AI 中文总结
研究由非局部算子驱动的Hartree型方程，在特定假设下证明山路解存在及能量水平与Pohozaev流形最小值一致，建立非负解有界性，还得到一般Pohozaev恒等式，核心方法包括使用Sobolev嵌入、精细处理积分及Kato型不等式。
AI 中文摘要
我们研究由非局部算子$\mathcal{L}_\mu$驱动的Hartree型方程，该算子通过有号Borel测度$\mu$定义为分数阶拉普拉斯算子的叠加。在Berestycki-Lions型假设下，我们证明了山路解的存在性，并表明其能量水平与Pohozaev流形上的最小值一致。我们还建立了非负解的有界性。证明需要在迭代论证中谨慎使用Sobolev嵌入，对估计中涉及的积分进行精细处理，以及在一般设置中使用Kato型不等式。最后，在合适的可和性假设下，我们为解建立了一个一般的Pohozaev恒等式。
英文摘要
We investigate Hartree-type equations driven by a nonlocal operator $\mathcal{L}_μ$, 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.
