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

2026/07/21 by Alessandro Cannone, Silvia Cingolani, Serena Dipierro
#math.AP

paper · pdf

Abstract

We investigate Hartree-type equations driven by a nonlocal operator L_μ, defined as a superposition of fractional Laplacians through a signed Borel measure μ. 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.

Citations

• Gradient regularity in mixed local and nonlocal problems
• Nonlinear scalar field equations, I existence of a ground state

Related

• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• On fractional Schrödinger equations with Hartree type nonlinearities
• Pohozaev identity for the fractional p-Laplacian on ℝ^N
• Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions
• Ground state solutions for a non-local type problem in fractional Orlicz Sobolev spaces