English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation:(P)(−Δ)su+λu=μ∣u∣p−2u+∣u∣2s∗−2u,x∈RN,u&gt;0,∫RN∣u∣2dx=a2,\left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right.where0&lt;s&lt;10\lt s\lt 1,aa,μ&gt;0\mu \gt 0,N≥2N\ge 2, and2&lt;p&lt;2s∗2\lt p\lt {2}_{s}^{\ast }. We consider theL2{L}^{2}-subcritical andL2{L}^{2}-supercritical cases. More precisely, inL2{L}^{2}-subcritical case, we obtain the multiplicity of the normalized solutions for problem(P)\left(P)by using the truncation technique, concentration-compactness principle, and genus theory. InL2{L}^{2}-supercritical case, we obtain a couple of normalized solution for(P)\left(P)by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

advances in nonlinear analysis

The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the  L  2 -subcritical and  L  2 -supercritical cases