Uniqueness for elliptic problems with locally Lipschitz continuous dependence on the solution
The paper under review is concerned with the nonlinear elliptic problem \[ -\mathrm{div}(a(x,u,\nabla u)= f-\mathrm{div} \, g \quad \text{in} \quad \Omega,\quad u=0 \quad \text{on} \quad \partial\Omega, \] where \(\Omega \subset \mathbb{R}^{^N}\) open, bounded, \(f\in L^{1}(\Omega)\), \(g\in (L^{p}(\Omega))^{N} \quad (p>1)\). Besides the standard growth and strong monotonicity conditions, the authors impose the following local Lipschitz condition on the function \(a\): \[ \begin{cases} \forall \, k>0 \, \exists \, L_{k}>0 \quad \text{such that}\\ \mid a(x,s,\xi) - a(x,r,\xi)\mid \leq L_{k}\mid s-r\mid \, \mid\xi\mid^{p-1}\\ \forall \, s,r,\in \mathbb{R}: \mid s\mid, \mid r\mid \leq k, \quad \forall \, \xi \in \mathbb{R}^{N} \end{cases} \] \[ \begin{cases} \mid a(x,s,\xi)-a(x,s,\xi')\mid \leq b(s)\mid \xi-\xi'\mid \; (\mid \xi\mid + \mid\xi'\mid)^{p-2} \\ \forall \, s\in \mathbb{R}, \quad \forall \, \xi, \xi' \in \mathbb{R}^{N} \end{cases} \] for a.e. \(x\in \Omega\), where \(b:\mathbb{R} \longrightarrow ]0,+\infty[\) is a continuous function such that \(b(s)\geq\) const \(>0\) for all \(s\in \mathbb{R}\). Under an additional structural condition on \((a(x,s,\xi)\cdot \xi'\) and the assumption that one of the conditions (A1) \(\quad 1<p\leq 2\), \(\quad\) (A2) \(\quad p>2\) and \(\text{sign}(f-\text{div} g) =\) const in \(\Omega\) holds, the authors prove the existence and uniqueness of a renormalized solution of (1).
locally Lipschitz continuous coefficients
scientific article; zbMATH DE number 6668261