http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
Multiplicity results of critical local equation related to the genus theory
Mohsen Alimohammady,Asieh Rezvani,Cemil Tunc 대한수학회 2023 대한수학회논문집 Vol.38 No.4
Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation \[ \begin{cases} -div [a(x, |\nabla u|) \nabla u] = \mu (b(x) |u|^{s(x) -2} - |u|^{r(x) -2})u & \text{in} ~~\Omega,\\ u=0 & \text{on}~~ \partial \Omega, \end{cases} \] where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain, $\mu$ is a positive real parameter, $p$, $r$ and $s$ are continuous real functions on $\bar{\Omega}$ and $a(x, \xi)$ is of type $|\xi|^{p(x) -2}$. Next, we study boundedness and simplicity of eigenfunction for the case $a(x, |\nabla u|) \nabla u= g(x) | \nabla u|^{p(x) -2}\nabla u$, where $g\in L^{\infty}(\Omega)$ and $g(x) \geq 0$ and the case $a(x, |\nabla u|) \nabla u= (1+ \nabla u|^2)^{\frac{p(x) -2}{2}} \nabla u$ such that $p(x) \equiv p$.