Asymptotic behavior of ground state solutions to nonlinear elliptic problems with the fractional Laplacian
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911475445006336 |
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| author | Yang, Jinge Yang, Jianfu |
| author_facet | Yang, Jinge Yang, Jianfu |
| contents | In this paper, we consider the asymptotic behavior of the ground state solution $u_s$ of the nonlinear fractional Laplacian equation
\begin{equation}\label{eq:0.1a}
(-Δ)^su+Vu=|u|^{p-2}u\quad x\in \mathbb{R}^n
\end{equation}
by taking $s$ as a parameter, where $n\geq 4$, $2<p<\frac{2n}{n-2}$, $V$ is a potential function. We show that for a fixed $p$, there exists $s_0\in(0,1)$ such that equation \eqref{eq:0.1a} admits a ground state solution $u_s$ if and only if $s_0<s<1$. Our main results give a description of the asymptotic behavior of $u_s$ as $s\uparrow1$ and $s\downarrow s_0$: $u_s$ converges to a function as $s\uparrow1$, and it blows up as $s\downarrow s_0$. Particularly, we prove that $u_s$ concentrates at a minimum point of the function $V$ as $s\downarrow s_0$. The local uniqueness of $u_s$ is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of ground state solutions to nonlinear elliptic problems with the fractional Laplacian Yang, Jinge Yang, Jianfu Analysis of PDEs In this paper, we consider the asymptotic behavior of the ground state solution $u_s$ of the nonlinear fractional Laplacian equation \begin{equation}\label{eq:0.1a} (-Δ)^su+Vu=|u|^{p-2}u\quad x\in \mathbb{R}^n \end{equation} by taking $s$ as a parameter, where $n\geq 4$, $2<p<\frac{2n}{n-2}$, $V$ is a potential function. We show that for a fixed $p$, there exists $s_0\in(0,1)$ such that equation \eqref{eq:0.1a} admits a ground state solution $u_s$ if and only if $s_0<s<1$. Our main results give a description of the asymptotic behavior of $u_s$ as $s\uparrow1$ and $s\downarrow s_0$: $u_s$ converges to a function as $s\uparrow1$, and it blows up as $s\downarrow s_0$. Particularly, we prove that $u_s$ concentrates at a minimum point of the function $V$ as $s\downarrow s_0$. The local uniqueness of $u_s$ is also given. |
| title | Asymptotic behavior of ground state solutions to nonlinear elliptic problems with the fractional Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.00681 |