Ground state solutions for Hartree type equations driven by superposition operators and Pohozaev Identity

Abstract

We investigate Hartree-type equations driven by a nonlocal operator ${\mathcal{L}}_{\hat{I}}¼$, defined as a superposition of fractional Laplacians through a signed Borel measure $\hat{I}¼$. Under Berestycki-Lions type assumptions, we prove the existence of a Mountain Pass solution and show that its energy level coincides with the minimum on the Pohozaev manifold. We also establish the boundedness of non-negative solutions. The proof of this fact requires a careful use of the Sobolev embedding in the iterative argument and a delicate treatment of the integrals involved in the estimates, as well as a Kato-type inequality in our general setting. Finally, we establish a general Pohozaev identity for solutions under a suitable summability assumption.