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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
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