An existence result for a class of variational inequalities with \(L^1\) data (Q1864691)

The authors prove the existence of a solution \(u\geq0\) of the variational inequality \[ \int_\Omega a(x,u,Du)DT_k(v-u)+[H(x,Du)+g(x,u)]T_k(v-u)\geq \int_\Omega fT_k(v-u),\forall k>0, \] for all \(v\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega),\;v\geq 0\), where \(\Omega\) is an open bounded set in \(\mathbb R^N\), \(N\geq 2\), \(1<p<N\), \(-\text{div}\,a(x,u,Du)\) is a Leray-Lions operator, \(| H(x,\eta)| \leq C| \eta| ^{p-1}\), \(0\leq g(x,\xi)\leq C| \xi| ^\delta\), \(\delta(N-p)<N(p-1)\), \(f\in L^1(\Omega)\) and \(T_k\) denotes the usual truncation operator. The proof is based on the approximation of \(f\) by smooth functions and a priori estimates of approximating solutions.