Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation
Shuai Yao, Haibo Chen, Vicenţiu D. Rădulescu, Juntao Sun
Abstract
We study normalized solutions for the following Choquard equations with lower critical exponent and a local perturbation $ -\Delta u+\lambda u=\gamma (I_{\alpha }\ast |u|^{\frac{\alpha }{N}+1})|u|^{\frac{\alpha }{N}-1}u+\mu |u|^{q-2}u \quad \text{in}\quad \mathbb{R}^{N}, \int_{\mathbb{R}^{N}}$ $|u|^{2}dx=c^{2},$ where $\gamma ,\,\mu,\,c$ are given positive numbers and $2<q\leq \frac{2N}{N-2}$. The frequency $\lambda $ appears as a real Lagrange parameter and is part of the unknowns. By introducing new arguments and under different assumptions on $q,\,c,\,\gamma $, and $\mu ,$ we prove several nonexistence and existence results. In particular, we consider the case $q=\frac{2N}{N-2},$ which corresponds to equations involving double critical exponents. We also describe some qualitative properties of the solutions with prescribed mass and of the associated Lagrange multipliers $\lambda$.