Regular and irregular solutions for a class of elliptic systems in the critical dimension

摘要: We study regularity properties of weak solutions in the Sobolev space \({W^{1,n}_0}\) to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains \({\mathbb{R}^n}\) , i. e. we investigate limiting situation embedding. Several counterexamples irregular are constructed cases, where additional structure conditions might have led regularity. Among others present both unbounded obeying one-sided condition, further construct extremals two-dimensional variational problems. These do not exclude existence regular solution. In fact, establish solutions—under standard assumptions principal part aforementioned inhomogeneity. This extends previous works for n = 2 more general including arbitrary dimensions. Moreover, this result is achieved by simplified proof invoking modern techniques.