Generalized limit theorem and bifurcation for problems with Pucci's operator
Generalized limit theorem and bifurcation for problems with Pucci's operator
复制标题
Pucci 算子问题的广义极限定理和分岔
DOI：
10.12775/tmna.2020.012
复制
发表时间：
2020-08
期刊：
影响因子：
--
通讯作者：
Guowei Dai
中科院分区：
文献类型：
--
作者：
Guowei Dai
我们建立了一个新的极限结果，推广了著名的Whyburn极限定理。作为应用，我们研究了以下问题的单号或变号解的存在性和多解性：开始{方程*}\开始{案例}-\数学{M}_{\lambda，\Lambda}^+\Left(D^2 u\right)=\muf(U)&\Text{in}\Omega，\\u=0&\Text{on}\Partial\Omega，\end{case}\end{方程*}其中$\mathcal{M}_{\lambda，^+$表示Pucci极值算子。将分歧方法与我们的广义极限定理相结合，根据$f$在$0$和$increty$的行为，以及$f$是否满足符号条件$f(S)S和gt；0$对$S\neq0$，确定了上述问题有一个或多个单号或变号解的参数$Mu$的范围。
We establish a new limiting result which extends the famous Whyburn's limit theorem. As applications, we study the existence and multiplicity of one-sign or sign-changing solutions with a prescribed number of simple zeros for the following problem \begin{equation*} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+\left(D^2 u\right)=\mu f(u) &\text{in } \Omega,\\ u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\mathcal{M}_{\lambda,\Lambda}^+$ denotes the Pucci extremal operator. Combining bifurcation approach with our generalized limit theorem, we determine the range of parameter $\mu$ in which the above problem has one or multiple one-sign or sign-changing solutions according to the behaviors of $f$ at $0$ and $\infty$, and whether $f$ satisfies the signum condition $f(s)s> 0$ for $s\neq0$.