## Injections de type Sobolev. (Injections of Sobolev type.).(French)Zbl 0602.46031

We consider some open subsets $$\Omega \hookrightarrow {\mathbb{R}}^ 3$$ with radial symmetry and a degeneracy at the boundary. We consider some Sobolev spaces of radially symmetric functions defined in the sequence of open subsets. We prove the following results: $W_ 0^{1,p}(\Omega,\quad axisymmetric)\subset L^ q(\Omega,\quad axisymmetric)\;if\;\frac{1}{q}>\frac{1}{p}-\frac{1}{\alpha}$ and $W_ 0^{1,p}(\Omega,\quad axisymmetric)\not\subset L^ q(\Omega,\quad axisymmetric)\;if\;\frac{1}{q}<\frac{1}{p}-\frac{1}{\beta}$ where $$\alpha$$ and $$\beta$$ depend on the open subset form and satisfy: $$2\leq \alpha \leq \beta \leq 3$$ (generally $$\beta <3)$$.

### MSC:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems