Eigenvalue problem for a p-Laplacian equation with trapping potentials

TL;DR

This paper investigates the eigenvalue problem for a p-Laplacian equation with trapping potentials, establishing existence, non-existence, and blow-up behavior of ground states depending on a parameter, with explicit thresholds and rates.

Contribution

It introduces explicit thresholds for the existence of ground states in a p-Laplacian eigenvalue problem with trapping potentials, and analyzes their asymptotic behavior and blow-up rates.

Findings

Existence of ground states for parameter a in [0, a*)

Non-existence of ground states for a ≥ a*

Ground states concentrate and blow up near minima of V(x) as a approaches a*

Abstract

Consider the following eigenvalue problem of p-Laplacian equation \begin{equation}\label{P} -\Delta_{p}u+V(x)|u|^{p-2}u=\mu|u|^{p-2}u+a| u|^{s-2}u, x\in \mathbb{R}^{n}, \tag{P} \end{equation} where $a\geq0$, $p\in (1,n)$ and $\mu\in\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\inf\limits_{x \in \mathbb{R}^n}V(x)< \lim\limits_{|x|\rightarrow+\infty}V(x)$. By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\ref{P}) has a ground state $u$ with $\|u\|_{L^p}=1$ for some $\mu \in \mathbb{R}$ and all $a\in [0,a^*)$, but (\ref{P}) has no this kind of ground state if $a\geq a^*$. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground states of problem (\ref{P}) approach to one of the global minima of $V(x)$ and blow up if $a\nearrow a^*$. The optimal rate of blowup is obtained for $V(x)$ being a polynomial type potential.