## Limit profiles and uniqueness of ground states to the nonlinear Choquard equations.(English)Zbl 1419.35066

Summary: Consider nonlinear Choquard equations $\begin{cases}-\Delta u+u=(I^\alpha * \vert u\vert^{p})\vert u\vert^{p-2}u\quad\text{in }\mathbb R^N,\\ \lim_{x\to\infty}u(x)=0,\end{cases}$ where $$I_{\alpha}$$ denotes the Riesz potential and $$\alpha\in(0,N)$$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $$\alpha\to 0$$ or $$\alpha\to N$$. This leads to the uniqueness and nondegeneracy of ground states when $$\alpha$$ is sufficiently close to 0 or close to $$N$$.

### MSC:

 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J20 Variational methods for second-order elliptic equations

### Keywords:

Choquard equations; ground states
Full Text:

### References:

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