A remark on uniqueness for quasilinear elliptic equations. (English) Zbl 0865.35047

Janeczko, Stanisław (ed.) et al., Singularities and differential equations. Proceedings of a symposium, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 33, 9-18 (1996).
The authors consider the problem \[ -{\partial\over\partial x_i}\Biggl(a(x,u){\partial u\over\partial x_i}\Biggr)=f\quad\text{in }\Omega,\quad u-g\in H^1_0(\Omega),\tag{1} \] with \(\Omega\) a bounded open subset of \(\mathbb{R}^n\), \(f\in H^{-1}(\Omega)\), \(g\in H^1(\Omega)\), and prove that this problem has a unique solution provided the following conditions hold: \[ |a(x,u)-a(y,u)|\leq C|x-y|,\quad|a(x,u)-a(x,v)|\leq C|u-v|\tag{2} \] (\(u,v\in\mathbb{R}\), \(x,y\in\Omega\)). The uniqueness can fail if (2) fails even if \(u\to a(x,u)\) is Hölder continuous. Counterexamples are given.
For the entire collection see [Zbl 0840.00028].
Reviewer: A.Kufner (Praha)


35J65 Nonlinear boundary value problems for linear elliptic equations