The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L2-subcritical and L2-supercritical cases

Abstract

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>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. where 00\mu \gt 0, N≥2N\ge 2, and 2<p<2s∗2\lt p\lt {2}_{s}^{\ast }. We consider the L2{L}^{2}-subcritical and L2{L}^{2}-supercritical cases. More precisely, in L2{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. In L2{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.

Keywords

• normalized solutions
• l2-subcritical
• l2-supercritical
• sobolev critical exponent
• 35j20
• 35j50
• 35j70