TY - JOUR
T1 - Entire Sign-Changing Solutions to the Fractional Critical Schrödinger Equation
AU - Tang , Xingdong
AU - Xu , Guixiang
AU - Zhang , Chunyan
AU - Zhang , Jihui
JO - Annals of Applied Mathematics
VL - 3
SP - 219
EP - 248
PY - 2024
DA - 2024/09
SN - 40
DO - http://doi.org/10.4208/aam.OA-2024-0006
UR - https://global-sci.org/intro/article_detail/aam/23421.html
KW - Fractional critical Schrödinger equation, sign-changing solution, the
concentration-compactness principle, the equivariant group action, the mountain pass theorem.
AB - <p style="text-align: justify;">In this paper, we consider the fractional critical Schrödinger equation (FCSE) $$(-\Delta)^su-|u|^{2^*_s-2}u=0,$$ where $u∈\dot{H}^s(\mathbb{R}^N),$ $N≥4,$ $0&lt;s&lt;1$ and $2^∗_s=\frac{2N}{N−2s}$ is the critical Sobolev exponent of order $s.$ By virtue of the variational method and the concentration compactness principle with the equivariant group action, we obtain some new type of nonradial, sign-changing solutions of (FCSE) in the energy space $\dot{H}^s(\mathbb{R}^N)$. The key component is that we take the equivariant group action to construct several subspace of $\dot{H}^s(\mathbb{R}^N)$&nbsp;with trivial intersection, then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of (FCSE) in $\dot{H}^s(\mathbb{R}^N).$</p>