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# Dynamics of Partial Differential Equations

## Volume 12 (2015)

### Number 3

### Sobolev spaces on time scales and applications to semilinear Dirichlet problems

Pages: 241 – 263

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n3.a3

#### Authors

#### Abstract

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.

Published 8 September 2015