## $n$-Laplacian in $\mathcal {H}^ 1_ \mathrm {loc}$ does not lead to regularity

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- by Nikan B. Firoozye
- Proc. Amer. Math. Soc.
**123**(1995), 3357-3360 - DOI: https://doi.org/10.1090/S0002-9939-1995-1277110-0
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## Abstract:

It is well known that in two space dimensions, if a solution to Poisson’s equation has right-hand side in $\mathcal {H}_{{\text {loc}}}^1$, then this solution is actually continuous. The corresponding result for*n*-Laplacians is shown to be false for $n \geq 3$; we construct two examples with right-hand sides in $\mathcal {H}_{{\text {loc}}}^1({\Re ^n})$ such that the corresponding solutions to the

*n*-Laplacian are unbounded in the first case, and bounded but discontinuous in the second.

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## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 3357-3360 - MSC: Primary 35J05; Secondary 42B30
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277110-0
- MathSciNet review: 1277110