﻿ 关于一类(p(u),q(u))-Laplacian问题

# 关于一类(p(u),q(u))-Laplacian问题On a Class of (p(u),q(u))-Laplacian Problem

Abstract: In this paper, we consider the existence of the following variable exponent elliptic problem when(p(u),q(u)) is a local quantity: where Ω ⊂ ℝd (d ≥ 2) is a smooth bounded domain, f ( x) is a given data, p,q : ℝ →[1,+∞) are exponent functions. We obtain the existence of weak solution of (p(u),q(u)) -Laplacian, (p(u),q(u)) is a local quantity by means of singular perturbation technique and Schauder fixed point theorem.

1. 绪论现状

1.1. 研究背景及现状

$\left\{\begin{array}{ll}u-div\left({|\nabla u|}^{p\left(u\right)-2}\nabla u\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}$ (1.1)

$\left\{\begin{array}{ll}-div\left({|\nabla u|}^{p\left(u\right)-2}\nabla u\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u=0,\hfill & x\in \Omega ,\hfill \end{array}$ (1.2)

2019年，Chipot、Oliverira [8] 研究了非局部情形下的方程

$\left\{\begin{array}{ll}-div\left({|\nabla u|}^{p\left(b\left(u\right)\right)-2}\nabla u\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}$ (1.3)

1.2. 预备知识

$\varsigma \left(\Omega \right)$ 表示所有Lebesgue可测函数 $h:=\Omega \to \left[1,\infty \right)$ 的集合，并定义：

${h}_{-}:=\underset{x\in \Omega }{\text{ess}}\mathrm{inf}h\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{+}:=\underset{x\in \Omega }{\text{ess}}\mathrm{sup}h\left( x \right)$

${\rho }_{h\left(x\right)}\left(u\right):={\int }_{\Omega }{|u\left(x\right)|}^{h\left(x\right)}\text{d}x<\infty$

$1\le {h}_{-}\le {h}_{+}<\infty$ (1.4)

${L}^{h\left(x\right)}\left(\Omega \right)$ 是可分的，且 ${C}_{0}^{\infty }\left(\Omega \right)$${L}^{h\left(x\right)}\left(\Omega \right)$ 中稠密。同时， ${L}^{\infty }\left(\Omega \right)\cap {L}^{h\left(x\right)}\left(\Omega \right)$${L}^{h\left(x\right)}\left(\Omega \right)$ 中也稠密。

$1<{h}_{-}\le {h}_{+}<\infty ,$ (1.5)

${L}^{h\left(x\right)}\left(\Omega \right)$ 是自反的。在方程(1.5)成立的情况下，定义 ${L}^{{h}^{\prime }\left(x\right)}\left(\Omega \right)$${L}^{h\left(x\right)}\left(\Omega \right)$ 的对偶空间，其中 ${h}^{\prime }\left(x\right)$$h\left(x\right)$ 的Holder共轭，且两者满足 $\frac{1}{h\left(x\right)}+\frac{1}{{h}^{\prime }\left(x\right)}=1$

${h}_{-},{h}_{+}$ 的定义及方程(1.5)中，我们可以得到

$\underset{x\in \Omega }{\text{ess}}\mathrm{inf}{h}^{\prime }\left(x\right)\le \underset{x\in \Omega }{\text{ess}}\mathrm{sup}h\left(x\right)\le {\left({h}_{-}\right)}^{\prime }<\infty ,$

$\mathrm{min}\left\{{‖u‖}_{h\left(x\right)}^{{h}_{-}},{‖u‖}_{h\left(x\right)}^{{h}_{+}}\right\}\le {\rho }_{h\left(\cdot \right)}\left(u\right)\le \mathrm{max}\left\{{‖u‖}_{h\left(x\right)}^{{h}_{-}},{‖u‖}_{h\left(x\right)}^{{h}_{+}}\right\},$

$\mathrm{min}\left\{{\rho }_{h\left(x\right)}{\left(u\right)}^{\frac{1}{{h}_{-}}},{\rho }_{h\left(x\right)}{\left(u\right)}^{\frac{1}{{h}_{+}}}\right\}\le {‖u‖}_{h\left(x\right)}\le \mathrm{max}\left\{{\rho }_{h\left(x\right)}{\left(u\right)}^{\frac{1}{{h}_{-}}},{\rho }_{h\left(x\right)}{\left(u\right)}^{\frac{1}{{h}_{+}}}\right\},$

${‖u‖}_{h\left(x\right)}^{{h}_{-}}-1\le {\rho }_{h\left(x\right)}\left(u\right)\le {‖u‖}_{h\left(x\right)}^{{h}_{+}}+1,$ (1.6)

Young不等式：对任意的 $u\in {L}^{h\left(x\right)}\left(\Omega \right)$$v\in {L}^{{h}^{\prime }\left(x\right)}\left(\Omega \right)$ 及正常数 $C\left(\delta \right)$，任意的都有

Holder不等式：对任意的，都有

(1.7)

(1.8)

i)

ii) 当时，在中有

iii) 当时，在弱收敛于

iv) 对常数C有

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

，这里，即

，则

2. 局部问题解的存在性

2.1. 引言

(2.1)

2.2. 准备知识

，对所有的，则这个集合为Banach空间。

2.3. 主要结论

(2.2)

2.4. 局部问题解的存在性

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

，令为方程组(2.5)的解，且令，也就是满足对任意

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

，其中。由方程(2.16)，可得

(2.17)

(2.18)

(2.19)

(2.20)

，得

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

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