K-limits of solutions of second-order quasilinear elliptic equations. (English. Russian original) Zbl 0702.35103

Kufarev, B. P.; Nikulina, N. G.; Sokolov, B. V.

Differ. Equations 25, No. 4, 473-478 (1989); translation from Differ. Uravn. 25, No. 4, 686-692 (1989).

It is proved that functions from \(L^ 1_{1,loc}(R^ n_+)\), satisfying the condition \[ \int_{\Omega}| \nabla u|^ px_ n^{\alpha} dx<\infty,\quad 1<p\leq n+\alpha,\quad \alpha <p-1 \] for any bounded \(\Omega \subset R^ n_+\), have the finite K-limit except a set of zero Bessel (1-\(\alpha\) /p,p)-capacity. The notions of the K-limit introduced by the first author [Sov. Math., Dokl. 26, 456-459 (1982); translation from Dokl. Akad. Nauk SSSR 266, 1052-1055 (1982; Zbl 0548.31005)] generalizes the notion of the angular (non-tangential) limit of a function.
Applications are given to solutions of Cordes-type equations, to free extremals of certain functionals of the calculus of variations and to elliptic mappings.

Reviewer: T.Shaposhnikova

MSC:

35J67	Boundary values of solutions to elliptic equations and elliptic systems
35J65	Nonlinear boundary value problems for linear elliptic equations
31B15	Potentials and capacities, extremal length and related notions in higher dimensions

Keywords:

Bessel capacity; K-limit; angular; Cordes-type equations; elliptic mappings

Citations:

Zbl 0548.31005