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 $u-v$.


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