具有超临界增长的非线性 Dirichlet 问题解的唯一性
偏微分方程分析
2019-12-30 v1
摘要
我们关注如下形式的 Dirichlet 问题：d i v ( ∣ D u ∣ p − 2 D u ) + f ( u ) = 0 \mbox i n Ω , u = 0 \mbox o n ∂ Ω , {\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, div ( ∣ D u ∣ p − 2 D u ) + f ( u ) = 0 \mbox in Ω , u = 0 \mbox o n ∂ Ω , Ω \Omega Ω R n \mathbb{R}^n R n n ≥ 2 n\ge 2 n ≥ 2 1 < p < n 1<p<n 1 < p < n f f f n = 2 n=2 n = 2 γ : [ a , b ] → R 2 \gamma:[a,b]\to\mathbb{R}^2 γ : [ a , b ] → R 2 t 1 ≠ t 2 t_1\neq t_2 t 1 = t 2 γ ( t 1 ) ≠ γ ( t 2 ) \gamma(t_1)\neq\gamma(t_2) γ ( t 1 ) = γ ( t 2 ) ε > 0 \varepsilon>0 ε > 0 Ω = Ω ε Γ = { ( x 1 , x 2 ) ∈ R 2 : d i s t ( ( x 1 , x 2 ) , Γ ) < ε } \Omega=\Omega^\Gamma_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2\ :\ \mathop{\rm dist}\big((x_1,x_2),\Gamma\big)<\varepsilon\} Ω = Ω ε Γ = {( x 1 , x 2 ) ∈ R 2 : dist ( ( x 1 , x 2 ) , Γ ) < ε } Γ = { γ ( t ) : t ∈ [ a , b ] } \Gamma=\{\gamma(t)\ :\ t\in[a,b]\} Γ = { γ ( t ) : t ∈ [ a , b ]} n > 2 n>2 n > 2 Ω \Omega Ω Ω = Ω ~ ε , s Γ = { ( x 1 , x 2 , y ) : ( x 1 , x 2 ) ∈ Ω ε Γ , y ∈ R n − 2 , ∣ y ∣ < s } . \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y)\ :\ (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|<s\}. Ω = Ω ε , s Γ = {( x 1 , x 2 , y ) : ( x 1 , x 2 ) ∈ Ω ε Γ , y ∈ R n − 2 , ∣ y ∣ < s } .
引用
@article{arxiv.1912.12243,
title = {Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth},
author = {Riccardo Molle and Donato Passaseo},
journal= {arXiv preprint arXiv:1912.12243},
year = {2019}
}