Abstract
We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation
where 𝜖 > 0, s ∈ (0,1), \(2^{*}_{s}:=\frac {2N}{N-2s}\) and N > 4s, as we deal with finite energy solutions. We show that the ground state u𝜖 blows up and precisely with the following rate \(\|u_{\epsilon }\|_{L^{\infty }(\mathbb {R}^{N})}\sim \epsilon ^{-\frac {N-2s}{4s}}\), as \(\epsilon \rightarrow 0^{+}\). We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.
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Cassani, D., Wang, Y. Asymptotic Behavior of Ground States and Local Uniqueness for Fractional Schrödinger Equations with Nearly Critical Growth. Potential Anal 59, 1–39 (2023). https://doi.org/10.1007/s11118-021-09959-4
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DOI: https://doi.org/10.1007/s11118-021-09959-4