## 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)

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

counterexamples; uniqueness