﻿ 整数阶Choquard方程三解的存在性 Existence of Three Solutions for a Choquard Equation

Pure Mathematics
Vol. 09  No. 03 ( 2019 ), Article ID: 30129 , 8 pages
10.12677/PM.2019.93039

Existence of Three Solutions for a Choquard Equation

Yue Li, Anran Hou

School of Mathematics, Yunnan Normal University, Kunming Yunnan

Received: Apr. 15th, 2019; accepted: Apr. 26th, 2019; published: May 9th, 2019

ABSTRACT

We study the following Choquard equation by the Theorem 1.1 in [1]

$\left\{\begin{array}{l}-\Delta u=\beta \left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)+\lambda u-{|u|}^{p-2}u+h\left(x\right),\text{}x\in \Omega \\ u=0,\text{}\text{\hspace{0.17em}}\text{}x\in \partial \Omega \end{array}$

where, $\Omega \subset {R}^{3}$ is an open, and bounded domain with a smooth boundary, $h\in {L}^{2}\left(\Omega \right)$ , $0<\mu <3$ , $4 , $\beta >0$ , $\lambda >0$ . Under suitable assumption $f\in C\left(ℝ,ℝ\right)$ , we prove this problem at least three weak solutions.

Keywords:Choquard Equation, Three Critical Points

$\left\{\begin{array}{l}-\Delta u=\beta \left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)+\lambda u-{|u|}^{p-2}u+h\left(x\right),\text{}x\in \Omega \\ u=0,\text{}\text{\hspace{0.17em}}\text{}x\in \partial \Omega \end{array}$

1. 引言

$\begin{array}{l}-{\epsilon }^{2}\Delta u+V\left(x\right)u\\ ={\epsilon }^{\mu -N}\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)+h\left(u\right),\text{}x\in {ℝ}^{N}\end{array}$ (1.1)

(1.2)

$N=3$$q=2$$\mu =1$ 时的情况，是1954年Pekar在 [2] 中用来描述极化子静止时的量子理论时提出的。(1.2)式是1976年Choquard在 [3] 中描述单组分等离子体的Hartree-Fock理论时提出的。Lions在 [4] 中由临界点定理得到方程在 ${H}^{1}\left({ℝ}^{N}\right)$ 中有无穷多镜像解的存在性。对于基态解的一些性质，L. Ma和L. Zhao在 [5] 中证明了对于 $q\ge 2$ 时，广义的Choquard方程(1.2)式的每个正解都是径向对称的，并且单调递减到某一点。后来Moroz和Schaftingen在 [6] [7] 中消除了这种限制，并得出最佳参数的、基态的正则性和径向对称性，并推导出这些解在无限远处渐近衰减。还有一些人专注于半经典问题，即(1.1)式中的 $\epsilon \to 0$ 。非局部问题(1.1)的半经典解的存在性已经在 [8] 中给出。

$\left\{\begin{array}{l}-\Delta u=\beta \left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)+\lambda u-{|u|}^{p-2}u+h\left(x\right),\text{}x\in \Omega \\ u=0,\text{}x\in \partial \Omega \end{array}$ (1.3)

(f1) $\underset{t\to 0}{\mathrm{lim}}\frac{f\left(t\right)}{t}=0$ .

(f2) $\exists q\in \left(\frac{6-\mu }{3},\mathrm{min}\left\{6-\mu ,\frac{p}{2}\right\}\right)\text{}s.t.\text{}\underset{t\to \infty }{\mathrm{lim}}\frac{f\left(t\right)}{{t}^{q-1}}=0$ .

2. 泛函设置

$\Omega \in {ℝ}^{3}$ 是具有光滑边界的有界开集，Sobolev空间 ${W}_{0}^{1,2}\left(\Omega \right)$ 的范数为

${‖u‖}_{1,2}={\left({\int }_{\Omega }{|\nabla u|}^{2}\text{d}x\right)}^{\frac{1}{2}}$

Lebesgue空间 ${L}^{p}\left(\Omega \right)\text{\hspace{0.17em}}\left(p\ge 1\right)$ 的范数为

${\iint }_{{ℝ}^{2N}}\frac{f\left(x\right)h\left(y\right)}{{|x-y|}^{\mu }}\text{d}x\text{d}y\le C\left(r,N,\mu ,t\right){‖f‖}_{r}{‖h‖}_{t}$

(R) $\exists \text{}R>0\text{}s.t.\text{}\underset{u\in \partial {B}_{Y}\left(R\right)}{\mathrm{max}}J\left(u\right)<\underset{u\in Z}{\mathrm{inf}}J\left(u\right)$

(PS) 对任意的序列 $\left\{{u}_{n}\right\}\subset X$ 使得 $\left\{J\left({u}_{n}\right)\right\}\subset ℝ$ 有界，并且 ${‖{J}^{\prime }\left({u}_{n}\right)‖}_{{X}^{\ast }}\to 0$ 有收敛子列。

$J\left(u\right)=\frac{1}{2}{\int }_{\Omega }{|\nabla u|}^{2}\text{d}x-\frac{\beta }{2}{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)F\left(u\right)\text{d}x-\frac{\lambda }{2}{\int }_{\Omega }{|u|}^{2}\text{d}x+\frac{1}{p}{\int }_{\Omega }{|u|}^{p}\text{d}x-\frac{1}{2}{\int }_{\Omega }{|h\left(x\right)|}^{2}\text{d}x$

a) $J\in {C}^{1}\left({W}_{0}^{1,2}\left(\Omega \right),ℝ\right)$ 并且满足

$〈{J}^{\prime }\left(u\right),\phi 〉={\int }_{\Omega }\nabla u\nabla \phi \text{d}x-\beta {\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)\phi \text{d}x-\lambda {\int }_{\Omega }u\phi \text{d}x+{\int }_{\Omega }{|u|}^{p-2}u\phi \text{d}x-{\int }_{\Omega }h\left(x\right)\phi \text{d}x$

b) $u\in {W}_{0}^{1,2}\left(\Omega \right)$ 是(1.3)的弱解，当且仅当 $u\in {W}_{0}^{1,2}\left(\Omega \right)$ 是J的临界点。

$f\left(t\right)\le \xi |t|+{C}_{\xi }{|t|}^{q-1},\text{}F\left(t\right)\le \xi {|t|}^{2}+{C}_{\xi }{|t|}^{q}$ (2.1)

$\begin{array}{c}|{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)F\left(u\right)\text{d}x|\le C{‖F\left(u\right)‖}_{t}{‖F\left(u\right)‖}_{t}\\ \le C{\left({\int }_{\Omega }{\left({|u|}^{2}+{|u|}^{q}\right)}^{t}\text{d}x\right)}^{\frac{2}{t}}\\ \le C\left({‖u‖}_{2t}^{4}+{‖u‖}_{qt}^{2q}\right)\end{array}$ (2.2)

$\begin{array}{c}J\left(u\right)=\frac{1}{2}{\int }_{\Omega }{|\nabla u|}^{2}\text{d}x-\frac{\beta }{2}{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)F\left(u\right)\text{d}x-\frac{\lambda }{2}{\int }_{\Omega }{|u|}^{2}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{p}{\int }_{\Omega }{|u|}^{p}\text{d}x-\frac{1}{2}{\int }_{\Omega }{|h\left(x\right)|}^{2}\text{d}x\\ \ge \frac{1}{2}{‖u‖}_{1,2}^{2}-C\left({‖u‖}_{2t}^{4}+{‖u‖}_{qt}^{2q}\right)-\frac{\lambda }{2}{‖u‖}_{2}^{2}+\frac{1}{p}{‖u‖}_{p}^{p}-\frac{1}{2}{‖h‖}_{2}^{2}\\ \ge \frac{1}{2}{‖u‖}_{1,2}^{2}-C\left({‖u‖}_{p}^{4}+{‖u‖}_{p}^{2q}+\frac{\lambda }{2}{‖u‖}_{p}^{2}\right)+\frac{1}{p}{‖u‖}_{p}^{p}-\frac{1}{2}{‖h‖}_{2}^{2}\end{array}$ (2.3)

${‖u‖}_{1,2}\to \infty$ 时，有以下两种情况:

i) 若 ${‖u‖}_{p}$ 有界，则有 $J\left(u\right)\to \infty$

ii) 若 ${‖u‖}_{p}\to \infty$ ，则由 $p>2q$ 以及 $p>4$ 可知~ $J\left(u\right)\to \infty$

$J\left(u\right)\ge -C\left({‖u‖}_{p}^{4}+{‖u‖}_{p}^{2q}+\frac{\lambda }{2}{‖u‖}_{p}^{2}\right)+\frac{1}{p}{‖u‖}_{p}^{p}-\frac{1}{2}{‖h‖}_{2}^{2}$

${u}_{n}\stackrel{弱}{\to }u\text{}于{W}_{0}^{1,2}，\text{}{u}_{n}\to u\text{}于\text{}{L}^{t}\left(\Omega \right)\text{}\forall t\in \left[1,{2}^{\ast }\right)$

$\begin{array}{c}{o}_{n}\left(1\right)=〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\\ ={‖{u}_{n}‖}_{1,2}^{2}-\beta {\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left({u}_{n}\right)\right)f\left({u}_{n}\right){u}_{n}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\lambda {\int }_{\Omega }{u}_{n}^{2}\text{d}x+{\int }_{\Omega }{|{u}_{n}|}^{p}\text{d}x-{\int }_{\Omega }h\left(x\right){u}_{n}\text{d}x\end{array}$

${‖{u}_{n}‖}_{1,2}^{2}=\beta {\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left({u}_{n}\right)\right)f\left({u}_{n}\right){u}_{n}\text{d}x+\lambda {‖{u}_{n}‖}_{2}^{2}-{‖{u}_{n}‖}_{p}^{p}+{\int }_{\Omega }h\left(x\right){u}_{n}\text{d}x+{o}_{n}\left(1\right)$ (2.4)

${‖u‖}_{1,2}^{2}=\beta {\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left({u}_{n}\right)\right)f\left({u}_{n}\right)u\text{d}x+\lambda {‖u‖}_{2}^{2}-{‖u‖}_{p}^{p}+{\int }_{\Omega }h\left(x\right)u\text{d}x+{o}_{n}\left(1\right)$ (2.5)

$\begin{array}{l}|{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left({u}_{n}\right)\right)f\left({u}_{n}\right){u}_{n}\text{d}x-{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left({u}_{n}\right)\right)f\left({u}_{n}\right)u\text{d}x|\\ \le {\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast |F\left({u}_{n}\right)|\right)|f\left({u}_{n}\right)||{u}_{n}-u|\text{d}x\\ \le C{\left({\int }_{\Omega }{|F\left({u}_{n}\right)|}^{t}\text{d}x\right)}^{\frac{1}{t}}{\left({\int }_{\Omega }{|f\left({u}_{n}\right)|}^{t}{|{u}_{n}-u|}^{t}\text{d}x\right)}^{\frac{1}{t}}\\ \le C{\left({\int }_{\Omega }{\left(\xi |{u}_{n}|+{C}_{\xi }{|{u}_{n}|}^{q-1}\right)}^{t}{|{u}_{n}-u|}^{t}\text{d}x\right)}^{\frac{1}{t}}\end{array}$

$\begin{array}{l}\le C{\left({\int }_{\Omega }{|{u}_{n}|}^{t}{|{u}_{n}-u|}^{t}\text{d}x+{\int }_{\Omega }{|{u}_{n}|}^{\left(q-1\right)t}{|{u}_{n}-u|}^{t}\text{d}x\right)}^{\frac{1}{t}}\\ \le C{\left({\left({\int }_{\Omega }{|{u}_{n}-u|}^{2t}\text{d}x\right)}^{\frac{1}{2}}+{\left({\int }_{\Omega }{|{u}_{n}-u|}^{qt}\text{d}x\right)}^{\frac{1}{q}}\right)}^{\frac{1}{t}}\\ ={o}_{n}\left(1\right)\end{array}$

3. 定理1.1的证明

$Β=\left\{{\phi }_{i}:i\in ℕ\right\}$

${W}_{0}^{1,2}\left(\Omega \right)$ 的规范正交基(参见文献 [11] 的Thm. 2.2.16)，并且 ${\lambda }_{k}<{\lambda }_{k+1}$ 。将 ${W}_{0}^{1,2}\left(\Omega \right)$ 分解为 $Y\oplus Z$ ，其中

$Y=\left\{\underset{i=1}{\overset{k}{\sum }}{a}_{i}{\phi }_{i}:{a}_{i}\in ℝ,{\phi }_{i}\in Β\right\},\text{}Z=\left\{\underset{i=k+1}{\overset{\infty }{\sum }}{a}_{i}{\phi }_{i}:{a}_{i}\in ℝ,{\phi }_{i}\in Β\right\}={Y}^{\perp }$ (3.1)

$u=\underset{i=k+1}{\overset{\infty }{\sum }}{a}_{i}{\phi }_{i},\text{}{‖u‖}_{1,2}^{2}=\underset{i=k+1}{\overset{\infty }{\sum }}{a}_{i}^{2}$

${\lambda }_{i}{\int }_{\Omega }{|{\phi }_{i}\left(x\right)|}^{2}\text{d}x={\int }_{\Omega }{|\nabla {\phi }_{i}\left(x\right)|}^{2}\text{d}x=1\text{}\forall i\in ℕ$ (3.2)

${\int }_{\Omega }{|\nabla u\left(x\right)|}^{2}\text{d}x-\lambda {\int }_{\Omega }{|u\left(x\right)|}^{2}\text{d}x=\underset{i=k+1}{\overset{\infty }{\sum }}{a}_{i}^{2}\left(1-\frac{\lambda }{{\lambda }_{i}}\right)\ge \left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){‖u‖}_{1,2}^{2}$ (3.3)

$\begin{array}{c}J\left(u\right)\ge \frac{1}{2}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){‖u‖}_{1,2}^{2}-\frac{\beta }{2}{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)F\left(u\right)\text{d}x+\frac{1}{p}{\int }_{\Omega }{|u|}^{2}\text{d}x-\frac{1}{2}{\int }_{\Omega }{|h\left(x\right)|}^{2}\text{d}x\\ \ge {C}_{1}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){‖u‖}_{p}^{2}-\beta {C}_{2}\left({‖u‖}_{p}^{2}+{‖u‖}_{p}^{2q}\right)+\frac{1}{p}{‖u‖}_{p}^{p}-\frac{1}{2}{‖h‖}_{2}^{2}\\ \ge -\frac{1}{2}{‖h‖}_{2}^{2}+{\alpha }_{\beta }\end{array}$ (3.4)

${g}_{\beta }\left(t\right)={C}_{1}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){t}^{2}+\frac{1}{p}{t}^{p}-\beta {C}_{2}\left({t}^{4}+{t}^{2q}\right)\text{,}t\ge 0$

${g}_{\beta }\left(t\right)\ge {C}_{1}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){t}^{2}+\frac{1}{p}{t}^{p}-{C}_{2}\left({t}^{2}+{t}^{2q}\right)>1$

${g}_{\beta }\left(t\right)$ 的最小值只能在区间 $\left[0,{t}_{0}\right]$ 上达到。因为 $\lambda <{\lambda }_{k+1}$ ，所以存在 ${\beta }_{0}>0$ ，当 $\beta \in \left(0,{\beta }_{0}\right)$ 时，对任意 $t\in \left(0,{t}_{0}\right)$

$\begin{array}{c}{g}_{\beta }\left(t\right)={t}^{2}\left({C}_{1}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right)+\frac{1}{p}{t}^{p-2}-\beta {C}_{2}\left({t}^{2}+{t}^{2q-2}\right)\right)\\ \ge {t}^{2}\left({C}_{1}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right)-\beta {C}_{2}\left({t}_{0}^{2}+{t}_{0}^{2q-2}\right)\right)\\ \ge 0\end{array}$ (3.5)

$u=\underset{i=1}{\overset{k}{\sum }}{a}_{i}{\phi }_{i},\text{}{‖u‖}_{1,2}^{2}=\underset{i=1}{\overset{k}{\sum }}{a}_{i}^{2}$

${\lambda }_{k}<\lambda$ ，以及(3.2)式可以推出

${\int }_{\Omega }{|\nabla u\left(x\right)|}^{2}\text{d}x-\lambda {\int }_{\Omega }{|u\left(x\right)|}^{2}\text{d}x=\underset{i=1}{\overset{k}{\sum }}{a}_{i}^{2}\left(1-\frac{\lambda }{{\lambda }_{i}}\right)\le \left(1-\frac{\lambda }{{\lambda }_{k}}\right){‖u‖}_{1,2}^{2}$ (3.6)

$\begin{array}{c}J\left(u\right)\le \frac{1}{2}\left(1-\frac{\lambda }{{\lambda }_{k}}\right){‖u‖}_{1,2}^{2}-\frac{\beta }{2}{\int }_{\Omega }\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)F\left(u\right)\text{d}x+\frac{1}{p}{‖u‖}_{p}^{p}-\frac{1}{2}{\int }_{\Omega }{|h\left(x\right)|}^{2}\text{d}x\\ \le \frac{1}{2}\left(1-\frac{\lambda }{{\lambda }_{k}}\right){‖u‖}_{1,2}^{2}+C{‖u‖}_{1,2}^{p}\end{array}$ (3.7)

$\frac{1}{2}\left(1-\frac{\lambda }{{\lambda }_{k+1}}\right){‖u‖}_{1,2}^{2}+C{‖u‖}_{1,2}^{p}<-\frac{1}{2}{‖h‖}_{2}^{2}$

${‖u‖}_{1,2}=r$ 整理得出下式

$\Lambda \left(r\right)=\frac{1}{2}\left(1-\frac{\lambda }{{\lambda }_{k}}\right){r}^{2}+C{r}^{p}$

$\Lambda \left(R\right)<-\frac{1}{2}{‖h‖}_{2}^{2}\text{}\forall {‖h‖}_{2}<{\alpha }_{0}$

$J\left(u\right)<-\frac{1}{2}{‖h‖}_{2}^{2}\le \underset{u\in Z}{\mathrm{inf}}J\left(u\right)$

Existence of Three Solutions for a Choquard Equation[J]. 理论数学, 2019, 09(03): 291-298. https://doi.org/10.12677/PM.2019.93039

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