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Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

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  • In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation

    $\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$

    where $\varepsilon$ is a positive parameter, $(-Δ)^{s}$ is the fractional Laplacian, $s ∈ (0,1)$ and $N> 2s$. Suppose that the potential $V(x) ∈\mathcal{C}(\mathbb{R}^{N})$ satisfies $\text{inf}_{\mathbb{R}^{N}} V(x)>0$, and there exist $k$ points $x^{j} ∈ \mathbb{R}^{N}$ such that for each $j = 1,···,k$, $V(x^{j})$ are strict global minimum. When $f$ is subcritical, we prove that the problem has at least $k$ positive solutions for $\varepsilon>0$ small. Moreover, we establish the concentration property of the solutions as $\varepsilon$ tends to zero.

    Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.


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