Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials

Abstract

In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.