We prove the existence and uniqueness of weak solutions of boundary problem value problems in an unbounded domain Ω ⊂ Rn for strongly nonlinear 2m order elliptic differential equations.

The asymptotic behaviour is studied for minima of regular variational problems with Neumann boundary conditions on noncompact part of boundary.

We consider local minimizers $u:{ℝ}^2\supset \Omega \to {ℝ}^N$ of variational integrals like ${\int }_{\Omega }\left[\right(1+|{\partial }_1{u|}^2{)}^{p/2}+(1+|{\partial }_2{u|}^2{)}^{q/2}]dx$ or its degenerate variant ${\int }_{\Omega }\left[\right|{\partial }_1{u|}^p+|{\partial }_{2}u{|}^q]dx$ with exponents $2\le p<q<\infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 16 (2003), 177–186. We prove interior $C^{1,\alpha }$- respectively $C^1$-regularity of $u$ under the condition that $q<2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. 31 (2006), 349–362.