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Article

MSC: 35D30, 35J25
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Keywords:
integrability; very weak solution; boundary value problem; $p$-harmonic equation
Summary:
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 \mbox {$p$-harmonic} equation $$ \begin {cases} -\mbox {div}(|\nabla u(x)|^{p-2} \nabla u(x))=0, & x\in \Omega , \\ u(x)=\theta (x), & x\in \partial \Omega . \end {cases} \eqno (*) $$ We show that, under the assumption $\theta \in W^{1,q}(\Omega )$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$ u\in \begin {cases} \theta +L_{\rm weak}^{q^*}(\Omega ) & \mbox {for } q<n, \\ \theta +L_{\rm weak}^\tau (\Omega ) & \mbox {for } q=n \mbox { and any } \tau <\infty , \\ \theta +L^\infty (\Omega ) & \mbox {for } q>n, \end {cases} $$ provided that $r$ is sufficiently close to $p$.
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