利用莫尔斯理论证明p-Laplace Benci-Cerami型问题（1<p<2）的多个正解
Multiple positive solutions for a p-Laplace Benci-Cerami type problem (1<p<2), via Morse theory
摘要：
考虑拟线性问题\[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p Δ_{p}u + u^{p-1} = f(u) & \hbox{in} \ Ω \newline u>0 & \hbox{in} \ Ω \newline u=0 & \hbox{on} \ \partial Ω \end{array} \right. \]，其中$Ω$是$\mathbb{R}^N$中具有光滑边界的有界区域，$N\geq 2$，$1< p < 2$，$\varepsilon >0$是参数，$f: \mathbb{R} \to \mathbb{R}$是具有次临界增长的连续函数.我们证明了存在$\varepsilon^* >0$，使得对于每个$\varepsilon \in (0, \varepsilon^*)$，$(P_\varepsilon)$至少有$2{\mathcal P}_1(Ω)-1$的解决方案，可能与他们的多重性计数，其中${\mathcal P}_t(Ω)$是$Ω$的庞加莱多项式。利用Morse技巧，我们提供了一个解的多重性的解释，在$Ω$上的一个拟线性方程的正的独特的解决方案，近似$(P_\varepsilon)$。
英文摘要：
Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p Δ_{p}u + u^{p-1} = f(u) & \hbox{in} \ Ω \newline u>0 & \hbox{in} \ Ω \newline u=0 & \hbox{on} \ \partial Ω \end{array} \right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ is a parameter and $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least $2{\mathcal P}_1(Ω)-1$ solutions, possibly counted with their multiplicities, where ${\mathcal P}_t(Ω)$ is the Poincaré polynomial of $Ω$. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on $Ω$, approximating $(P_\varepsilon)$.
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