Concentrating standing waves for the fractional nonlinear Schr\"odinger equation

By J. D'avila, Manuel del Pino, Jun-cheng Wei. 2013

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We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0 0$, and $\epsilon>0$ is a small number. Letting $w_\lambda$ be the radial ground state of $(-\Delta)^s w_\lambda + \lambda w_\lambda - w_\lambda^p=0$ in $H^{2s}(\R^N)$, we build solutions of the form $$ u_\epsilon(x) \sim \sum_{i=1}^k w_{\lambda_i} ((x-\xi_i^\epsilon)/\epsilon),$$ where $\lambda_i = V(\xi_i^\epsilon)$ and the $\xi_i^\epsilon $ approach suitable critical points of $V$. Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case $s=1$. In particular such a solution exists around $k$ nondegenerate critical points of $V$. For $s=1$ this corresponds to the classical results by Floer-Weinstein and Oh.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38fd21063120e706476b50c81b7d93f1ada4e850

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method (2015, background): “.f. [20]. For instance, the truncation argument has to handle carefully in the present situation. We would like tocite [20, 22] forthe existence ofpositive solution, imposing aglobalcondition on V.In [11] is studied the existence and concentration phenomena for potential verifying local condition (V”
The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system (2017, unknown): “The concentration phenomena was considered independently in [9,12] via a Lyapunov–Schmidt reduction argument.”
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, unknown): “Notice that the arguments of [13,15,16,21] heavily rely on the uniqueness and non-degeneracy of ground state solutions to the corresponding autonomous problems which have been proved in [22,23].”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “4) in [10] for the existence of such solution and to Lemma 2.”