Ground State and Bounded State Solutions for a Critical Stationary Maxwell System Arising in Electromagnetism

The aim of this paper is to establish the existence and multiplicity of bounded state solutions for a critical (p(x), q(x))-curl system which arises in electromagnetism. The existence of nontrivial bounded state solutions is first obtained by applying the mountain pass lemma. Then the existence of ground state solutions is studied by restricting the analysis to the Nehari manifold. Furthermore, under some suitable assumptions, we prove that the mountain pass solution is actually a ground state solution. Finally, the existence of infinitely many solutions is investigated by using the genus theory combined with a truncated argument. The results obtained in this paper develop and complement several contributions concerning the p-curl operator and we focus on new existence results which are due to the presence of nonhomogeneous (p(x), q(x))-curl operator and critical nonlinearity. To the best of our knowledge, our results are new even in the semilinear case.