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Asymptotic behavior of least energy solutions for a fractional Laplacian eigenvalue problem on $R^N$
Analysis of PDEs · Submitted 10 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that there is a threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution $u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow a_s^{*}$, at some point $x_0 \in \mathbb{R}^N$, which makes $V(x_0)$ be the minimum and $m(x_0)$ be the maximum.
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The abstract, sentence by sentence
We are interested in the existence and asymptotical behavior for the least energy solutions of the following fractional eigenvalue problem \begin{equation*} (P)\quad (-\Delta)^{s}u+V(x)u=\mu u+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 $N\geq2$. We prove that there is a threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution $u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow a_s^{*}$, 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 $m(x)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We are interested in the existence and asymptotical behavior for the least energy solutions of the following fractional eigenvalue problem \begin{equation*} (P)\quad (-\Delta)^{s}u+V(x)u=\mu u+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 $N\geq2$. | background 0.83 | 0.22 | 0.03 | unclear 0.54 |
| s1 | We prove that there is a threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution $u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow a_s^{*}$, at some point $x_0 \in \mathbb{R}^N$, which makes $V(x_0)$ be the minimum and $m(x_0)$ be the maximum. | result 0.99 | 0.96 | 0.87 | asserted 1.00 |
| s2 | Moreover, the precise blowup rates for $u_a$ are obtained under suitable conditions on $V(x)$ and $m(x)$. | result 0.95 | 0.94 | 0.12 | asserted 0.99 |
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Original 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 (-\Delta)^{s}u+V(x)u=\mu u+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 $N\geq2$. We prove that there is a threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution $u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow a_s^{*}$, 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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