Sobolev spaces on time scales and applications to semilinear Dirichlet problems

In this paper, we present some theoretical results of Sobolev spaces of functions defined on an open subset of an arbitrary time scale $\mathbb{T}^n$, where $n \geq 1$ is a positive integer. As an application, we consider a class of semilinear Dirichlet problems on time scales $\mathbb{T}^n$ of the form\[\begin{cases}-\Delta u + {\lambda u}^{\sigma} = \vert u^{\sigma} {\vert}^{p-2} u^{\sigma} , \\u \geq 2 , u \in H^{1}_{0, \Delta} (\Omega_{\mathbb{T}}) ,\end{cases}\]where $\Omega_{\mathbb{T}}$ is a domain of ${(\mathbb{T}^{\kappa})}^n$ and $\Delta u = {\sum}^{n}_{i=1} D^{2}_{i, \Delta} u$ is the Laplace operator. Under certain conditions, the sufficient and necessary condition of the existence of a nontrivial solution is established by using the mountain pass theorem.

Keywords

compact embedding theorem, time scales, semilinear Dirichlet problem, mountain pass theorem, critical point

2010 Mathematics Subject Classification

Primary 34N05, 35J05, 37J45. Secondary 37C25.

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Published 8 September 2015