Article
MSC:
35B45,
35B65,
35J15,
35J25,
35J60,
35J92 | MR 3959951 | Zbl 07088791 | DOI: 10.21136/CMJ.2018.0322-17
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Keywords:
bounded solution; $p$-Laplacian; renormalized solution; measure data
bounded solution; $p$-Laplacian; renormalized solution; measure data
Summary:
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation \[ -{\rm div}(a(x,\nabla u))=h(x,u)+\mu ,\quad \text {in} \ \Omega \subset \mathbb R^N, \] where the left-hand side is a Leray-Lions operator from $W_0^{1,p} (\Omega )$ into $W^{-1,p'}(\Omega )$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like $|s|^{p-1}$ and $\mu $ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu $.
References:
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