On viscosity and weak solutions for non-homogeneous \(p\)-Laplace equations. (English) Zbl 1419.35075
Summary: In this manuscript, we study the relation between viscosity and weak solutions for non-homogeneous \(p\)-Laplace equations with lower-order term depending on \(x\), \(u\) and \(\nabla u\). More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending results presented in [P. Juutinen et al., SIAM J. Math. Anal. 33, No. 3, 699–717 (2001; Zbl 0997.35022)]
and [V. Julin and P. Juutinen, Commun. Partial Differ. Equations 37, No. 4–6, 934–946 (2012; Zbl 1260.35069)]. Moreover, we provide a converse statement in the full case under extra assumptions on the data.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35D40 | Viscosity solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
References:
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