Optimal control problems governed by non-well-posed semilinear elliptic equation (Q5960812)

The paper considers an optimal control problem for the semilinear elliptic equation NEWLINE\[NEWLINE\Delta y(x)- y(x)+ f(x, y(x))+ u(x)= 0\quad\text{in }\Omega,\quad y\in H^1_0(\Omega),\tag{1}NEWLINE\]NEWLINE with small positive controls \(u\) and the state \(y\) as the minimal positive solution of (1). Under some rather strong assumptions on the function \(f\) (convexity, subcritical growth, etc.) the author proves the continuous dependence of the state on the control and gives necessary optimality conditions in the form of Pontryagin's maximum principle.