We introduce the Lebesgue space and the exterior Sobolev space for differential forms on Riemannian manifold 𝑀 which are the Lebesgue space and the Sobolev space of functions on 𝑀, respectively, when the degree of differential forms to be zero. After discussing the properties of these spaces, we obtain the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous 𝑝(𝑚)-harmonic equations with variable growth in 𝑊01,𝑝(𝑚)(Λ𝑘𝑀).

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations \begin{document}$ \partial_{t}^{\beta}u+M(\|u\|_{H_0^{s}(\Omega)}^2)(-\Delta)^{s}u = \gamma|u|^{\rho}u+g(t,x) $\end{document} with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let \begin{document}$ 0<\rho<\frac{4s}{N-2s} $\end{document} when \begin{document}$ \gamma<0 $\end{document}, and \begin{document}$ 0<\rho<\min\{\frac{4s}{N},\frac{2s}{N-2s}\} $\end{document} when \begin{document}$ \gamma>0 $\end{document}, then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in \begin{document}$ L^2(\Omega) $\end{document} and \begin{document}$ L^{\rho+2}(\Omega) $\end{document}. Remarkably, the decay rate differs from that in the case \begin{document}$ \beta = 1 $\end{document}.

In this paper, we mainly discuss the embedding theory of variable exponent fractional Sobolev space $ W^{s(\cdot), p(\cdot)} (\Omega) $, and apply this theory to study the $ s(x) $-$ p(x) $-Laplacian equation: $ (-\varDelta)_{p(\cdot)}^{s(\cdot)}u+V(x)|u|^{p(x)-2}u = f(x,u)+g(x) $ where $ x\in\Omega\subset \mathbb{R}^n $, $ (-\varDelta)_{p(\cdot)}^{s(\cdot)} $ is $ s(x) $-$ p(x) $-Laplacian operator with $ 0 < s(x) < 1 < p(x) < \infty $ and $ p(x)s(x) < n $, the nonlinear term $ f: \Omega \times \mathbb{R} \to \mathbb{R} $ is a Carathéodory function, $ V:\mathbb{R}^n\to \mathbb{R} $ is a potential function and $ g:\mathbb{R}^n\to \mathbb{R} $ is a perturbation term.

In this paper we consider the following Dirichlet problem for elliptic systems: