Weak and renormalized solutions for anisotropic Neumann problems with degenerate coercivity
In this work, we study the following quasilinear Neumann boundary-value problem $$\left\{\begin{array}{ll} \displaystyle -\sum^{N}_{i=1} D^{i}(a_{i}(x,u,\nabla u))+|u|^{p_{0}-2} u= f(x,u,\nabla u) & \mbox{in } \ \quad \Omega,\\ \displaystyle \sum^{N}_{i=1} a_{i}(x,u,\nabla u)\cdot n_{i} = g(x)...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Sociedade Brasileira de Matemática
2022-12-01
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| Series: | Boletim da Sociedade Paranaense de Matemática |
| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/62362 |
| Summary: | In this work, we study the following quasilinear Neumann boundary-value problem
$$\left\{\begin{array}{ll}
\displaystyle -\sum^{N}_{i=1} D^{i}(a_{i}(x,u,\nabla u))+|u|^{p_{0}-2} u= f(x,u,\nabla u) & \mbox{in } \ \quad \Omega,\\
\displaystyle \sum^{N}_{i=1} a_{i}(x,u,\nabla u)\cdot n_{i} = g(x) & \mbox{on } \ \quad \partial\Omega,
\end{array}\right.$$
where $\Omega$ is a bounded open domain in $\>I\!\!R^{N}$, $(N\geq 2)$. We prove the existence of a weak solution for $f \in L^{\infty}(\Omega)$ and $g\in L^{\infty}(\partial\Omega)$ and the existence of renormalized solutions for $L^{1}$-data $f$ and $g$. The functional setting involves anisotropic Sobolev spaces with constants exponents.
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| ISSN: | 0037-8712 2175-1188 |