Fractional Schrödinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states
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| Format: | Preprint |
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2024
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| author | Bhakta, Mousomi Das, Paramananda Ganguly, Debdip |
| author_facet | Bhakta, Mousomi Das, Paramananda Ganguly, Debdip |
| contents | We study the fractional Schrödinger equations with a vanishing parameter: $$
(-Δ)^s u+u =|u|^{p-2}u+λ|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq 2^*_s=\frac{2N}{N-2s}$ are fixed parameters and $λ>0$ is a vanishing parameter. We investigate the asymptotic behaviour of positive ground state solutions for $λ$ 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 $(-Δ)^s u+u=u^p$, whereas for $p=2_s^*$ the asymptotic behaviour of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for $λ>0$ small, we show the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional Schrödinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states Bhakta, Mousomi Das, Paramananda Ganguly, Debdip Analysis of PDEs Primary 35J60, 35B08, 35B20, 35B40, 35B44, 35J10, 35J20 We study the fractional Schrödinger equations with a vanishing parameter: $$ (-Δ)^s u+u =|u|^{p-2}u+λ|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq 2^*_s=\frac{2N}{N-2s}$ are fixed parameters and $λ>0$ is a vanishing parameter. We investigate the asymptotic behaviour of positive ground state solutions for $λ$ 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 $(-Δ)^s u+u=u^p$, whereas for $p=2_s^*$ the asymptotic behaviour of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for $λ>0$ small, we show the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions. |
| title | Fractional Schrödinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states |
| topic | Analysis of PDEs Primary 35J60, 35B08, 35B20, 35B40, 35B44, 35J10, 35J20 |
| url | https://arxiv.org/abs/2410.03233 |