An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

We study an optimal control problem associated to a Dirichlet boundary value problem of the type

$ \begin{equation*} \mbox{(BVP) }\,\,\,\,\,\,\,\, {\rm{div}}\; \left[ \beta(x)\nabla u (x) + \left( A\frac {x}{|x|^2 } +g(x) \right)u(x)\right] = {\rm{div}}\; \mathcal F,\quad u\in W^{1,p}_0( \Omega ) , \end{equation*} $

$ 1<p\leqslant 2, $ where $ \Omega $ is a bounded regular domain of $ \mathbb{R}^N $, $ 0\in \Omega , $ $ \beta: \Omega \rightarrow {\mathbb R} $ is an unbounded function satisfying $ \beta(x)\geqslant\lambda_0>0 $ a.e., $ A $ is a suitably small constant, and $ g\in L^\infty( \Omega ; \mathbb{R}^N ) $.

We consider the vector field $ \mathcal F $ as the control and the corresponding weak solution $ u $ to (BVP) as the state. Our aim is to find the optimal vector field $ \mathcal F\in L^p( \Omega ) $ so that the corresponding state $ u\in W^{1,p}_0( \Omega ) $ is close to the desired profile in $ L^p( \Omega ) $ while the norm of $ u $ in $ W^{1,p}( \Omega ) $ is not too large.

We prove that, for every $ p $ less than $ 2 $ and suitably close to $ 2 $, (BVP) admits an unique weak solution and for such values of $ p $, we prove the existence of optimal pairs.