Integrability for very weak solutions to boundary value problems of p-harmonic equation

Abstract

The paper deals with very weak solutions $u \in \theta + W_0^{1,r}(\Omega )$, max{1, p − 1} < r < p < n, to boundary value problems of the p-harmonic equation

$$\left\{ {\begin{array}{*{20}c} { - div\left( {\left| {\nabla u(x)} \right|^{p - 2} \nabla u(x)} \right) = 0,} & {x \in \Omega ,} \\ {u(x) = \theta (x),} & {x \in \partial \Omega .} \\ \end{array} } \right.$$

((*))

We show that, under the assumption θ ∈ W ^1,q(Ω), q > r, any very weak solution u to the boundary value problem (*) is integrable with

$$u \in \left\{ {\begin{array}{*{20}c} {\theta + L_{weak}^{q*} (\Omega )} & {for q < n,} \\ {\theta + L_{weak}^\tau (\Omega )} & {for q = n and any \tau < \infty ,} \\ {\theta + L^\infty (\Omega )} & {for q > n,} \\ \end{array} } \right.$$

provided that r is sufficiently close to p.