The classical linear duality method in some semilinear and noncoercive Dirichlet problems

We study the following semilinear system:

$$(\ast )\{\begin{array}{l}u\in W_0^{1,2}(\mathrm{\Omega }):-\mathrm{d}\mathrm{i}\mathrm{v}(M(x)Du)+u=\theta \psi |\psi {|}^{p^{\prime }-2}+f(x);\\ \psi \in W_0^{1,2}(\mathrm{\Omega }):-\mathrm{d}\mathrm{i}\mathrm{v}(M(x)D\psi )+\psi =u|u{|}^{p-2}\end{array}$$

and we prove the existence of bounded weak solutions in $W_0^{1,2}(\mathrm{\Omega })$. Even if the system is nonlinear, we use a duality method.

We dedicate this paper to Patrizia Pucci certain that she will help us for the study of (*) with nonlinear principal part.