Uniqueness of weak solution for nonlinear elliptic equations in divergence form (Q1573702)

The author proves that there exists at most one weak solution \(u\in H^1_0(\Omega)\cap L^\infty(\Omega)\) to the problem \[ -\sum^n_{i=1} {\partial\over\partial x_i} (a_i(x, u,\nabla_u))+ b(x,u)= 0\quad\text{in }\Omega,\quad u|_{\partial\Omega}= 0. \] Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain, and it is supposed that for all \(x\in\Omega\), \(u,v,w\in \mathbb{R}\) and \(\xi,\mu\in \mathbb{R}^n\) \[ \sum^n_{i,j=1} {\partial a_i\over\partial\mu_j} (x,u,\mu)\xi_i\xi_j\geq \|\xi\|^2, \] \[ (b(x,v)- b(x,w))(v- w)\geq 0, \] \[ |a_i(x,v,\mu)- a_i(x,w,\mu)|\leq|a(v,w)||v- w|(1+ \|\mu\|) \] with a function \(a\in L^\infty_{\text{loc}}(\mathbb{R}^2)\). Moreover, it is shown that there exists at least one weak solution \(u\in H^1_0(\Omega)\) if \(a\in L^\infty(\mathbb{R}^2)\). Finally, some counterexamples are given to show that the results cannot be improved.