On Euler equations in higher order Sobolev spaces
The author considers the functional \[ I(u) = \int_\Omega A(x, u, \dots, D^m u) dx \] in the space \(W_0^{m, p}(\Omega)\) \((\Omega\) an \(n\)-dimensional domain) under appropriate smoothness and growth conditions on \(A(x, \zeta).\) It is known that if \(\mu \in W^{-m, p'}(\Omega),\) \(f(x, u)\) is a suitably restricted Carathéodory function and \(u \to F(x, u)\) is a primitive of \(u \to f(x, u),\) a minimum \(\bar u\) of the functional \[ J(u) = I(u) + \int_\Omega F(x, u) dx - \langle \mu, u \rangle \] satisfies the weak version of the Euler equation under the critical growth condition \(|f(x, u)|\leq a|u|^{p^* - 1} + b(x)\) (\(p^*\) the critical Sobolev exponent) that makes the functional everywhere defined and differentiable. In this paper, growth conditions are abolished by constructing a test space \(\mathcal W\) dense in \(W^{m, p}(\Omega)\) where \(J\) is differentiable, which gives a weak interpretation of the Euler equation. This generalizes existing results for the case \(m = 1\), \(p = 2.\)
scientific article; zbMATH DE number 1565490