Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator

Yuxia Guo; Jianjun  Nie

doi:10.3934/dcds.2016099

Article Contents

2016, Volume 36, Issue 12: 6873-6898. Doi: 10.3934/dcds.2016099

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Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator

Yuxia Guo^1, and
Jianjun Nie^2,

1.
Department of Mathematics, Tsinghua University, Beijing, 100084
2.
Department of Mathematics, Tsinghua University, Beijing 100084

Received: November 30, 2015

Revised: March 31, 2016

Published: May 2017

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Abstract

We consider the following problem: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. (P) \end{equation*} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.

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Mathematics Subject Classification: 35B09, 35J60.

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