Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers

Zhongyuan Liu

doi:10.3934/dcds.2017230

Article Contents

2017, Volume 37, Issue 10: 5299-5317. Doi: 10.3934/dcds.2017230

This issue Previous Article A general law of large permanent Next Article Relations between Bohl and general exponents

Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers

Zhongyuan Liu^,

School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, China

* Corresponding author: Zhongyuan Liu
* Corresponding author: Zhongyuan Liu

Received: February 2017

Revised: May 2017

Published: October 2017

The author is supported by NSFC under grant No.11501166.

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Abstract

In this paper, we consider the following semilinear elliptic problem

$\begin{equation*}\begin{cases}-Δ u=|u|^{p-1}u-|u|^{q-1}u~\text{in}~\mathbb{R}^N, \\ u(x)\to 0, ~~\text{as}~|x|\to ∞, \end{cases}\end{equation*}$

where $\frac{N}{N-2} < q < p < p^*$ or $q>p>p^*$, $p^*=\frac{N+2}{N-2}$, $N≥3$. We show that if $q$ is fixed and $p$ is close enough to $\frac{N+2}{N-2}$, the above problem has radial nodal bubble tower solutions, which behave like a superposition of bubbles with different orders and blow up at the origin.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 58C15, 35J61.

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