Weak solutions for anisotropic nonlinear elliptic equations with variable exponents. (English) Zbl 1182.35092
Summary: We study the anisotropic boundary-value problem
\[ -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}a_{i}(x,\frac{\partial}{\partial x_{i}}u)=f \quad \text{in } \Omega, \qquad u=0 \quad\text{on }\partial \Omega, \]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^{N}\) \((N\geq 3)\). We obtain the existence and uniqueness of a weak energy solution for \(f\in L^{\infty}(\Omega)\), and the existence of weak energy solution for general data \(f\) dependent on \(u\).
\[ -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}a_{i}(x,\frac{\partial}{\partial x_{i}}u)=f \quad \text{in } \Omega, \qquad u=0 \quad\text{on }\partial \Omega, \]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^{N}\) \((N\geq 3)\). We obtain the existence and uniqueness of a weak energy solution for \(f\in L^{\infty}(\Omega)\), and the existence of weak energy solution for general data \(f\) dependent on \(u\).
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35D30 | Weak solutions to PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35J60 | Nonlinear elliptic equations |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
