An approximation of Hedberg's type in Sobolev spaces with variable exponent and application
Summary: The aim of this paper is to extend the usual framework of PDE with \(A(u)=-\operatorname{div}a(x,u,\nabla u)\) to include a large class of cases with \(A(u)=\sum_{|\beta|\leq\alpha}(-1)^{|\beta|}D^\beta A_\beta (x,u,\nabla u,\dots ,\nabla^\alpha u)\), whose coefficient \(A_\beta\) satisfies conditions (including growth conditions) which guarantee the solvability of the problem \(A(u)=f\). This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and \(p\)-Laplace equation.
scientific article; zbMATH DE number 6356652
scientific article