## Rocky Mountain Journal of Mathematics

### Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth

Lixia Wang

#### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1371-1390.

Dates
First available in Project Euclid: 29 August 2019

https://projecteuclid.org/euclid.rmjm/1567044044

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1371

Mathematical Reviews number (MathSciNet)
MR3998926

Zentralblatt MATH identifier
07104722

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Wang, Lixia. Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth. Rocky Mountain J. Math. 49 (2019), no. 4, 1371--1390. doi:10.1216/RMJ-2019-49-4-1371. https://projecteuclid.org/euclid.rmjm/1567044044

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