Optimal control problems governed by an elliptic differential equation with critical exponent (Q2712216)

The paper considers the problem of minimization of a convex functional over all pairs \((y,u)\in L_2(\Omega)\times L_2(\Omega)\) such that the controls \(u\) are functions nonnegative and bounded from above, the state \(y\) is the minimal solution of the equation NEWLINE\[NEWLINE\Delta y(x)+|y(x)|^p+ u(x)= 0\quad\text{in }\Omega,\quad y\in H^1_0(\Omega),NEWLINE\]NEWLINE and, in addition, \(F(y)\in Q\) where \(Q\) is a closed convex set of some Banach space. Here \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), is a bounded domain and \(p= (n+2)/(n- 2)\). Under some additional assumptions a necessary optimality condition in the form of the maximum principle is derived.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].