## Existence and Asymptotic Behavior of Ground State for Fractional p-Laplacian Equations

Han Yaling,1, Xiang Jianlin,2

 基金资助: 国家自然科学基金.  61503415

 Fund supported: the NSFC.  61503415

Abstract

In this paper, existence and asymptotic behavior of a fractional p-Laplacian equation with biharmonic potential and critical nonlinear was considered. First, variational method is used to obtain the existence and nonexistence of ground state under different parameter β. Then, the technique of energy estimates was used to get the asymptotic properties of ground state as parameter β tends to critical case.

Keywords： Ground states ; Fractional p-Laplacian equations ; Energy estimates

Han Yaling, Xiang Jianlin. Existence and Asymptotic Behavior of Ground State for Fractional p-Laplacian Equations. Acta Mathematica Scientia[J], 2020, 40(6): 1622-1633 doi:

## 1 引言

$$$(-\Delta)_p^su+|x|^2|u|^{p-2}u-\beta|u|^{p+\frac{sp^2}{N}-2}u = \nu|u|^{p-2}u,$$$

$$$\alpha = \lim\limits_{ m \rightarrow\infty}J_{s, p}(u_m) = J_{s, q}(v_m) = \lim\limits_{ m \rightarrow\infty}\inf\limits_{u\in W^{s, p}({{\Bbb R}} ^N)}J_{s, p}(u).$$$

(ⅰ) $v_m^*\geq 0$, $\forall x\in {{\Bbb R}} ^N$;

(ⅱ) $v_m^*$是径向对称函数;

(ⅲ) 对任意$r\in [1, \infty]$, 若$v_m\in L^r({{\Bbb R}} ^N)$, 则$|v_m^*|_r = |v_m|_r$;

(ⅳ) 若$v_m\in W^{s, p}({{\Bbb R}} ^N)$, 则

$$$|v_m^*|\leq |x|^{-\frac{N}{p}},$$$

$|v_m^*|_p = 1$.利用条件(ⅲ)和(ⅳ), 可得

$\{v_m^*\}$在空间$W^{s, p}({{\Bbb R}} ^N)$中有界.则存在序列$\{v_m^*\}$的一个子列, 仍记作$\{v_m^*\}$, 存在$v\in W^{s, p}({{\Bbb R}} ^N)$, 使得

(2) 当$\beta>\beta^*$时.对常数$\lambda$, 设

$$$u_{\lambda}(x) = \frac{\lambda^{\frac{N}{p}}}{|Q|_{p}}Q(\lambda x).$$$

$\beta = \beta^*$.此时

$\begin{eqnarray} m(\beta_n)& = &E_{\beta_n}(u_{\beta_n}) = \frac{1}{p}[u_{\beta_n}]_{W^{s, p}}^p+\frac{1}{p}\int_{{{\Bbb R}} ^N} |x|^2|u_{\beta_n}|^p{\rm d}x-\frac{ N\beta_n}{p(N+sp)}\int_{{{\Bbb R}} ^N} |u_{\beta_n}|^{p+\frac{sp^2}{N}}{\rm d}x\\ & = &\frac{1}{p\varepsilon_{\beta_n}^{sp}}[v_{\beta_n}]_{W^{s, p}}^p+\frac{\varepsilon_{\beta_n}^{2}}{p}\int_{{{\Bbb R}} ^N} |x|^2|v_{\beta_n}|^p{\rm d}x-\frac{ N\beta_n}{p(N+sp)\varepsilon_{\beta_n}^{sp}}\int_{{{\Bbb R}} ^N} |v_{\beta_n}|^{p+\frac{sp^2}{N}}{\rm d}x\\ &\geq& \frac{\varepsilon_{\beta_n}^{2}}{p}\int_{{{\Bbb R}} ^N} |x|^2|v_{\beta_n}|^p{\rm d}x+\frac{N\varepsilon_{\beta_n}^2}{p(N+sp)}\int_{{{\Bbb R}} ^N} |v_{\beta_n}|^{p+\frac{sp^2}{N}}{\rm d}x. \end{eqnarray}$

$v$代入方程(3.9)并计算可得$\tilde{Q}$是方程(1.3)的解, 由于$v$满足方程(3.9), 结合Pohozaev恒等式可知$\tilde{Q}$是方程(1.3)的基态解.因此存在$x_0\in {{\Bbb R}} ^N$, 使得$\tilde{Q} = Q(x+x_0)$, 这意味着

$$$v = \left(\frac{N\nu}{sp\beta^*}\right)^{\frac{N}{sp^2}}Q\left(\left(\frac{N\nu}{sp}\right)^{\frac{1}{sp}}(x+x_0)\right).$$$

$\begin{eqnarray} &&\liminf\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^N} |x|^2|v_{\beta_n}|^p{\rm d}x\geq\int_{{{\Bbb R}} ^N} |x|^2|v|^p{\rm d}x \end{eqnarray}$

$\begin{eqnarray} & = &(\beta^*)^{-\frac{N}{sp}}\left(\frac{N\nu}{sp}\right)^{-\frac{2}{sp}}\int_{{{\Bbb R}} ^N} |x-\left(\frac{N\nu}{sp}\right)^{\frac{1}{sp}}x_0|^2|Q|^p{\rm d}x\\ &\geq& (\beta^*)^{-\frac{N}{sp}}\left(\frac{N\nu}{sp}\right)^{-\frac{2}{sp}}\int_{{{\Bbb R}} ^N} |x|^2|Q|^p{\rm d}x, \end{eqnarray}$

$$$\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^N} |v_{\beta_n}|^{p+\frac{sp^2}{N}}{\rm d}x = \int_{{{\Bbb R}} ^N} |v|^{p+\frac{sp^2}{N}}{\rm d}x = \frac{\nu(N+sp)}{sp\beta^*}.$$$

$\mu = \left(\frac{2\int_{{{\Bbb R}} ^N} |x|^2|Q|^p{\rm d}x}{sp}\right)^{\frac{1}{2+sp}}$, 将$\nu$$x_0$代入表达式(3.11)可得

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