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2019 Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth
Lixia Wang
Rocky Mountain J. Math. 49(4): 1371-1390 (2019). DOI: 10.1216/RMJ-2019-49-4-1371

Abstract

In this paper, we consider the existence of infinitely many sign-changing solutions for the following Hardy-Sobolev-Maz'ya equation \[ \left \{\begin{aligned} &-\triangle u-\frac {\mu u}{|y|^2}=\lambda u+\frac {|u|^{2^{\ast }(t)-2}u}{|y|^t} \quad \mbox {in } \Omega , \\ &u=0 \quad \qquad \qquad \qquad \qquad \qquad \qquad \,\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega $ is an open bounded domain in $\mathbb {R}^N, \mathbb {R}^N=\mathbb {R}^k\times \mathbb {R}^{N-k}$, $\lambda >0$, $0\leq \mu \lt {(k-2)^2}/{4}$ when $k>2$, $\mu =0$ when $k=2$ and $2^{\ast }(t)={2(N-t)}/({N-2})$. A point $x\in \mathbb {R}^N$ is denoted as $x=(y,z)\in \mathbb {R}^k\times \mathbb {R}^{N-k}$, and the points $x^0=(0,z^0)$ are contained in $\Omega $. By using a compactness result obtained previously, we prove the existence of infinitely many sign changing solutions by a combination of the invariant sets method and the Ljusternik-Schnirelman type minimax method.

Citation

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Lixia Wang. "Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth." Rocky Mountain J. Math. 49 (4) 1371 - 1390, 2019. https://doi.org/10.1216/RMJ-2019-49-4-1371

Information

Published: 2019
First available in Project Euclid: 29 August 2019

zbMATH: 07104722
MathSciNet: MR3998926
Digital Object Identifier: 10.1216/RMJ-2019-49-4-1371

Subjects:
Primary: 35J20
Secondary: 35J60

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 4 • 2019
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