非光滑型 Ricceri 变分原理在 p(x)-Laplacian 方程中的应用 Alternative Title The applications of nonsmooth version Ricceri's variational principle to the $p(x)$-Laplacian equation 代国伟 Thesis Advisor 范先令 2009-05-18 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 变指数 Sobolev 空间 非光滑版 Ricceri 变分原理 非光滑版 Ricceri 变分原理 Neumann 边值问题 p(x)-Laplacian Abstract 在这篇文章中，我们在有界域 \Omega 上分别考虑了包含 p(x)-Laplacian 算子的 Neumann 型的微分包含问题 $$\left\{ \begin{array}{l} -\text{div}\left(\vert \nabla u\vert^{p(x)-2}\nabla u\right)+\lambda(x)\vert u\vert^{p(x)-2}u\in \partial F(x,u)+\partial G(x,u) \text{ in} \,\,\Omega, \\ \frac{\partial u}{\partial \gamma }=0 \text{ \ on }\partial \Omega,% \end{array}% \right.\nonumber$$ 和 ~Dirichlet 型的微分包含问题 $$\left\{ \begin{array}{l} -\text{div}\left(\vert \nabla u\vert^{p(x)-2}\nabla u\right)\in \partial F(x,u) \text{ in} \,\,\Omega,\\ u=0 \quad \text{ on }\partial \Omega. \end{array} \right. \nonumber$$ 在对非线性项作适当假设后, 我们分别在变指数 ~Sobolev 空间 W^{1,p(x)}(\Omega) 和 W_0^{1,p(x)}(\Omega) 中, 利用非光滑型 Ricceri 变分原理得到了两类问题的无穷多解性. Other Abstract In this paper, we consider differential inclusion problem in a bounded domain $\Omega$ involving $p(x)$-Laplacian of Neumann-type $$\left\{ \begin{array}{l} -\text{div}\left(\vert \nabla u\vert^{p(x)-2}\nabla u\right)+\lambda(x)\vert u\vert^{p(x)-2}u\in \partial F(x,u)+\partial G(x,u) \text{ in} \,\,\Omega, \\ \frac{\partial u}{\partial \gamma }=0 \text{ \ on }\partial \Omega,% \end{array}% \right.\nonumber$$ and Dirichlet-type $$\left\{ \begin{array}{l} -\text{div}\left(\vert \nabla u\vert^{p(x)-2}\nabla u\right)\in \partial F(x,u) \text{ in} \,\,\Omega,\\ u=0 \quad \text{ on }\partial \Omega. \end{array} \right. \nonumber$$ With some suitable assumptions on nonlinearities, the existences of infinitely many solutions are obtained by using nonsmooth version Ricceri's variational principle in variable exponent Sobolev spaces W^{1,p(x)}(\Omega) and W_0^{1,p(x)}(\Omega), respectively. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225476 Collection 数学与统计学院 Recommended CitationGB/T 7714 代国伟. 非光滑型 Ricceri 变分原理在 p(x)-Laplacian 方程中的应用[D]. 兰州. 兰州大学,2009.
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