# Variational problems living in two different Sobolev spaces

Is there a general reference concerning variational problems living in $$W^{h,p}\times W^{k,p}$$, with $$h, k\in\mathbb{N}_0$$ not coinciding? I'm thinking to problems of type: $$\inf_{u,v}\int_{\Omega} f_1\left(\frac{\partial^{|\alpha|} u}{\partial x_1^{\alpha_1}\dots \partial x_n^{\alpha_n}}\right)+f_2\left(\frac{\partial^{|\beta|} v}{\partial x_1^{\beta_1}\dots \partial x_n^{\beta_n}}\right)+ f_3\left(u-v\right)+V(x,u,v),$$ where $$\Omega$$ is a bounded (or even compact) domain in $$\mathbb{R}^n$$ ($$n\ge 1$$), $$|\alpha|=h$$ and $$|\beta|=k$$. You can take $$f_{1,2,3}$$ and $$V$$ as nice as you wish ($$V$$ can also be assumed to depend only on the less regular between $$u$$ and $$v$$). You can include terms depending on other partial derivatives. What's important for me is that there's a term depending on $$u-v$$.