Bbb R}^N$中$v_n(x)\to v_0(x)\ge 0$几乎处处成立以及$v_0\in Y_r$.现在证明$v_0\in {\cal M}$和$J(v_0)=d_4.$要求在$Y$中$v_0\not=0$.否则,$v_n\to 0$,$n\to\infty$.按照(2.39)式,这是不可能的.于是有$\|v_0\|_{Y}>0$.由$v_n\in {\cal M}$,弱下半连续和Fatou引理得

$\begin{eqnarray}&&\|\nabla v_0\|_N^N+\int_{{\Bbb R}^N}|f(v_0)|^{N-2}f(v_0)f'(v_0)v_0{\rm d}x\\&\le&\liminf_{n\to\infty}\bigg(\|\nabla v_n\|_N^N+\int_{{\Bbb R}^N}|f(v_n)|^{N-2}f(v_n)f'(v_n)v_n{\rm d}x\bigg)\\&=&\liminf_{n\to\infty}\int_{{\Bbb R}^N}h(f(v_n))f'(v_n)v_n{\rm d}x.\end{eqnarray}$

(2.51)

同理会用到(待证)

$\begin{eqnarray}\begin{array}{l} \lim_{n\to\infty}\int_{{\Bbb R}^N}h(f(v_n))f'(v_n)v_n{\rm d}x=\int_{{\Bbb R}^N}h(f(v_0))f'(v_0)v_0{\rm d}x,\\[3mm]\lim_{n\to\infty}\int_{{\Bbb R}^N}H(f(v_n)){\rm d}x=\int_{{\Bbb R}^N}H(f(v_0)){\rm d}x.\end{array}\end{eqnarray}$

(2.52)

若$\|\nabla v_0\|_N^N+\int_{{\Bbb R}^N}|f(v_0)|^{N-2}f(v_0)f'(v_0)v_0{\rm d}x=\int_{{\Bbb R}^N}h(f(v_0))f'(v_0)v_0{\rm d}x,$则$v_0\in {\cal M}.$假设不成立,则有

$\begin{eqnarray}\|\nabla v_0\|_N^N+\int_{{\Bbb R}^N}|f(v_0)|^{N-2}f(v_0)f'(v_0)v_0{\rm d}x<\int_{{\Bbb R}^N}h(f(v_0))f'(v_0)v_0{\rm d}x.\end{eqnarray}$

(2.53)

证法与前面相同,于是有$J(v_0)=d_4 >0$.下面证明式(2.52).因$\{v_n\}$在$Y$中有界,可设$\|v_n\|_Y\le M(n\ge 1)$,$M>0.$与(2.47)式同样的证明方法可得

$\begin{eqnarray}\lim_{n\to\infty}\int_{B_r}h(f(v_n))f'(v_n)v_n{\rm d}x=\int_{B_r}h(f(v_0))f'(v_0)v_0{\rm d}x,\end{eqnarray}$

(2.54)

又由

$\begin{eqnarray} \int_{B^c_r}|h(f(v_n))f'(v_n)v_n|{\rm d}x&\le&\int_{B^c_r}h(f(v_n))f(v_n){\rm d}x\\&\le& \sum_{k=N-1}^{\infty}\frac{b_2\alpha_0^k}{k!}(2\alpha)^{\frac{k}{N-1}}\int_{B^c_r}|v_n(x)|^{q_0+q_k}{\rm d}x,\end{eqnarray}$

(2.55)

其中$b_2,q_0,q_k$在(2.25)式中已知,又

$$\begin{eqnarray*}\int_{B^c_r}|v_n(x)|^{q_k+q_0}{\rm d}x&\le&M^{q_k+q_0}\omega_N^{-\frac{q_k+q_0}{N}}\int_r^{\infty}\int_{|\omega|=1}\rho^{N-1-q_k-q_0}{\rm d}\omega {\rm d}\rho\\&=&M^{q_k+q_0}N\omega_N^{1-\frac{q_k+q_0}{N}}\frac{r^{N-q_k-q_0}}{q_k+q_0-N}\\&\le&\frac{r^{-q_0}}{q_0}M^{q_0}N\omega_N^{\frac{N-q_0}{N}}(M^{\frac{N}{N-1}}\omega_N^{-\frac{1}{N-1}})^k\\&\equiv&A^k\frac{r^{-q_0}}{q_0}M^{q_0}N\omega_N^{\frac{N-q_0}{N}}.\end{eqnarray*}$$

于是有$r\to \infty$时

$$\begin{eqnarray*}&&\int_{B^c_r}|h(f(u_n))f'(v_n)v_n|{\rm d}x\\&\le&\frac{r^{-q_0}}{q_0}b_2NM^{q_0}\omega_N^{\frac{N-q_0}{N}}\sum_{k=N-1}^{\infty}\frac{((2\alpha)^{1/(N-1)}A\alpha_0)^k}{k!}\to 0.\end{eqnarray*}$$

说明对$\forall\epsilon>0$,$\exists r_0>1$使得$r>r_0$和

$\begin{equation}\int_{B^c_r}|h(f(v_n))f'(v_n)v_n|{\rm d}x<\epsilon,~~ \forall n\ge 1.\end{equation}$

(2.56)

应用(2.54),(2.56)式可推出(2.52)式的第一个极限成立,类似的可推出第二个极限成立.证毕.

证 首先考虑$(A_1)$.显然若$v_0$为$J$在$X$上的临界点,即对所有$v\in X$,有$J'(v_0)v=0$和在$X^*$上$J'(v_0)=0$.对$\forall v\in X$,选取$\varepsilon>0$使得对所有$s\in(-\varepsilon,\varepsilon)$,有$v_0+sv\not=0$.定义函数$\varphi:(-\varepsilon,\varepsilon)\times(0,\infty)\to{\Bbb R}$为

$\begin{eqnarray}\varphi(s,t)&=&J'(t(v_0+sv))t(v_0+sv)\\&=&\int_{{\Bbb R}^N}|f(t(v_0+sv))|^{p-2}f(t(v_0+sv))f'(t(v_0+sv))t(v_0+sv){\rm d}x+\|\nabla t(v_0+sv)\|_p^p\\&&-\lambda\int_{{\Bbb R}^N}|f(t(v_0+sv))|^{m-2}f(t(v_0+sv))f'(t(v_0+sv))t(v_0+sv){\rm d}x.\end{eqnarray}$

(3.1)

有$\varphi(0,1)=J'(v_0)v_0$,

$$\begin{eqnarray*}\frac{\partial\varphi}{\partial t}(0,1)&=&(p-1)\int_{{\Bbb R}^N}|f(v_0)|^{p-2}((f'(v_0))^2v^2_0-f(v_0)f'(v_0)v_0){\rm d}x\\&&+\int_{{\Bbb R}^N}|f(v_0)|^{p-1}f"(v_0)v_0^2{\rm d}x+\lambda\int_{{\Bbb R}^N}F(v_0){\rm d}x,\end{eqnarray*}$$

其中$(f'(v_0))^2v^2_0-f(v_0)f'(v_0)v_0 \le 0,f"(v_0)\le 0$和在条件$m>2\alpha p$下

$$ \begin{eqnarray*}F(v_0)&=&|f(v_0)|^{m-2}\left[(p-1)f(v_0)f'(v_0)v_0-(m-1)(f'(v_0))^2v_0^2-f(v_0)f"(v_0)v_0^2\right]\\&\le&|f(v_0)|^{m-2}\left[2\alpha(p-1)(f'(v_0))^2v_0^2-(m-1)(f'(v_0))^2v_0^2-f(v_0)f"(v_0)v_0^2\right]\\&=&\frac{|f(v_0)|^{m-2}v_0^2}{\left[1+(2\alpha)^{p-1}|f(v_0)|^{(2\alpha-1)p}\right]^{\frac{2}{p}}}\left[2\alpha(p-1)-m+1+\frac{(2\alpha-1)(2\alpha)^{p-1}|f(v_0)|^{(2\alpha-1)p}}{1+(2\alpha)^{p-1}|f(v_0)|^{(2\alpha-1)p}}\right]\\&<&0.\end{eqnarray*}$$

于是$\frac{\partial\varphi}{\partial t}(0,1)<0$.由隐函数存在定理,存在一$C^1$函数$t:(-\varepsilon_0,\varepsilon_0)(\subseteq(-\varepsilon,\varepsilon))\to {\Bbb R}$使得$t(0)=1$和$\varphi(s,t(s))=0$,$s\in(-\varepsilon_0,\varepsilon_0)$.同时至少对充分小的$\varepsilon_0$,有$t(s)\not=0$.因此有$t(s)(v_0+sv)\in{\cal N}$.记$t=t(s)$以及

$\begin{eqnarray}\phi(s)=J(t(v_0+sv))&=&\frac{1}{p}\|\nabla t(v_0+sv)\|_p^{p}+\frac{1}{p}\int_{{\Bbb R}^N}|f(t(v_0+sv))|^{p}{\rm d}x\\&&-\frac{\lambda}{m}\int_{{\Bbb R}^N}|f(t(v_0+sv))|^m{\rm d}x.\end{eqnarray}$

(3.2)

可看出$\phi(s)$可微且在$s=0$处取得极小值点,则

$\begin{equation}0=\phi'(0)=t'(0)J'(v_0)v_0+J'(v_0)v.\end{equation}$

(3.3)

因为$v_0\in {\cal N}$,$J'(v_0)v_0=0$,由(3.3)式可得对任一$v\in X$都有$J'(v_0)v=0$也就有$X^*$中$J'(v_0)=0$.因此$v_0$为$J$的临界点,$v_0$为方程(2.9)在$X$中的弱解,相应的$u_0=f(v_0)$为(1.1)式的弱解.又由$J(v_0)=J(|v_0|)=d_2>0$,假设$v_0\ge 0$在${\Bbb R}^N$中几乎处处成立.由极值原理可得出$v_0(x)>0$和$u_0=f(v_0)>0$.

接下来考虑$(A_2)-(A_3)$.对每一$v\in Y$,同样的定义函数$\varphi(s,t)$为

$$\begin{eqnarray*}\varphi(s,t)&=&J'(t(v_0+sv))t(v_0+sv)\\&=&\int_{R^N}|f(t(v_0+sv))|^{N-2}f(t(v_0+sv))f'(t(v_0+sv))t(v_0+sv){\rm d}x\\&&+\|\nabla t(v_0+sv)\|_N^N-\int_{{\Bbb R}^N}h(f(t(v_0+sv)))f'(t(v_0+sv))t(v_0+sv){\rm d}x.\end{eqnarray*}$$

有$\varphi(0,1)=J'(v_0)v_0=0$,

$$\begin{eqnarray*}\frac{\partial\varphi}{\partial t}(0,1)&=&(N-1)\int_{{\Bbb R}^N}|f(v_0)|^{N-2}((f'(v_0))^2v^2_0-f(v_0)f'(v_0)v_0){\rm d}x\\&&+\int_{{\Bbb R}^N}|f(v_0)|^{N-1}f"(v_0)v_0^2{\rm d}x+\int_{{\Bbb R}^N}G(v_0){\rm d}x\end{eqnarray*}$$

以及$(f'(v_0))^2v^2_0-f(v_0)f'(v_0)v_0 \le 0,f"(v_0)\le 0$和在$q>0$的前提下

$$\begin{eqnarray*} G(v_0)&=&(N-1)h(f(v_0))f'(v_0)v_0-h'(f(v_0))(f'(v_0))^2v_0^2-h(f(v_0))f''(v_0)v_0^2 \\ &\le&\frac{h(f(v_0))}{f(v_0)}\left[(N-1)f(v_0)f'(v_0)v_0-(q-1+2\alpha N)(f'(v_0))^2v_0^2-f(v_0)f''(v_0)v_0^2\right] \\ &\le &\frac{h(f(v_0))}{f(v_0)} \left[(2\alpha(N-1)-(q-1+2\alpha N))(f'(v_0))^2v_0^2-f(v_0)f''(v_0)v_0^2\right] \\ &=& \frac{h(f(v_0))v_0^2}{f(v_0)\left[1+(2\alpha)^{N-1}f^{(2\alpha-1)N}(v_0)\right]^{2/N}} \left[1-q-2\alpha+\frac{(2\alpha-1)(2\alpha)^{N-1}f^{(2\alpha-1)N}(v_0)}{1+(2\alpha)^{N-1}f^{(2\alpha-1)N}(v_0)}\right] \\ &\le& 0. \end{eqnarray*}$$

于是$\frac{\partial\varphi}{\partial t}(0,1)<0$.与前面作法相同,可推出$v_0$为$J$的临界点,即$v_0$为(2.9)式在$Y$中的弱解,相应的$u_0=f(v_0)$为(1.1)式的弱解.由$J(v_0)=J(|v_0|)=d_4>0$,假设$v_0\ge 0$在${\Bbb R}^N$中几乎处处成立.最后由Harnack不等式,有$v_0(x)>0$.于是$u_0(x)=f(v_0(x))>0$在${\Bbb R}^N$中.定理1.1证毕.