On very weak entire solutions on nonlinear partial differential equations. (English. French, Italian summaries) Zbl 0954.35060
Summary: We consider the variational equation in \(\mathbb{R}^n\),
\[
\text{div}\Biggl(a(x) F'(|\nabla u|){\nabla u\over|\nabla u|}\Biggr)= 0,
\]
where \(0< \lambda_0\leq a(x)\leq \Lambda_0< \infty\) and \(F\) is a convex increasing function such that \(pF(t)\leq tF'(t)\leq qF(t)\) where \(1< p\leq q<\infty\). We prove that the very weak solutions of such equations, belonging to a suitable Orlicz-Sobolev space, must be zero almost everywhere.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 42B25 | Maximal functions, Littlewood-Paley theory |
Keywords:
Orlicz-Sobolev spaceReferences:
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