Asymptotic behavior of least energy solutions for a fractional Laplacian eigenvalue problem on $R^N$

Abstract

We are interested in the existence and asymptotical behavior for the least energy solutions of the following fractional eigenvalue problem \begin{equation*} (P)\quad (-Δ)^{s}u+V(x)u=μu+am(x)|u|^{\frac{4s}{N}}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=1,\ u\in H^{s}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1)$, $\mu \in \mathbb{R}$, $a>0$, $V(x)$ and $m(x)$ are $L^{\infty }({\mathbb{R}}^N)$ functions with $N\ge 2$. We prove that there is a threshold $a_s^{\ast }>0$ such that problem $(P)$ has a least energy solution $u_a(x)$ for each $a\in (0,a_s^{\ast })$ and $u_a$ blows up, as $a\nearrow a_s^{\ast }$, at some point $x_0\in {\mathbb{R}}^N$, which makes $V(x_0)$ be the minimum and $m(x_0)$ be the maximum. Moreover, the precise blowup rates for $u_a$ are obtained under suitable conditions on $V(x)$ and $m(x)$.

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