| Abstract: |
| In this talk I will discuss fractional Schr\"odinger equations with a vanishing parameter, namely
$$
(-\Delta)^s u+u=|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N, \quad u \in H^s(\mathbb{R}^N),
$$
where $ s \in (0,1)$, $N > 2s$, $2 < q < p\leq 2^*_s=\frac{2N}{N-2s} $ are fixed parameters and $\lambda > 0$ is a vanishing parameter. We investigate the asymptotic behavior of positive ground state solutions for $\lambda$ small, when $p$ is subcritical, or critical Sobolev exponent $2^*_s$. For $p < 2_s^*$, the ground state solution asymptotically coincides with unique positive ground state solution of $(-\Delta)^s u+u=u^p$, whereas for $p=2_s^*$ the asymptotic behavior of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for $\lambda>0$ small, we will discuss the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions. |
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