EXISTENCE OF SOLUTIONS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITH Lm-DATA AND NEUMANN BOUNDARY CONDITION
Abstract
In this paper, we study the following nonhomogenous Neumann problem
$$\left\{\begin{array}{ll}
\displaystyle -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}(a_{i}(x,u,D^{i}u)) + d(|u|)|\nabla u|^{p_{i}} + |u|^{s_{0}-2}u = f & \mbox{in } \ \quad \Omega,\\
\displaystyle \sum^{N}_{i=1}a_{i}(x,u,D^{i}u).n_{i}= 0 & \mbox{in } \ \quad \partial\Omega,
\end{array}\right.$$
where $\Omega$ is a bounded open domain in \ ${\!\!R}^{N}$ ($N\geq 2$) of Lipschitz boundary $\partial\Omega$ and $n_{i}$ signifies components of the unit outward normal vector $\vec{n}$ to $\partial\Omega$. Anisotropic Sobolev spaces with constant exponents are involved in the functional context.
$$\left\{\begin{array}{ll}
\displaystyle -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}(a_{i}(x,u,D^{i}u)) + d(|u|)|\nabla u|^{p_{i}} + |u|^{s_{0}-2}u = f & \mbox{in } \ \quad \Omega,\\
\displaystyle \sum^{N}_{i=1}a_{i}(x,u,D^{i}u).n_{i}= 0 & \mbox{in } \ \quad \partial\Omega,
\end{array}\right.$$
where $\Omega$ is a bounded open domain in \ ${\!\!R}^{N}$ ($N\geq 2$) of Lipschitz boundary $\partial\Omega$ and $n_{i}$ signifies components of the unit outward normal vector $\vec{n}$ to $\partial\Omega$. Anisotropic Sobolev spaces with constant exponents are involved in the functional context.
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