Necessary optimality conditions for a certain class of nonlinear singular elliptic systems (Q1204704)

The following optimization problem is considered: minimize \[ I(v,y):= (1/q) \| y- y^*\|^ q_ q+ (\nu/q') \| v\|^{q'}_{q'} \] subject to \(\Delta y+ | y|^ \rho y= v+f\), where \(f\in H^{-1}(\Omega)\) and \(y^*\in L_ q(\Omega)\) are given functions, \(\rho\), \(\nu\) are given positive numbers, \(\Omega\) is an open region in \(\mathbb{R}^ n\) with a sufficiently smooth boundary, \(\|\cdot\|_ q\) denotes the norm in \(L_ q(\Omega)\), and \(1/q+ 1/q'= 1\). It is proved that this problem has a solution; then some necessary optimality conditions are established.