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Abstract
Abstract Consider nonlinear Choquard equations { - Δ u + u = ( I α * | u | p ) | u | p - 2 u in ℝ N , lim x → ∞ u ( x ) = 0 , \left\{\begin{aligned} \displaystyle-\Delta u+u&\displaystyle=(I_{\alpha}*% \lvert u\rvert^{p})\lvert u\rvert^{p-2}u&&\displaystyle\phantom{}\text{in }% \mathbb{R}^{N},\\ \displaystyle\lim_{x\to\infty}u(x)&\displaystyle=0,\end{aligned}\right. where I α {I_{\alpha}} denotes the Riesz potential and α ∈ ( 0 , N ) {\alpha\in(0,N)} . In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 {\alpha\to 0} or α → N {\alpha\to N} . This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N.
Highlights
We investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 or α → N
In [13], Moroz and Van Schaftingen proved the existence of a ground state solution to in the range of p
It is well known that the Euler–Lagrange equation (equation (1.3) below) of J0 admits a unique positive radial ground state solution
Summary
When N = 3, α = 2 and p = 2, the existence of a radial positive solution is proved in [7, 9, 11] by variational methods and in [1, 12, 18] by ODE approaches. In [13], Moroz and Van Schaftingen proved the existence of a ground state solution to in the range of p. We are interested in limit behaviors of ground state to (1.1) as either α → 0 or α → N These shall play essential roles to prove the uniqueness and nondegeneracy of a positive radial ground state to (1.1). It is well known that the Euler–Lagrange equation (equation (1.3) below) of J0 admits a unique positive radial ground state solution. Be a family of positive radial ground states to for α close to and let u0 be a unique positive radial ground state of the equation. It is easy to see that for p ∈ (2, N2−N2 ) the limit functional JN is C1 on H1(RN ) and its Euler–Lagrange equation is
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