A weak solution to quasilinear elliptic problems with perturbed gradient. (English) Zbl 1465.35189
Summary: We consider weak solutions to the Dirichlet problem
\begin{align*}
\left\{\begin{array}{ll}
-\operatorname{div}A\big (x,Du-\varTheta (u)\big)=f &\text{ in }\varOmega,\\ u=0 &\text{ on }\partial\varOmega,
\end{array} \right.
\end{align*}
where \(\varTheta:\mathbb{R}^m\rightarrow\mathbb{M}^{m\times n}\) is a continuous function assumed to satisfy a Lipschitz condition. Based on the theory of Young measures, we prove the existence result when \(f\in W^{-1,p'}(\varOmega ;\mathbb{R}^m)\).
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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