论文标题
超级线性椭圆方程的解决方案及其摩尔斯
Solutions of super-linear elliptic equations and their Morse indices
论文作者
论文摘要
我们在这里调查了退化的双谐波方程：$$δ_{m}^2 u = f（x，u）\; \ \; \ mbox {in}Ø，\ quad u =ΔU= 0 \ quad \ mbox {on} \; \pΩ，$$，带有$ m \ ge 2，$，以及退化的tri -harmonic方程：$$-Δ__{m}^3 u = f（x，x，u）\; \; \; \ mbox {in}Ø，\ quad u = \ frac {\ p u} {\pν} = \ frac {\ p^{2} u} {\pν^{2}} = 0 \ quad \ quad \ quad \ mbox {on} \; \pΩ,$$ where $Ω\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ resp, and $f \in \mathrm{C}^{1}(Ω\times \mathbb{R})$ satisfying suitable m-superlinear and subcritical growth conditions.我们的主要目的是建立$ l^{p} $和$ l^{\ infty} $通过Morse索引为弱解决方案的显式界限。我们的结果扩展了以前在\ cite {c，hhf，hyf，lec}中获得的显式估计。
We investigate here the degenerate bi-harmonic equation: $$Δ_{m}^2 u=f(x,u)\; \;\;\mbox{in} Ø,\quad u = Δu = 0\quad \mbox{on }\; \pΩ,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -Δ_{m}^3 u=f(x,u)\;\;\; \mbox{in} Ø,\quad u = \frac{\p u}{\p ν} = \frac{\p^{2} u}{\pν^{2}} = 0\quad \mbox{on }\; \pΩ,$$ where $Ω\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ resp, and $f \in \mathrm{C}^{1}(Ω\times \mathbb{R})$ satisfying suitable m-superlinear and subcritical growth conditions. Our main purpose is to establish $L^{p}$ and $L^{\infty}$ explicit bounds for weak solutions via the Morse index. Our results extend previous explicit estimate obtained in \cite{c, HHF, hyf, lec}.
